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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories</title>   <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2403.08892v3/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S1" title="In Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S2" title="In Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span>Link models in <math alttext="2+1" class="ltx_Math" display="inline"><semantics><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow><annotation-xml encoding="MathML-Content"><apply><plus></plus><cn type="integer">2</cn><cn type="integer">1</cn></apply></annotation-xml><annotation encoding="application/x-tex">2+1</annotation><annotation encoding="application/x-llamapun">2 + 1</annotation></semantics></math>D</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S2.SS1" title="In II Link models in 2+1D ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II.1 </span>Non-Integrability of the Spin-1 Ladder</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3" title="In Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span>Zero-mode scars</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.SS1" title="In III Zero-mode scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III.1 </span>QMBS in TLM With Arbitrary Integer Spin</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.SS2" title="In III Zero-mode scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III.2 </span>Beyond Zero-Mode Building Blocks</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.SS2.SSS0.Px1" title="In III.2 Beyond Zero-Mode Building Blocks ‣ III Zero-mode scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title">Diagonal Tiling</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.SS2.SSS0.Px2" title="In III.2 Beyond Zero-Mode Building Blocks ‣ III Zero-mode scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title">Non-Tiling Scar</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S4" title="In Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span>Numerical results and discussion</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S5" title="In Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span>Conclusions and outlook</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S1a" title="In Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span>Symmetries of the systems</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S2a" title="In Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span>Index Theorem and an Exponential Number of zero-modes</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3a" title="In Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span>Low Entropy Zero-Modes in Truncated Link Models</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S4a" title="In Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span>The tiling product</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S5a" title="In Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span>A <math alttext="4\times 4" class="ltx_Math" display="inline"><semantics><mrow><mn>4</mn><mo lspace="0.222em" rspace="0.222em">×</mo><mn>4</mn></mrow><annotation-xml encoding="MathML-Content"><apply><times></times><cn type="integer">4</cn><cn type="integer">4</cn></apply></annotation-xml><annotation encoding="application/x-tex">4\times 4</annotation><annotation encoding="application/x-llamapun">4 × 4</annotation></semantics></math> Scar for the <math alttext="E^{2}" class="ltx_Math" display="inline"><semantics><msup><mi>E</mi><mn>2</mn></msup><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝐸</ci><cn type="integer">2</cn></apply></annotation-xml><annotation encoding="application/x-tex">E^{2}</annotation><annotation encoding="application/x-llamapun">italic_E start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> potential</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S6" title="In Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">VI </span>Amplitudes of Scars</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Thea Budde </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Marina Krstic Marinkovic </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Joao C. Pinto Barros </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Institut für Theoretische Physik, ETH Zürich, Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland </span></span></span> </div> <div class="ltx_dates">(September 2, 2024)</div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id5.5">The existence of Quantum Many-Body Scars, which prevents thermalization from certain initial states after a long time, has been established across different quantum many-body systems. These include gauge theories corresponding to spin-1/2 quantum link models. Establishing quantum scars in gauge theories with high spin is not accessible with existing numerical methods, which rely on exact diagonalization. We systematically identify scars for pure gauge theories with arbitrarily large integer spin <math alttext="S" class="ltx_Math" display="inline" id="id1.1.m1.1"><semantics id="id1.1.m1.1a"><mi id="id1.1.m1.1.1" xref="id1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="id1.1.m1.1b"><ci id="id1.1.m1.1.1.cmml" xref="id1.1.m1.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="id1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="id1.1.m1.1d">italic_S</annotation></semantics></math> in <math alttext="2+1" class="ltx_Math" display="inline" id="id2.2.m2.1"><semantics id="id2.2.m2.1a"><mrow id="id2.2.m2.1.1" xref="id2.2.m2.1.1.cmml"><mn id="id2.2.m2.1.1.2" xref="id2.2.m2.1.1.2.cmml">2</mn><mo id="id2.2.m2.1.1.1" xref="id2.2.m2.1.1.1.cmml">+</mo><mn id="id2.2.m2.1.1.3" xref="id2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="id2.2.m2.1b"><apply id="id2.2.m2.1.1.cmml" xref="id2.2.m2.1.1"><plus id="id2.2.m2.1.1.1.cmml" xref="id2.2.m2.1.1.1"></plus><cn id="id2.2.m2.1.1.2.cmml" type="integer" xref="id2.2.m2.1.1.2">2</cn><cn id="id2.2.m2.1.1.3.cmml" type="integer" xref="id2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id2.2.m2.1c">2+1</annotation><annotation encoding="application/x-llamapun" id="id2.2.m2.1d">2 + 1</annotation></semantics></math>D, where the electric field is restricted to <math alttext="2S+1" class="ltx_Math" display="inline" id="id3.3.m3.1"><semantics id="id3.3.m3.1a"><mrow id="id3.3.m3.1.1" xref="id3.3.m3.1.1.cmml"><mrow id="id3.3.m3.1.1.2" xref="id3.3.m3.1.1.2.cmml"><mn id="id3.3.m3.1.1.2.2" xref="id3.3.m3.1.1.2.2.cmml">2</mn><mo id="id3.3.m3.1.1.2.1" xref="id3.3.m3.1.1.2.1.cmml">⁢</mo><mi id="id3.3.m3.1.1.2.3" xref="id3.3.m3.1.1.2.3.cmml">S</mi></mrow><mo id="id3.3.m3.1.1.1" xref="id3.3.m3.1.1.1.cmml">+</mo><mn id="id3.3.m3.1.1.3" xref="id3.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="id3.3.m3.1b"><apply id="id3.3.m3.1.1.cmml" xref="id3.3.m3.1.1"><plus id="id3.3.m3.1.1.1.cmml" xref="id3.3.m3.1.1.1"></plus><apply id="id3.3.m3.1.1.2.cmml" xref="id3.3.m3.1.1.2"><times id="id3.3.m3.1.1.2.1.cmml" xref="id3.3.m3.1.1.2.1"></times><cn id="id3.3.m3.1.1.2.2.cmml" type="integer" xref="id3.3.m3.1.1.2.2">2</cn><ci id="id3.3.m3.1.1.2.3.cmml" xref="id3.3.m3.1.1.2.3">𝑆</ci></apply><cn id="id3.3.m3.1.1.3.cmml" type="integer" xref="id3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id3.3.m3.1c">2S+1</annotation><annotation encoding="application/x-llamapun" id="id3.3.m3.1d">2 italic_S + 1</annotation></semantics></math> states per link. Through an explicit analytic construction, we show that the presence of scars is widespread in <math alttext="2+1" class="ltx_Math" display="inline" id="id4.4.m4.1"><semantics id="id4.4.m4.1a"><mrow id="id4.4.m4.1.1" xref="id4.4.m4.1.1.cmml"><mn id="id4.4.m4.1.1.2" xref="id4.4.m4.1.1.2.cmml">2</mn><mo id="id4.4.m4.1.1.1" xref="id4.4.m4.1.1.1.cmml">+</mo><mn id="id4.4.m4.1.1.3" xref="id4.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="id4.4.m4.1b"><apply id="id4.4.m4.1.1.cmml" xref="id4.4.m4.1.1"><plus id="id4.4.m4.1.1.1.cmml" xref="id4.4.m4.1.1.1"></plus><cn id="id4.4.m4.1.1.2.cmml" type="integer" xref="id4.4.m4.1.1.2">2</cn><cn id="id4.4.m4.1.1.3.cmml" type="integer" xref="id4.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id4.4.m4.1c">2+1</annotation><annotation encoding="application/x-llamapun" id="id4.4.m4.1d">2 + 1</annotation></semantics></math>D gauge theories for arbitrary integer spin. We confirm these findings numerically for small truncated spin and <math alttext="S=1" class="ltx_Math" display="inline" id="id5.5.m5.1"><semantics id="id5.5.m5.1a"><mrow id="id5.5.m5.1.1" xref="id5.5.m5.1.1.cmml"><mi id="id5.5.m5.1.1.2" xref="id5.5.m5.1.1.2.cmml">S</mi><mo id="id5.5.m5.1.1.1" xref="id5.5.m5.1.1.1.cmml">=</mo><mn id="id5.5.m5.1.1.3" xref="id5.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="id5.5.m5.1b"><apply id="id5.5.m5.1.1.cmml" xref="id5.5.m5.1.1"><eq id="id5.5.m5.1.1.1.cmml" xref="id5.5.m5.1.1.1"></eq><ci id="id5.5.m5.1.1.2.cmml" xref="id5.5.m5.1.1.2">𝑆</ci><cn id="id5.5.m5.1.1.3.cmml" type="integer" xref="id5.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id5.5.m5.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="id5.5.m5.1d">italic_S = 1</annotation></semantics></math> quantum link models. Our analytic construction establishes the presence of scars far beyond volumes and spins that can be probed with existing numerical methods and can guide quantum simulation experiments toward interesting non-equilibrium phenomena, inaccessible otherwise.</p> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Evolving an isolated quantum system over long times might suggest that the unitary evolution of the system would prevent thermalization. This puzzle is addressed by the Eigenstate Thermalization Hypothesis (ETH) for quantum many-body systems <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib5" title="">5</a>]</cite>. According to the ETH, for non-integrable systems, the high-energy states and the observables of interest will eventually converge to a description following equilibrium statistical mechanics, achieving thermal equilibrium. In contrast to the ETH, many-body localized systems defy this trend due to their emergent integrability <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib6" title="">6</a>]</cite>. Systems exhibiting Quantum Many-Body Scars (QMBS) offer a contrasting example <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib9" title="">9</a>]</cite>; they also evade the ETH, but only in an exponentially small fraction of states in the Hilbert space.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">The effect of QMBS was first observed in a Rydberg-atom quantum simulator, marked by persistent revivals for particular initial states, in contrast to the vast majority of other high-energy initial states <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib11" title="">11</a>]</cite>. In parallel with the exact construction of highly excited eigenstates in a non-integrable model <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib12" title="">12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib13" title="">13</a>]</cite>, the experiment has sparked intense research in a variety of quantum many-body systems, where a <em class="ltx_emph ltx_font_italic" id="S1.p2.1.1">weak</em> breaking of the ETH takes place <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib18" title="">18</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib20" title="">20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib22" title="">22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib23" title="">23</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib25" title="">25</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib26" title="">26</a>]</cite>. In these systems, the presence of <em class="ltx_emph ltx_font_italic" id="S1.p2.1.2">few</em> anomalous eigenstates can leave an imprint on thermalization. Among the special features of the anomalous states is that they are characterized by atypically low entanglement entropy, compared with other states arbitrarily close in the spectrum. In particular, QMBS may exhibit an area law of bipartite entanglement entropy. This means that the entanglement between a subsystem and its complement is proportional to the area of the boundary that divides them. This contradicts the ETH, which predicts that highly excited states have an entanglement that grows linearly with the volume of the subsystem.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.5">The interplay between gauge symmetry and scarring phenomena is not fully understood. Gauge theories naturally lead to constraints, and specific constraints can give rise to QMBS, as observed in the PXP model <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib28" title="">28</a>]</cite>. The constraints in gauge theories stem from local symmetries, which segment the Hamiltonian into distinct sectors that do not mix under time evolution. Supplied by a condition on physical states, this leads to local constraints in the form of Gauss’ law. Beyond the PXP model, QMBS have also been observed in other gauge theories, such as the Abelian case in the presence of matter in <math alttext="1+1" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mrow id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml"><mn id="S1.p3.1.m1.1.1.2" xref="S1.p3.1.m1.1.1.2.cmml">1</mn><mo id="S1.p3.1.m1.1.1.1" xref="S1.p3.1.m1.1.1.1.cmml">+</mo><mn id="S1.p3.1.m1.1.1.3" xref="S1.p3.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><apply id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1"><plus id="S1.p3.1.m1.1.1.1.cmml" xref="S1.p3.1.m1.1.1.1"></plus><cn id="S1.p3.1.m1.1.1.2.cmml" type="integer" xref="S1.p3.1.m1.1.1.2">1</cn><cn id="S1.p3.1.m1.1.1.3.cmml" type="integer" xref="S1.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">1+1</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">1 + 1</annotation></semantics></math>D <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib29" title="">29</a>]</cite>, pure gauge in <math alttext="2+1" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mrow id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml"><mn id="S1.p3.2.m2.1.1.2" xref="S1.p3.2.m2.1.1.2.cmml">2</mn><mo id="S1.p3.2.m2.1.1.1" xref="S1.p3.2.m2.1.1.1.cmml">+</mo><mn id="S1.p3.2.m2.1.1.3" xref="S1.p3.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><apply id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1"><plus id="S1.p3.2.m2.1.1.1.cmml" xref="S1.p3.2.m2.1.1.1"></plus><cn id="S1.p3.2.m2.1.1.2.cmml" type="integer" xref="S1.p3.2.m2.1.1.2">2</cn><cn id="S1.p3.2.m2.1.1.3.cmml" type="integer" xref="S1.p3.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">2+1</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">2 + 1</annotation></semantics></math>D <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib30" title="">30</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib32" title="">32</a>]</cite> and non-Abelian in <math alttext="1+1" class="ltx_Math" display="inline" id="S1.p3.3.m3.1"><semantics id="S1.p3.3.m3.1a"><mrow id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1.cmml"><mn id="S1.p3.3.m3.1.1.2" xref="S1.p3.3.m3.1.1.2.cmml">1</mn><mo id="S1.p3.3.m3.1.1.1" xref="S1.p3.3.m3.1.1.1.cmml">+</mo><mn id="S1.p3.3.m3.1.1.3" xref="S1.p3.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.3.m3.1b"><apply id="S1.p3.3.m3.1.1.cmml" xref="S1.p3.3.m3.1.1"><plus id="S1.p3.3.m3.1.1.1.cmml" xref="S1.p3.3.m3.1.1.1"></plus><cn id="S1.p3.3.m3.1.1.2.cmml" type="integer" xref="S1.p3.3.m3.1.1.2">1</cn><cn id="S1.p3.3.m3.1.1.3.cmml" type="integer" xref="S1.p3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.3.m3.1c">1+1</annotation><annotation encoding="application/x-llamapun" id="S1.p3.3.m3.1d">1 + 1</annotation></semantics></math>D <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib33" title="">33</a>]</cite> and <math alttext="2+1" class="ltx_Math" display="inline" id="S1.p3.4.m4.1"><semantics id="S1.p3.4.m4.1a"><mrow id="S1.p3.4.m4.1.1" xref="S1.p3.4.m4.1.1.cmml"><mn id="S1.p3.4.m4.1.1.2" xref="S1.p3.4.m4.1.1.2.cmml">2</mn><mo id="S1.p3.4.m4.1.1.1" xref="S1.p3.4.m4.1.1.1.cmml">+</mo><mn id="S1.p3.4.m4.1.1.3" xref="S1.p3.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.4.m4.1b"><apply id="S1.p3.4.m4.1.1.cmml" xref="S1.p3.4.m4.1.1"><plus id="S1.p3.4.m4.1.1.1.cmml" xref="S1.p3.4.m4.1.1.1"></plus><cn id="S1.p3.4.m4.1.1.2.cmml" type="integer" xref="S1.p3.4.m4.1.1.2">2</cn><cn id="S1.p3.4.m4.1.1.3.cmml" type="integer" xref="S1.p3.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.4.m4.1c">2+1</annotation><annotation encoding="application/x-llamapun" id="S1.p3.4.m4.1d">2 + 1</annotation></semantics></math>D <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib34" title="">34</a>]</cite>. So far, the research on QMBS in Abelian pure gauge theories has exclusively focused on spin-<math alttext="1/2" class="ltx_Math" display="inline" id="S1.p3.5.m5.1"><semantics id="S1.p3.5.m5.1a"><mrow id="S1.p3.5.m5.1.1" xref="S1.p3.5.m5.1.1.cmml"><mn id="S1.p3.5.m5.1.1.2" xref="S1.p3.5.m5.1.1.2.cmml">1</mn><mo id="S1.p3.5.m5.1.1.1" xref="S1.p3.5.m5.1.1.1.cmml">/</mo><mn id="S1.p3.5.m5.1.1.3" xref="S1.p3.5.m5.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.5.m5.1b"><apply id="S1.p3.5.m5.1.1.cmml" xref="S1.p3.5.m5.1.1"><divide id="S1.p3.5.m5.1.1.1.cmml" xref="S1.p3.5.m5.1.1.1"></divide><cn id="S1.p3.5.m5.1.1.2.cmml" type="integer" xref="S1.p3.5.m5.1.1.2">1</cn><cn id="S1.p3.5.m5.1.1.3.cmml" type="integer" xref="S1.p3.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.5.m5.1c">1/2</annotation><annotation encoding="application/x-llamapun" id="S1.p3.5.m5.1d">1 / 2</annotation></semantics></math> Quantum Link Models (QLM) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib30" title="">30</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib32" title="">32</a>]</cite>, which also exhibit other interesting phenomena such as crystalline confining phases or confining strings with fractionalized electric flux stands <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib35" title="">35</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib36" title="">36</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib37" title="">37</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib38" title="">38</a>]</cite>. This work demonstrates, for the first time, the existence of QMBS in link models with arbitrary integer spins.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">Addressing questions related to real-time dynamics falls outside the capabilities of conventional lattice gauge theory methods, as Monte Carlo simulations face inefficiencies due to severe sign problems (see e.g. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib39" title="">39</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib40" title="">40</a>]</cite>). Sign problems have been a driving force in the development of quantum simulations of gauge theories <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib41" title="">41</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib42" title="">42</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib43" title="">43</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib44" title="">44</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib45" title="">45</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib46" title="">46</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib47" title="">47</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib48" title="">48</a>]</cite>. Formulations like QLMs are essential for preserving gauge symmetry while ensuring a finite Hilbert space for a finite lattice volume. Recovering the original theory may take different routes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib49" title="">49</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib50" title="">50</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib51" title="">51</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib52" title="">52</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib53" title="">53</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib54" title="">54</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib55" title="">55</a>]</cite>. Regardless of the chosen method, significant contributions to theories such as Quantum Chromodynamics (QCD) are still far off. Therefore, it is crucial to identify intriguing phenomena that can guide experimental efforts in this direction. The study of QMBS is a meritorious example, as they appear in simple gauge theories and probe fundamental aspects of quantum many-body theory.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">Demonstrating the presence of QMBS for large Hilbert spaces per gauge link, beyond spin-<math alttext="1/2" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><mrow id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml"><mn id="S1.p5.1.m1.1.1.2" xref="S1.p5.1.m1.1.1.2.cmml">1</mn><mo id="S1.p5.1.m1.1.1.1" xref="S1.p5.1.m1.1.1.1.cmml">/</mo><mn id="S1.p5.1.m1.1.1.3" xref="S1.p5.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><apply id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1"><divide id="S1.p5.1.m1.1.1.1.cmml" xref="S1.p5.1.m1.1.1.1"></divide><cn id="S1.p5.1.m1.1.1.2.cmml" type="integer" xref="S1.p5.1.m1.1.1.2">1</cn><cn id="S1.p5.1.m1.1.1.3.cmml" type="integer" xref="S1.p5.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">1/2</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">1 / 2</annotation></semantics></math>, serves several fundamental purposes: it directly addresses how widespread QMBS are across many-body systems; it reveals novel mechanisms for their formation, illuminating the role played by gauge symmetry; and it helps to guide quantum simulation experiments toward interesting questions before the complexity of QCD is achieved.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.2">In this work, we demonstrate the extensive presence of QMBS across <math alttext="2+1" class="ltx_Math" display="inline" id="S1.p6.1.m1.1"><semantics id="S1.p6.1.m1.1a"><mrow id="S1.p6.1.m1.1.1" xref="S1.p6.1.m1.1.1.cmml"><mn id="S1.p6.1.m1.1.1.2" xref="S1.p6.1.m1.1.1.2.cmml">2</mn><mo id="S1.p6.1.m1.1.1.1" xref="S1.p6.1.m1.1.1.1.cmml">+</mo><mn id="S1.p6.1.m1.1.1.3" xref="S1.p6.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.1.m1.1b"><apply id="S1.p6.1.m1.1.1.cmml" xref="S1.p6.1.m1.1.1"><plus id="S1.p6.1.m1.1.1.1.cmml" xref="S1.p6.1.m1.1.1.1"></plus><cn id="S1.p6.1.m1.1.1.2.cmml" type="integer" xref="S1.p6.1.m1.1.1.2">2</cn><cn id="S1.p6.1.m1.1.1.3.cmml" type="integer" xref="S1.p6.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.1.m1.1c">2+1</annotation><annotation encoding="application/x-llamapun" id="S1.p6.1.m1.1d">2 + 1</annotation></semantics></math>D <math alttext="U(1)" class="ltx_Math" display="inline" id="S1.p6.2.m2.1"><semantics id="S1.p6.2.m2.1a"><mrow id="S1.p6.2.m2.1.2" xref="S1.p6.2.m2.1.2.cmml"><mi id="S1.p6.2.m2.1.2.2" xref="S1.p6.2.m2.1.2.2.cmml">U</mi><mo id="S1.p6.2.m2.1.2.1" xref="S1.p6.2.m2.1.2.1.cmml">⁢</mo><mrow id="S1.p6.2.m2.1.2.3.2" xref="S1.p6.2.m2.1.2.cmml"><mo id="S1.p6.2.m2.1.2.3.2.1" stretchy="false" xref="S1.p6.2.m2.1.2.cmml">(</mo><mn id="S1.p6.2.m2.1.1" xref="S1.p6.2.m2.1.1.cmml">1</mn><mo id="S1.p6.2.m2.1.2.3.2.2" stretchy="false" xref="S1.p6.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.2.m2.1b"><apply id="S1.p6.2.m2.1.2.cmml" xref="S1.p6.2.m2.1.2"><times id="S1.p6.2.m2.1.2.1.cmml" xref="S1.p6.2.m2.1.2.1"></times><ci id="S1.p6.2.m2.1.2.2.cmml" xref="S1.p6.2.m2.1.2.2">𝑈</ci><cn id="S1.p6.2.m2.1.1.cmml" type="integer" xref="S1.p6.2.m2.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.2.m2.1c">U(1)</annotation><annotation encoding="application/x-llamapun" id="S1.p6.2.m2.1d">italic_U ( 1 )</annotation></semantics></math> gauge theories without matter. Concretely, we explicitly construct mid-spectrum states that satisfy area law entanglement for arbitrary integer spin, specifically for a simply truncated Hilbert space per link. We further verify the existence of these states by numerically determining the system’s eigenstates and calculating their entanglement and Shannon entropy for spin 1 and 2. Our findings unveil the presence of QMBS for single-leg ladders of the spin-1 QLM, which are more accessible to experiments than wider systems. Furthermore, our analytical approach allows us to identify QMBS for arbitrary integer spin and system size, circumventing the limitations of existing numerical methods when applied to large dimensional Hilbert spaces.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span>Link models in <math alttext="2+1" class="ltx_Math" display="inline" id="S2.1.m1.1"><semantics id="S2.1.m1.1b"><mrow id="S2.1.m1.1.1" xref="S2.1.m1.1.1.cmml"><mn id="S2.1.m1.1.1.2" xref="S2.1.m1.1.1.2.cmml">2</mn><mo id="S2.1.m1.1.1.1" xref="S2.1.m1.1.1.1.cmml">+</mo><mn id="S2.1.m1.1.1.3" xref="S2.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.1.m1.1c"><apply id="S2.1.m1.1.1.cmml" xref="S2.1.m1.1.1"><plus id="S2.1.m1.1.1.1.cmml" xref="S2.1.m1.1.1.1"></plus><cn id="S2.1.m1.1.1.2.cmml" type="integer" xref="S2.1.m1.1.1.2">2</cn><cn id="S2.1.m1.1.1.3.cmml" type="integer" xref="S2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.1.m1.1d">2+1</annotation><annotation encoding="application/x-llamapun" id="S2.1.m1.1e">2 + 1</annotation></semantics></math>D</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">We consider models on square <math alttext="L_{1}\times L_{2}" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><mrow id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml"><msub id="S2.p1.1.m1.1.1.2" xref="S2.p1.1.m1.1.1.2.cmml"><mi id="S2.p1.1.m1.1.1.2.2" xref="S2.p1.1.m1.1.1.2.2.cmml">L</mi><mn id="S2.p1.1.m1.1.1.2.3" xref="S2.p1.1.m1.1.1.2.3.cmml">1</mn></msub><mo id="S2.p1.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.p1.1.m1.1.1.1.cmml">×</mo><msub id="S2.p1.1.m1.1.1.3" xref="S2.p1.1.m1.1.1.3.cmml"><mi id="S2.p1.1.m1.1.1.3.2" xref="S2.p1.1.m1.1.1.3.2.cmml">L</mi><mn id="S2.p1.1.m1.1.1.3.3" xref="S2.p1.1.m1.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><apply id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1"><times id="S2.p1.1.m1.1.1.1.cmml" xref="S2.p1.1.m1.1.1.1"></times><apply id="S2.p1.1.m1.1.1.2.cmml" xref="S2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.p1.1.m1.1.1.2.1.cmml" xref="S2.p1.1.m1.1.1.2">subscript</csymbol><ci id="S2.p1.1.m1.1.1.2.2.cmml" xref="S2.p1.1.m1.1.1.2.2">𝐿</ci><cn id="S2.p1.1.m1.1.1.2.3.cmml" type="integer" xref="S2.p1.1.m1.1.1.2.3">1</cn></apply><apply id="S2.p1.1.m1.1.1.3.cmml" xref="S2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p1.1.m1.1.1.3.1.cmml" xref="S2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S2.p1.1.m1.1.1.3.2.cmml" xref="S2.p1.1.m1.1.1.3.2">𝐿</ci><cn id="S2.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S2.p1.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">L_{1}\times L_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> lattices with bosonic degrees of freedom living on the links</p> <table class="ltx_equation ltx_eqn_table" id="S2.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H=\sum_{n}\left(U_{n1}^{\dagger}U_{n+\hat{1}2}^{\dagger}U_{n2}U_{n+\hat{2}1}+% \mathrm{h.c.}\right)+V," class="ltx_math_unparsed" display="block" id="S2.E1.m1.1"><semantics id="S2.E1.m1.1a"><mrow id="S2.E1.m1.1b"><mi id="S2.E1.m1.1.1">H</mi><mo id="S2.E1.m1.1.2" rspace="0.111em">=</mo><munder id="S2.E1.m1.1.3"><mo id="S2.E1.m1.1.3.2" movablelimits="false" rspace="0em">∑</mo><mi id="S2.E1.m1.1.3.3">n</mi></munder><mrow id="S2.E1.m1.1.4"><mo id="S2.E1.m1.1.4.1">(</mo><msubsup id="S2.E1.m1.1.4.2"><mi id="S2.E1.m1.1.4.2.2.2">U</mi><mrow id="S2.E1.m1.1.4.2.2.3"><mi id="S2.E1.m1.1.4.2.2.3.2">n</mi><mo id="S2.E1.m1.1.4.2.2.3.1">⁢</mo><mn id="S2.E1.m1.1.4.2.2.3.3">1</mn></mrow><mo id="S2.E1.m1.1.4.2.3">†</mo></msubsup><msubsup id="S2.E1.m1.1.4.3"><mi id="S2.E1.m1.1.4.3.2.2">U</mi><mrow id="S2.E1.m1.1.4.3.2.3"><mi id="S2.E1.m1.1.4.3.2.3.2">n</mi><mo id="S2.E1.m1.1.4.3.2.3.1">+</mo><mrow id="S2.E1.m1.1.4.3.2.3.3"><mover accent="true" id="S2.E1.m1.1.4.3.2.3.3.2"><mn id="S2.E1.m1.1.4.3.2.3.3.2.2">1</mn><mo id="S2.E1.m1.1.4.3.2.3.3.2.1">^</mo></mover><mo id="S2.E1.m1.1.4.3.2.3.3.1">⁢</mo><mn id="S2.E1.m1.1.4.3.2.3.3.3">2</mn></mrow></mrow><mo id="S2.E1.m1.1.4.3.3">†</mo></msubsup><msub id="S2.E1.m1.1.4.4"><mi id="S2.E1.m1.1.4.4.2">U</mi><mrow id="S2.E1.m1.1.4.4.3"><mi id="S2.E1.m1.1.4.4.3.2">n</mi><mo id="S2.E1.m1.1.4.4.3.1">⁢</mo><mn id="S2.E1.m1.1.4.4.3.3">2</mn></mrow></msub><msub id="S2.E1.m1.1.4.5"><mi id="S2.E1.m1.1.4.5.2">U</mi><mrow id="S2.E1.m1.1.4.5.3"><mi id="S2.E1.m1.1.4.5.3.2">n</mi><mo id="S2.E1.m1.1.4.5.3.1">+</mo><mrow id="S2.E1.m1.1.4.5.3.3"><mover accent="true" id="S2.E1.m1.1.4.5.3.3.2"><mn id="S2.E1.m1.1.4.5.3.3.2.2">2</mn><mo id="S2.E1.m1.1.4.5.3.3.2.1">^</mo></mover><mo id="S2.E1.m1.1.4.5.3.3.1">⁢</mo><mn id="S2.E1.m1.1.4.5.3.3.3">1</mn></mrow></mrow></msub><mo id="S2.E1.m1.1.4.6">+</mo><mi id="S2.E1.m1.1.4.7" mathvariant="normal">h</mi><mo id="S2.E1.m1.1.4.8" lspace="0em" rspace="0.167em">.</mo><mi id="S2.E1.m1.1.4.9" mathvariant="normal">c</mi><mo id="S2.E1.m1.1.4.10" lspace="0em" rspace="0.167em">.</mo><mo id="S2.E1.m1.1.4.11">)</mo></mrow><mo id="S2.E1.m1.1.5">+</mo><mi id="S2.E1.m1.1.6">V</mi><mo id="S2.E1.m1.1.7">,</mo></mrow><annotation encoding="application/x-tex" id="S2.E1.m1.1c">H=\sum_{n}\left(U_{n1}^{\dagger}U_{n+\hat{1}2}^{\dagger}U_{n2}U_{n+\hat{2}1}+% \mathrm{h.c.}\right)+V,</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.1d">italic_H = ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_n 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_U start_POSTSUBSCRIPT italic_n + over^ start_ARG 1 end_ARG 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_U start_POSTSUBSCRIPT italic_n 2 end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_n + over^ start_ARG 2 end_ARG 1 end_POSTSUBSCRIPT + roman_h . roman_c . ) + italic_V ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.8">where the indices <math alttext="n\equiv\left(n_{1},n_{2}\right)" class="ltx_Math" display="inline" id="S2.p1.2.m1.2"><semantics id="S2.p1.2.m1.2a"><mrow id="S2.p1.2.m1.2.2" xref="S2.p1.2.m1.2.2.cmml"><mi id="S2.p1.2.m1.2.2.4" xref="S2.p1.2.m1.2.2.4.cmml">n</mi><mo id="S2.p1.2.m1.2.2.3" xref="S2.p1.2.m1.2.2.3.cmml">≡</mo><mrow id="S2.p1.2.m1.2.2.2.2" xref="S2.p1.2.m1.2.2.2.3.cmml"><mo id="S2.p1.2.m1.2.2.2.2.3" xref="S2.p1.2.m1.2.2.2.3.cmml">(</mo><msub id="S2.p1.2.m1.1.1.1.1.1" xref="S2.p1.2.m1.1.1.1.1.1.cmml"><mi id="S2.p1.2.m1.1.1.1.1.1.2" xref="S2.p1.2.m1.1.1.1.1.1.2.cmml">n</mi><mn id="S2.p1.2.m1.1.1.1.1.1.3" xref="S2.p1.2.m1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.p1.2.m1.2.2.2.2.4" xref="S2.p1.2.m1.2.2.2.3.cmml">,</mo><msub id="S2.p1.2.m1.2.2.2.2.2" xref="S2.p1.2.m1.2.2.2.2.2.cmml"><mi id="S2.p1.2.m1.2.2.2.2.2.2" xref="S2.p1.2.m1.2.2.2.2.2.2.cmml">n</mi><mn id="S2.p1.2.m1.2.2.2.2.2.3" xref="S2.p1.2.m1.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.p1.2.m1.2.2.2.2.5" xref="S2.p1.2.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.2.m1.2b"><apply id="S2.p1.2.m1.2.2.cmml" xref="S2.p1.2.m1.2.2"><equivalent id="S2.p1.2.m1.2.2.3.cmml" xref="S2.p1.2.m1.2.2.3"></equivalent><ci id="S2.p1.2.m1.2.2.4.cmml" xref="S2.p1.2.m1.2.2.4">𝑛</ci><interval closure="open" id="S2.p1.2.m1.2.2.2.3.cmml" xref="S2.p1.2.m1.2.2.2.2"><apply id="S2.p1.2.m1.1.1.1.1.1.cmml" xref="S2.p1.2.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p1.2.m1.1.1.1.1.1.1.cmml" xref="S2.p1.2.m1.1.1.1.1.1">subscript</csymbol><ci id="S2.p1.2.m1.1.1.1.1.1.2.cmml" xref="S2.p1.2.m1.1.1.1.1.1.2">𝑛</ci><cn id="S2.p1.2.m1.1.1.1.1.1.3.cmml" type="integer" xref="S2.p1.2.m1.1.1.1.1.1.3">1</cn></apply><apply id="S2.p1.2.m1.2.2.2.2.2.cmml" xref="S2.p1.2.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.p1.2.m1.2.2.2.2.2.1.cmml" xref="S2.p1.2.m1.2.2.2.2.2">subscript</csymbol><ci id="S2.p1.2.m1.2.2.2.2.2.2.cmml" xref="S2.p1.2.m1.2.2.2.2.2.2">𝑛</ci><cn id="S2.p1.2.m1.2.2.2.2.2.3.cmml" type="integer" xref="S2.p1.2.m1.2.2.2.2.2.3">2</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m1.2c">n\equiv\left(n_{1},n_{2}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m1.2d">italic_n ≡ ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> represent lattice sites, while the labels <math alttext="i\in\left\{1,2\right\}" class="ltx_Math" display="inline" id="S2.p1.3.m2.2"><semantics id="S2.p1.3.m2.2a"><mrow id="S2.p1.3.m2.2.3" xref="S2.p1.3.m2.2.3.cmml"><mi id="S2.p1.3.m2.2.3.2" xref="S2.p1.3.m2.2.3.2.cmml">i</mi><mo id="S2.p1.3.m2.2.3.1" xref="S2.p1.3.m2.2.3.1.cmml">∈</mo><mrow id="S2.p1.3.m2.2.3.3.2" xref="S2.p1.3.m2.2.3.3.1.cmml"><mo id="S2.p1.3.m2.2.3.3.2.1" xref="S2.p1.3.m2.2.3.3.1.cmml">{</mo><mn id="S2.p1.3.m2.1.1" xref="S2.p1.3.m2.1.1.cmml">1</mn><mo id="S2.p1.3.m2.2.3.3.2.2" xref="S2.p1.3.m2.2.3.3.1.cmml">,</mo><mn id="S2.p1.3.m2.2.2" xref="S2.p1.3.m2.2.2.cmml">2</mn><mo id="S2.p1.3.m2.2.3.3.2.3" xref="S2.p1.3.m2.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.3.m2.2b"><apply id="S2.p1.3.m2.2.3.cmml" xref="S2.p1.3.m2.2.3"><in id="S2.p1.3.m2.2.3.1.cmml" xref="S2.p1.3.m2.2.3.1"></in><ci id="S2.p1.3.m2.2.3.2.cmml" xref="S2.p1.3.m2.2.3.2">𝑖</ci><set id="S2.p1.3.m2.2.3.3.1.cmml" xref="S2.p1.3.m2.2.3.3.2"><cn id="S2.p1.3.m2.1.1.cmml" type="integer" xref="S2.p1.3.m2.1.1">1</cn><cn id="S2.p1.3.m2.2.2.cmml" type="integer" xref="S2.p1.3.m2.2.2">2</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m2.2c">i\in\left\{1,2\right\}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m2.2d">italic_i ∈ { 1 , 2 }</annotation></semantics></math> represent the two directions. The first terms of the Hamiltonian are plaquette terms, constructed by acting on each of the four links of a plaquette with operators represented by <math alttext="U" class="ltx_Math" display="inline" id="S2.p1.4.m3.1"><semantics id="S2.p1.4.m3.1a"><mi id="S2.p1.4.m3.1.1" xref="S2.p1.4.m3.1.1.cmml">U</mi><annotation-xml encoding="MathML-Content" id="S2.p1.4.m3.1b"><ci id="S2.p1.4.m3.1.1.cmml" xref="S2.p1.4.m3.1.1">𝑈</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.m3.1c">U</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.m3.1d">italic_U</annotation></semantics></math>. The term <math alttext="V" class="ltx_Math" display="inline" id="S2.p1.5.m4.1"><semantics id="S2.p1.5.m4.1a"><mi id="S2.p1.5.m4.1.1" xref="S2.p1.5.m4.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S2.p1.5.m4.1b"><ci id="S2.p1.5.m4.1.1.cmml" xref="S2.p1.5.m4.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m4.1c">V</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m4.1d">italic_V</annotation></semantics></math>, which we will call generically <em class="ltx_emph ltx_font_italic" id="S2.p1.8.1">potential</em>, will always be diagonal in the electric field <math alttext="E_{ni}" class="ltx_Math" display="inline" id="S2.p1.6.m5.1"><semantics id="S2.p1.6.m5.1a"><msub id="S2.p1.6.m5.1.1" xref="S2.p1.6.m5.1.1.cmml"><mi id="S2.p1.6.m5.1.1.2" xref="S2.p1.6.m5.1.1.2.cmml">E</mi><mrow id="S2.p1.6.m5.1.1.3" xref="S2.p1.6.m5.1.1.3.cmml"><mi id="S2.p1.6.m5.1.1.3.2" xref="S2.p1.6.m5.1.1.3.2.cmml">n</mi><mo id="S2.p1.6.m5.1.1.3.1" xref="S2.p1.6.m5.1.1.3.1.cmml">⁢</mo><mi id="S2.p1.6.m5.1.1.3.3" xref="S2.p1.6.m5.1.1.3.3.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.p1.6.m5.1b"><apply id="S2.p1.6.m5.1.1.cmml" xref="S2.p1.6.m5.1.1"><csymbol cd="ambiguous" id="S2.p1.6.m5.1.1.1.cmml" xref="S2.p1.6.m5.1.1">subscript</csymbol><ci id="S2.p1.6.m5.1.1.2.cmml" xref="S2.p1.6.m5.1.1.2">𝐸</ci><apply id="S2.p1.6.m5.1.1.3.cmml" xref="S2.p1.6.m5.1.1.3"><times id="S2.p1.6.m5.1.1.3.1.cmml" xref="S2.p1.6.m5.1.1.3.1"></times><ci id="S2.p1.6.m5.1.1.3.2.cmml" xref="S2.p1.6.m5.1.1.3.2">𝑛</ci><ci id="S2.p1.6.m5.1.1.3.3.cmml" xref="S2.p1.6.m5.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m5.1c">E_{ni}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m5.1d">italic_E start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. The commutation relations between these variables are <math alttext="\left[E_{mi},U_{nj}\right]=U_{mi}\delta_{mn}\delta_{ij}" class="ltx_Math" display="inline" id="S2.p1.7.m6.2"><semantics id="S2.p1.7.m6.2a"><mrow id="S2.p1.7.m6.2.2" xref="S2.p1.7.m6.2.2.cmml"><mrow id="S2.p1.7.m6.2.2.2.2" xref="S2.p1.7.m6.2.2.2.3.cmml"><mo id="S2.p1.7.m6.2.2.2.2.3" xref="S2.p1.7.m6.2.2.2.3.cmml">[</mo><msub id="S2.p1.7.m6.1.1.1.1.1" xref="S2.p1.7.m6.1.1.1.1.1.cmml"><mi id="S2.p1.7.m6.1.1.1.1.1.2" xref="S2.p1.7.m6.1.1.1.1.1.2.cmml">E</mi><mrow id="S2.p1.7.m6.1.1.1.1.1.3" xref="S2.p1.7.m6.1.1.1.1.1.3.cmml"><mi id="S2.p1.7.m6.1.1.1.1.1.3.2" xref="S2.p1.7.m6.1.1.1.1.1.3.2.cmml">m</mi><mo id="S2.p1.7.m6.1.1.1.1.1.3.1" xref="S2.p1.7.m6.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.p1.7.m6.1.1.1.1.1.3.3" xref="S2.p1.7.m6.1.1.1.1.1.3.3.cmml">i</mi></mrow></msub><mo id="S2.p1.7.m6.2.2.2.2.4" xref="S2.p1.7.m6.2.2.2.3.cmml">,</mo><msub id="S2.p1.7.m6.2.2.2.2.2" xref="S2.p1.7.m6.2.2.2.2.2.cmml"><mi id="S2.p1.7.m6.2.2.2.2.2.2" xref="S2.p1.7.m6.2.2.2.2.2.2.cmml">U</mi><mrow id="S2.p1.7.m6.2.2.2.2.2.3" xref="S2.p1.7.m6.2.2.2.2.2.3.cmml"><mi id="S2.p1.7.m6.2.2.2.2.2.3.2" xref="S2.p1.7.m6.2.2.2.2.2.3.2.cmml">n</mi><mo id="S2.p1.7.m6.2.2.2.2.2.3.1" xref="S2.p1.7.m6.2.2.2.2.2.3.1.cmml">⁢</mo><mi id="S2.p1.7.m6.2.2.2.2.2.3.3" xref="S2.p1.7.m6.2.2.2.2.2.3.3.cmml">j</mi></mrow></msub><mo id="S2.p1.7.m6.2.2.2.2.5" xref="S2.p1.7.m6.2.2.2.3.cmml">]</mo></mrow><mo id="S2.p1.7.m6.2.2.3" xref="S2.p1.7.m6.2.2.3.cmml">=</mo><mrow id="S2.p1.7.m6.2.2.4" xref="S2.p1.7.m6.2.2.4.cmml"><msub id="S2.p1.7.m6.2.2.4.2" xref="S2.p1.7.m6.2.2.4.2.cmml"><mi id="S2.p1.7.m6.2.2.4.2.2" xref="S2.p1.7.m6.2.2.4.2.2.cmml">U</mi><mrow id="S2.p1.7.m6.2.2.4.2.3" xref="S2.p1.7.m6.2.2.4.2.3.cmml"><mi id="S2.p1.7.m6.2.2.4.2.3.2" xref="S2.p1.7.m6.2.2.4.2.3.2.cmml">m</mi><mo id="S2.p1.7.m6.2.2.4.2.3.1" xref="S2.p1.7.m6.2.2.4.2.3.1.cmml">⁢</mo><mi id="S2.p1.7.m6.2.2.4.2.3.3" xref="S2.p1.7.m6.2.2.4.2.3.3.cmml">i</mi></mrow></msub><mo id="S2.p1.7.m6.2.2.4.1" xref="S2.p1.7.m6.2.2.4.1.cmml">⁢</mo><msub id="S2.p1.7.m6.2.2.4.3" xref="S2.p1.7.m6.2.2.4.3.cmml"><mi id="S2.p1.7.m6.2.2.4.3.2" xref="S2.p1.7.m6.2.2.4.3.2.cmml">δ</mi><mrow id="S2.p1.7.m6.2.2.4.3.3" xref="S2.p1.7.m6.2.2.4.3.3.cmml"><mi id="S2.p1.7.m6.2.2.4.3.3.2" xref="S2.p1.7.m6.2.2.4.3.3.2.cmml">m</mi><mo id="S2.p1.7.m6.2.2.4.3.3.1" xref="S2.p1.7.m6.2.2.4.3.3.1.cmml">⁢</mo><mi id="S2.p1.7.m6.2.2.4.3.3.3" xref="S2.p1.7.m6.2.2.4.3.3.3.cmml">n</mi></mrow></msub><mo id="S2.p1.7.m6.2.2.4.1a" xref="S2.p1.7.m6.2.2.4.1.cmml">⁢</mo><msub id="S2.p1.7.m6.2.2.4.4" xref="S2.p1.7.m6.2.2.4.4.cmml"><mi id="S2.p1.7.m6.2.2.4.4.2" xref="S2.p1.7.m6.2.2.4.4.2.cmml">δ</mi><mrow id="S2.p1.7.m6.2.2.4.4.3" xref="S2.p1.7.m6.2.2.4.4.3.cmml"><mi id="S2.p1.7.m6.2.2.4.4.3.2" xref="S2.p1.7.m6.2.2.4.4.3.2.cmml">i</mi><mo id="S2.p1.7.m6.2.2.4.4.3.1" xref="S2.p1.7.m6.2.2.4.4.3.1.cmml">⁢</mo><mi id="S2.p1.7.m6.2.2.4.4.3.3" xref="S2.p1.7.m6.2.2.4.4.3.3.cmml">j</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.7.m6.2b"><apply id="S2.p1.7.m6.2.2.cmml" xref="S2.p1.7.m6.2.2"><eq id="S2.p1.7.m6.2.2.3.cmml" xref="S2.p1.7.m6.2.2.3"></eq><interval closure="closed" id="S2.p1.7.m6.2.2.2.3.cmml" xref="S2.p1.7.m6.2.2.2.2"><apply id="S2.p1.7.m6.1.1.1.1.1.cmml" xref="S2.p1.7.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p1.7.m6.1.1.1.1.1.1.cmml" xref="S2.p1.7.m6.1.1.1.1.1">subscript</csymbol><ci id="S2.p1.7.m6.1.1.1.1.1.2.cmml" xref="S2.p1.7.m6.1.1.1.1.1.2">𝐸</ci><apply id="S2.p1.7.m6.1.1.1.1.1.3.cmml" xref="S2.p1.7.m6.1.1.1.1.1.3"><times id="S2.p1.7.m6.1.1.1.1.1.3.1.cmml" xref="S2.p1.7.m6.1.1.1.1.1.3.1"></times><ci id="S2.p1.7.m6.1.1.1.1.1.3.2.cmml" xref="S2.p1.7.m6.1.1.1.1.1.3.2">𝑚</ci><ci id="S2.p1.7.m6.1.1.1.1.1.3.3.cmml" xref="S2.p1.7.m6.1.1.1.1.1.3.3">𝑖</ci></apply></apply><apply id="S2.p1.7.m6.2.2.2.2.2.cmml" xref="S2.p1.7.m6.2.2.2.2.2"><csymbol cd="ambiguous" 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id="S2.p1.7.m6.2.2.4.4.3.1.cmml" xref="S2.p1.7.m6.2.2.4.4.3.1"></times><ci id="S2.p1.7.m6.2.2.4.4.3.2.cmml" xref="S2.p1.7.m6.2.2.4.4.3.2">𝑖</ci><ci id="S2.p1.7.m6.2.2.4.4.3.3.cmml" xref="S2.p1.7.m6.2.2.4.4.3.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.7.m6.2c">\left[E_{mi},U_{nj}\right]=U_{mi}\delta_{mn}\delta_{ij}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.7.m6.2d">[ italic_E start_POSTSUBSCRIPT italic_m italic_i end_POSTSUBSCRIPT , italic_U start_POSTSUBSCRIPT italic_n italic_j end_POSTSUBSCRIPT ] = italic_U start_POSTSUBSCRIPT italic_m italic_i end_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_m italic_n end_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. Under these general conditions, the Hamiltonian has a set of local symmetries. Concretely, there is one generator of local gauge transformations <math alttext="G_{n}" class="ltx_Math" display="inline" id="S2.p1.8.m7.1"><semantics id="S2.p1.8.m7.1a"><msub id="S2.p1.8.m7.1.1" xref="S2.p1.8.m7.1.1.cmml"><mi id="S2.p1.8.m7.1.1.2" xref="S2.p1.8.m7.1.1.2.cmml">G</mi><mi id="S2.p1.8.m7.1.1.3" xref="S2.p1.8.m7.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.8.m7.1b"><apply id="S2.p1.8.m7.1.1.cmml" xref="S2.p1.8.m7.1.1"><csymbol cd="ambiguous" id="S2.p1.8.m7.1.1.1.cmml" xref="S2.p1.8.m7.1.1">subscript</csymbol><ci id="S2.p1.8.m7.1.1.2.cmml" xref="S2.p1.8.m7.1.1.2">𝐺</ci><ci id="S2.p1.8.m7.1.1.3.cmml" xref="S2.p1.8.m7.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.8.m7.1c">G_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.8.m7.1d">italic_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> per lattice site, which commutes with the Hamiltonian</p> 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id="S2.E2.m1.2c">G_{n}=E_{n1}+E_{n2}-E_{n-\hat{1}1}-E_{n-\hat{2}2},\quad\left[H,G_{n}\right]=0.</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.2d">italic_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_E start_POSTSUBSCRIPT italic_n 1 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_n 2 end_POSTSUBSCRIPT - italic_E start_POSTSUBSCRIPT italic_n - over^ start_ARG 1 end_ARG 1 end_POSTSUBSCRIPT - italic_E start_POSTSUBSCRIPT italic_n - over^ start_ARG 2 end_ARG 2 end_POSTSUBSCRIPT , [ italic_H , italic_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] = 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.11">We will use the electric field basis, denoted by <math alttext="\left|\varepsilon\right>" class="ltx_Math" display="inline" id="S2.p1.9.m1.1"><semantics id="S2.p1.9.m1.1a"><mrow id="S2.p1.9.m1.1.2.2" xref="S2.p1.9.m1.1.2.1.cmml"><mo id="S2.p1.9.m1.1.2.2.1" xref="S2.p1.9.m1.1.2.1.1.cmml">|</mo><mi id="S2.p1.9.m1.1.1" xref="S2.p1.9.m1.1.1.cmml">ε</mi><mo id="S2.p1.9.m1.1.2.2.2" xref="S2.p1.9.m1.1.2.1.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.9.m1.1b"><apply id="S2.p1.9.m1.1.2.1.cmml" xref="S2.p1.9.m1.1.2.2"><csymbol cd="latexml" id="S2.p1.9.m1.1.2.1.1.cmml" xref="S2.p1.9.m1.1.2.2.1">ket</csymbol><ci id="S2.p1.9.m1.1.1.cmml" xref="S2.p1.9.m1.1.1">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.9.m1.1c">\left|\varepsilon\right></annotation><annotation encoding="application/x-llamapun" id="S2.p1.9.m1.1d">| italic_ε ⟩</annotation></semantics></math> for a single link. The generators <math alttext="G_{n}" class="ltx_Math" display="inline" id="S2.p1.10.m2.1"><semantics id="S2.p1.10.m2.1a"><msub id="S2.p1.10.m2.1.1" xref="S2.p1.10.m2.1.1.cmml"><mi id="S2.p1.10.m2.1.1.2" xref="S2.p1.10.m2.1.1.2.cmml">G</mi><mi id="S2.p1.10.m2.1.1.3" xref="S2.p1.10.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p1.10.m2.1b"><apply id="S2.p1.10.m2.1.1.cmml" xref="S2.p1.10.m2.1.1"><csymbol cd="ambiguous" id="S2.p1.10.m2.1.1.1.cmml" xref="S2.p1.10.m2.1.1">subscript</csymbol><ci id="S2.p1.10.m2.1.1.2.cmml" xref="S2.p1.10.m2.1.1.2">𝐺</ci><ci id="S2.p1.10.m2.1.1.3.cmml" xref="S2.p1.10.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.10.m2.1c">G_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.10.m2.1d">italic_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> are diagonal in this basis. The Hilbert space breaks into many different sectors. 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id="S2.p2.2.m2.2.2.2.2.1.1.3.cmml" type="integer" xref="S2.p2.2.m2.2.2.2.2.1.1.3">2</cn></apply></interval><cn id="S2.p2.2.m2.2.2.2.4.cmml" type="integer" xref="S2.p2.2.m2.2.2.2.4">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.m2.2c">W_{2}=\sum_{n_{2}=0}^{L_{2}-1}E_{\left(m,n_{2}\right)1}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.2.m2.2d">italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT italic_E start_POSTSUBSCRIPT ( italic_m , italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) 1 end_POSTSUBSCRIPT</annotation></semantics></math> commute with the Hamiltonian. In the physical sector, their eigenvalues are independent of <math alttext="m" class="ltx_Math" display="inline" id="S2.p2.3.m3.1"><semantics id="S2.p2.3.m3.1a"><mi id="S2.p2.3.m3.1.1" xref="S2.p2.3.m3.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S2.p2.3.m3.1b"><ci id="S2.p2.3.m3.1.1.cmml" xref="S2.p2.3.m3.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.3.m3.1c">m</annotation><annotation encoding="application/x-llamapun" id="S2.p2.3.m3.1d">italic_m</annotation></semantics></math>. We focus on the zero-winding sector <math alttext="W_{i}\left|\psi\right>=0" class="ltx_Math" display="inline" id="S2.p2.4.m4.1"><semantics id="S2.p2.4.m4.1a"><mrow id="S2.p2.4.m4.1.2" xref="S2.p2.4.m4.1.2.cmml"><mrow id="S2.p2.4.m4.1.2.2" xref="S2.p2.4.m4.1.2.2.cmml"><msub id="S2.p2.4.m4.1.2.2.2" xref="S2.p2.4.m4.1.2.2.2.cmml"><mi id="S2.p2.4.m4.1.2.2.2.2" xref="S2.p2.4.m4.1.2.2.2.2.cmml">W</mi><mi id="S2.p2.4.m4.1.2.2.2.3" xref="S2.p2.4.m4.1.2.2.2.3.cmml">i</mi></msub><mo id="S2.p2.4.m4.1.2.2.1" xref="S2.p2.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S2.p2.4.m4.1.2.2.3.2" xref="S2.p2.4.m4.1.2.2.3.1.cmml"><mo id="S2.p2.4.m4.1.2.2.3.2.1" xref="S2.p2.4.m4.1.2.2.3.1.1.cmml">|</mo><mi id="S2.p2.4.m4.1.1" xref="S2.p2.4.m4.1.1.cmml">ψ</mi><mo id="S2.p2.4.m4.1.2.2.3.2.2" xref="S2.p2.4.m4.1.2.2.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S2.p2.4.m4.1.2.1" xref="S2.p2.4.m4.1.2.1.cmml">=</mo><mn id="S2.p2.4.m4.1.2.3" xref="S2.p2.4.m4.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.4.m4.1b"><apply id="S2.p2.4.m4.1.2.cmml" xref="S2.p2.4.m4.1.2"><eq id="S2.p2.4.m4.1.2.1.cmml" xref="S2.p2.4.m4.1.2.1"></eq><apply id="S2.p2.4.m4.1.2.2.cmml" xref="S2.p2.4.m4.1.2.2"><times id="S2.p2.4.m4.1.2.2.1.cmml" xref="S2.p2.4.m4.1.2.2.1"></times><apply id="S2.p2.4.m4.1.2.2.2.cmml" xref="S2.p2.4.m4.1.2.2.2"><csymbol cd="ambiguous" id="S2.p2.4.m4.1.2.2.2.1.cmml" xref="S2.p2.4.m4.1.2.2.2">subscript</csymbol><ci id="S2.p2.4.m4.1.2.2.2.2.cmml" xref="S2.p2.4.m4.1.2.2.2.2">𝑊</ci><ci id="S2.p2.4.m4.1.2.2.2.3.cmml" xref="S2.p2.4.m4.1.2.2.2.3">𝑖</ci></apply><apply id="S2.p2.4.m4.1.2.2.3.1.cmml" xref="S2.p2.4.m4.1.2.2.3.2"><csymbol cd="latexml" id="S2.p2.4.m4.1.2.2.3.1.1.cmml" xref="S2.p2.4.m4.1.2.2.3.2.1">ket</csymbol><ci id="S2.p2.4.m4.1.1.cmml" xref="S2.p2.4.m4.1.1">𝜓</ci></apply></apply><cn id="S2.p2.4.m4.1.2.3.cmml" type="integer" xref="S2.p2.4.m4.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.4.m4.1c">W_{i}\left|\psi\right>=0</annotation><annotation encoding="application/x-llamapun" id="S2.p2.4.m4.1d">italic_W start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_ψ ⟩ = 0</annotation></semantics></math> for both <math alttext="i=1" class="ltx_Math" display="inline" id="S2.p2.5.m5.1"><semantics id="S2.p2.5.m5.1a"><mrow id="S2.p2.5.m5.1.1" xref="S2.p2.5.m5.1.1.cmml"><mi id="S2.p2.5.m5.1.1.2" xref="S2.p2.5.m5.1.1.2.cmml">i</mi><mo id="S2.p2.5.m5.1.1.1" xref="S2.p2.5.m5.1.1.1.cmml">=</mo><mn id="S2.p2.5.m5.1.1.3" xref="S2.p2.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.5.m5.1b"><apply id="S2.p2.5.m5.1.1.cmml" xref="S2.p2.5.m5.1.1"><eq id="S2.p2.5.m5.1.1.1.cmml" xref="S2.p2.5.m5.1.1.1"></eq><ci id="S2.p2.5.m5.1.1.2.cmml" xref="S2.p2.5.m5.1.1.2">𝑖</ci><cn id="S2.p2.5.m5.1.1.3.cmml" type="integer" xref="S2.p2.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.5.m5.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S2.p2.5.m5.1d">italic_i = 1</annotation></semantics></math> and <math alttext="i=2" class="ltx_Math" display="inline" id="S2.p2.6.m6.1"><semantics id="S2.p2.6.m6.1a"><mrow id="S2.p2.6.m6.1.1" xref="S2.p2.6.m6.1.1.cmml"><mi id="S2.p2.6.m6.1.1.2" xref="S2.p2.6.m6.1.1.2.cmml">i</mi><mo id="S2.p2.6.m6.1.1.1" xref="S2.p2.6.m6.1.1.1.cmml">=</mo><mn id="S2.p2.6.m6.1.1.3" xref="S2.p2.6.m6.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.6.m6.1b"><apply id="S2.p2.6.m6.1.1.cmml" xref="S2.p2.6.m6.1.1"><eq id="S2.p2.6.m6.1.1.1.cmml" xref="S2.p2.6.m6.1.1.1"></eq><ci id="S2.p2.6.m6.1.1.2.cmml" xref="S2.p2.6.m6.1.1.2">𝑖</ci><cn id="S2.p2.6.m6.1.1.3.cmml" type="integer" xref="S2.p2.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.6.m6.1c">i=2</annotation><annotation encoding="application/x-llamapun" id="S2.p2.6.m6.1d">italic_i = 2</annotation></semantics></math>. This is the largest sector of the theory. There are further global symmetries that depend on the choice of the potential and are used to reduce the size of the Hilbert space in exact diagonalization calculations. They are described in the Supplementary Material <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib56" title="">56</a>]</cite>. We will be interested in two specific versions of this model that ensure that we will have a finite-dimensional Hilbert space per link.</p> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.6">In Quantum Link Models (QLM) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib57" title="">57</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib58" title="">58</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib51" title="">51</a>]</cite> <math alttext="U_{ni}/U_{ni}^{\dagger}" class="ltx_Math" display="inline" id="S2.p3.1.m1.1"><semantics id="S2.p3.1.m1.1a"><mrow id="S2.p3.1.m1.1.1" xref="S2.p3.1.m1.1.1.cmml"><msub id="S2.p3.1.m1.1.1.2" xref="S2.p3.1.m1.1.1.2.cmml"><mi id="S2.p3.1.m1.1.1.2.2" xref="S2.p3.1.m1.1.1.2.2.cmml">U</mi><mrow id="S2.p3.1.m1.1.1.2.3" xref="S2.p3.1.m1.1.1.2.3.cmml"><mi id="S2.p3.1.m1.1.1.2.3.2" xref="S2.p3.1.m1.1.1.2.3.2.cmml">n</mi><mo id="S2.p3.1.m1.1.1.2.3.1" xref="S2.p3.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.p3.1.m1.1.1.2.3.3" xref="S2.p3.1.m1.1.1.2.3.3.cmml">i</mi></mrow></msub><mo id="S2.p3.1.m1.1.1.1" xref="S2.p3.1.m1.1.1.1.cmml">/</mo><msubsup id="S2.p3.1.m1.1.1.3" xref="S2.p3.1.m1.1.1.3.cmml"><mi id="S2.p3.1.m1.1.1.3.2.2" xref="S2.p3.1.m1.1.1.3.2.2.cmml">U</mi><mrow id="S2.p3.1.m1.1.1.3.2.3" xref="S2.p3.1.m1.1.1.3.2.3.cmml"><mi id="S2.p3.1.m1.1.1.3.2.3.2" xref="S2.p3.1.m1.1.1.3.2.3.2.cmml">n</mi><mo id="S2.p3.1.m1.1.1.3.2.3.1" xref="S2.p3.1.m1.1.1.3.2.3.1.cmml">⁢</mo><mi id="S2.p3.1.m1.1.1.3.2.3.3" xref="S2.p3.1.m1.1.1.3.2.3.3.cmml">i</mi></mrow><mo id="S2.p3.1.m1.1.1.3.3" xref="S2.p3.1.m1.1.1.3.3.cmml">†</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.1b"><apply id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1"><divide id="S2.p3.1.m1.1.1.1.cmml" xref="S2.p3.1.m1.1.1.1"></divide><apply id="S2.p3.1.m1.1.1.2.cmml" xref="S2.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.p3.1.m1.1.1.2.1.cmml" xref="S2.p3.1.m1.1.1.2">subscript</csymbol><ci id="S2.p3.1.m1.1.1.2.2.cmml" xref="S2.p3.1.m1.1.1.2.2">𝑈</ci><apply id="S2.p3.1.m1.1.1.2.3.cmml" xref="S2.p3.1.m1.1.1.2.3"><times id="S2.p3.1.m1.1.1.2.3.1.cmml" xref="S2.p3.1.m1.1.1.2.3.1"></times><ci id="S2.p3.1.m1.1.1.2.3.2.cmml" xref="S2.p3.1.m1.1.1.2.3.2">𝑛</ci><ci id="S2.p3.1.m1.1.1.2.3.3.cmml" xref="S2.p3.1.m1.1.1.2.3.3">𝑖</ci></apply></apply><apply id="S2.p3.1.m1.1.1.3.cmml" xref="S2.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p3.1.m1.1.1.3.1.cmml" xref="S2.p3.1.m1.1.1.3">superscript</csymbol><apply id="S2.p3.1.m1.1.1.3.2.cmml" xref="S2.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p3.1.m1.1.1.3.2.1.cmml" xref="S2.p3.1.m1.1.1.3">subscript</csymbol><ci id="S2.p3.1.m1.1.1.3.2.2.cmml" xref="S2.p3.1.m1.1.1.3.2.2">𝑈</ci><apply id="S2.p3.1.m1.1.1.3.2.3.cmml" xref="S2.p3.1.m1.1.1.3.2.3"><times id="S2.p3.1.m1.1.1.3.2.3.1.cmml" xref="S2.p3.1.m1.1.1.3.2.3.1"></times><ci id="S2.p3.1.m1.1.1.3.2.3.2.cmml" xref="S2.p3.1.m1.1.1.3.2.3.2">𝑛</ci><ci id="S2.p3.1.m1.1.1.3.2.3.3.cmml" xref="S2.p3.1.m1.1.1.3.2.3.3">𝑖</ci></apply></apply><ci id="S2.p3.1.m1.1.1.3.3.cmml" xref="S2.p3.1.m1.1.1.3.3">†</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.1c">U_{ni}/U_{ni}^{\dagger}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.1d">italic_U start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT / italic_U start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT</annotation></semantics></math> are spin raising/lowering operators. <math alttext="U_{ni}" class="ltx_Math" display="inline" id="S2.p3.2.m2.1"><semantics id="S2.p3.2.m2.1a"><msub id="S2.p3.2.m2.1.1" xref="S2.p3.2.m2.1.1.cmml"><mi id="S2.p3.2.m2.1.1.2" xref="S2.p3.2.m2.1.1.2.cmml">U</mi><mrow id="S2.p3.2.m2.1.1.3" xref="S2.p3.2.m2.1.1.3.cmml"><mi id="S2.p3.2.m2.1.1.3.2" xref="S2.p3.2.m2.1.1.3.2.cmml">n</mi><mo id="S2.p3.2.m2.1.1.3.1" xref="S2.p3.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.p3.2.m2.1.1.3.3" xref="S2.p3.2.m2.1.1.3.3.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.p3.2.m2.1b"><apply id="S2.p3.2.m2.1.1.cmml" xref="S2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p3.2.m2.1.1.1.cmml" xref="S2.p3.2.m2.1.1">subscript</csymbol><ci id="S2.p3.2.m2.1.1.2.cmml" xref="S2.p3.2.m2.1.1.2">𝑈</ci><apply id="S2.p3.2.m2.1.1.3.cmml" xref="S2.p3.2.m2.1.1.3"><times id="S2.p3.2.m2.1.1.3.1.cmml" xref="S2.p3.2.m2.1.1.3.1"></times><ci id="S2.p3.2.m2.1.1.3.2.cmml" xref="S2.p3.2.m2.1.1.3.2">𝑛</ci><ci id="S2.p3.2.m2.1.1.3.3.cmml" xref="S2.p3.2.m2.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.2.m2.1c">U_{ni}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.2.m2.1d">italic_U start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="U^{\dagger}_{ni}" class="ltx_Math" display="inline" id="S2.p3.3.m3.1"><semantics id="S2.p3.3.m3.1a"><msubsup id="S2.p3.3.m3.1.1" xref="S2.p3.3.m3.1.1.cmml"><mi id="S2.p3.3.m3.1.1.2.2" xref="S2.p3.3.m3.1.1.2.2.cmml">U</mi><mrow id="S2.p3.3.m3.1.1.3" xref="S2.p3.3.m3.1.1.3.cmml"><mi id="S2.p3.3.m3.1.1.3.2" xref="S2.p3.3.m3.1.1.3.2.cmml">n</mi><mo id="S2.p3.3.m3.1.1.3.1" xref="S2.p3.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.p3.3.m3.1.1.3.3" xref="S2.p3.3.m3.1.1.3.3.cmml">i</mi></mrow><mo id="S2.p3.3.m3.1.1.2.3" xref="S2.p3.3.m3.1.1.2.3.cmml">†</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.p3.3.m3.1b"><apply id="S2.p3.3.m3.1.1.cmml" xref="S2.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p3.3.m3.1.1.1.cmml" xref="S2.p3.3.m3.1.1">subscript</csymbol><apply id="S2.p3.3.m3.1.1.2.cmml" xref="S2.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p3.3.m3.1.1.2.1.cmml" xref="S2.p3.3.m3.1.1">superscript</csymbol><ci id="S2.p3.3.m3.1.1.2.2.cmml" xref="S2.p3.3.m3.1.1.2.2">𝑈</ci><ci id="S2.p3.3.m3.1.1.2.3.cmml" xref="S2.p3.3.m3.1.1.2.3">†</ci></apply><apply id="S2.p3.3.m3.1.1.3.cmml" xref="S2.p3.3.m3.1.1.3"><times id="S2.p3.3.m3.1.1.3.1.cmml" xref="S2.p3.3.m3.1.1.3.1"></times><ci id="S2.p3.3.m3.1.1.3.2.cmml" xref="S2.p3.3.m3.1.1.3.2">𝑛</ci><ci id="S2.p3.3.m3.1.1.3.3.cmml" xref="S2.p3.3.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.3.m3.1c">U^{\dagger}_{ni}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.3.m3.1d">italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT</annotation></semantics></math> do not commute and act according to <math alttext="U\left|\varepsilon\right>\propto\sqrt{S\left(S+1\right)-\varepsilon\left(% \varepsilon+1\right)}\ket{\varepsilon+1}" class="ltx_Math" display="inline" id="S2.p3.4.m4.4"><semantics id="S2.p3.4.m4.4a"><mrow id="S2.p3.4.m4.4.5" xref="S2.p3.4.m4.4.5.cmml"><mrow id="S2.p3.4.m4.4.5.2" xref="S2.p3.4.m4.4.5.2.cmml"><mi 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id="S2.p3.4.m4.3.3.2.2.1.1.1.1" xref="S2.p3.4.m4.3.3.2.2.1.1.1.1.cmml">+</mo><mn id="S2.p3.4.m4.3.3.2.2.1.1.1.3" xref="S2.p3.4.m4.3.3.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S2.p3.4.m4.3.3.2.2.1.1.3" xref="S2.p3.4.m4.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></msqrt><mo id="S2.p3.4.m4.4.5.3.1" xref="S2.p3.4.m4.4.5.3.1.cmml">⁢</mo><mrow id="S2.p3.4.m4.1.1.3" xref="S2.p3.4.m4.1.1.2.cmml"><mo id="S2.p3.4.m4.1.1.3.1" stretchy="false" xref="S2.p3.4.m4.1.1.2.1.cmml">|</mo><mrow id="S2.p3.4.m4.1.1.1.1" xref="S2.p3.4.m4.1.1.1.1.cmml"><mi id="S2.p3.4.m4.1.1.1.1.2" xref="S2.p3.4.m4.1.1.1.1.2.cmml">ε</mi><mo id="S2.p3.4.m4.1.1.1.1.1" xref="S2.p3.4.m4.1.1.1.1.1.cmml">+</mo><mn id="S2.p3.4.m4.1.1.1.1.3" xref="S2.p3.4.m4.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.p3.4.m4.1.1.3.2" stretchy="false" xref="S2.p3.4.m4.1.1.2.1.cmml">⟩</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.4.m4.4b"><apply id="S2.p3.4.m4.4.5.cmml" xref="S2.p3.4.m4.4.5"><csymbol cd="latexml" 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xref="S2.p3.4.m4.2.2.1.1.2"></times><ci id="S2.p3.4.m4.2.2.1.1.3.cmml" xref="S2.p3.4.m4.2.2.1.1.3">𝑆</ci><apply id="S2.p3.4.m4.2.2.1.1.1.1.1.cmml" xref="S2.p3.4.m4.2.2.1.1.1.1"><plus id="S2.p3.4.m4.2.2.1.1.1.1.1.1.cmml" xref="S2.p3.4.m4.2.2.1.1.1.1.1.1"></plus><ci id="S2.p3.4.m4.2.2.1.1.1.1.1.2.cmml" xref="S2.p3.4.m4.2.2.1.1.1.1.1.2">𝑆</ci><cn id="S2.p3.4.m4.2.2.1.1.1.1.1.3.cmml" type="integer" xref="S2.p3.4.m4.2.2.1.1.1.1.1.3">1</cn></apply></apply><apply id="S2.p3.4.m4.3.3.2.2.cmml" xref="S2.p3.4.m4.3.3.2.2"><times id="S2.p3.4.m4.3.3.2.2.2.cmml" xref="S2.p3.4.m4.3.3.2.2.2"></times><ci id="S2.p3.4.m4.3.3.2.2.3.cmml" xref="S2.p3.4.m4.3.3.2.2.3">𝜀</ci><apply id="S2.p3.4.m4.3.3.2.2.1.1.1.cmml" xref="S2.p3.4.m4.3.3.2.2.1.1"><plus id="S2.p3.4.m4.3.3.2.2.1.1.1.1.cmml" xref="S2.p3.4.m4.3.3.2.2.1.1.1.1"></plus><ci id="S2.p3.4.m4.3.3.2.2.1.1.1.2.cmml" xref="S2.p3.4.m4.3.3.2.2.1.1.1.2">𝜀</ci><cn id="S2.p3.4.m4.3.3.2.2.1.1.1.3.cmml" type="integer" xref="S2.p3.4.m4.3.3.2.2.1.1.1.3">1</cn></apply></apply></apply></apply><apply id="S2.p3.4.m4.1.1.2.cmml" xref="S2.p3.4.m4.1.1.3"><csymbol cd="latexml" id="S2.p3.4.m4.1.1.2.1.cmml" xref="S2.p3.4.m4.1.1.3.1">ket</csymbol><apply id="S2.p3.4.m4.1.1.1.1.cmml" xref="S2.p3.4.m4.1.1.1.1"><plus id="S2.p3.4.m4.1.1.1.1.1.cmml" xref="S2.p3.4.m4.1.1.1.1.1"></plus><ci id="S2.p3.4.m4.1.1.1.1.2.cmml" xref="S2.p3.4.m4.1.1.1.1.2">𝜀</ci><cn id="S2.p3.4.m4.1.1.1.1.3.cmml" type="integer" xref="S2.p3.4.m4.1.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.4.m4.4c">U\left|\varepsilon\right>\propto\sqrt{S\left(S+1\right)-\varepsilon\left(% \varepsilon+1\right)}\ket{\varepsilon+1}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.4.m4.4d">italic_U | italic_ε ⟩ ∝ square-root start_ARG italic_S ( italic_S + 1 ) - italic_ε ( italic_ε + 1 ) end_ARG | start_ARG italic_ε + 1 end_ARG ⟩</annotation></semantics></math>. The electric field operators <math alttext="E" class="ltx_Math" display="inline" id="S2.p3.5.m5.1"><semantics id="S2.p3.5.m5.1a"><mi id="S2.p3.5.m5.1.1" xref="S2.p3.5.m5.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S2.p3.5.m5.1b"><ci id="S2.p3.5.m5.1.1.cmml" xref="S2.p3.5.m5.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.5.m5.1c">E</annotation><annotation encoding="application/x-llamapun" id="S2.p3.5.m5.1d">italic_E</annotation></semantics></math> correspond to the <math alttext="z" class="ltx_Math" display="inline" id="S2.p3.6.m6.1"><semantics id="S2.p3.6.m6.1a"><mi id="S2.p3.6.m6.1.1" xref="S2.p3.6.m6.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.p3.6.m6.1b"><ci id="S2.p3.6.m6.1.1.cmml" xref="S2.p3.6.m6.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.6.m6.1c">z</annotation><annotation encoding="application/x-llamapun" id="S2.p3.6.m6.1d">italic_z</annotation></semantics></math> component spin operators.</p> </div> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.11">In Truncated Link Models (TLM) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib59" title="">59</a>]</cite> (see also <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib29" title="">29</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib60" title="">60</a>]</cite>) the electric field is simply truncated, meaning <math alttext="U\left|S\right>=0" class="ltx_Math" display="inline" id="S2.p4.1.m1.1"><semantics id="S2.p4.1.m1.1a"><mrow id="S2.p4.1.m1.1.2" xref="S2.p4.1.m1.1.2.cmml"><mrow id="S2.p4.1.m1.1.2.2" xref="S2.p4.1.m1.1.2.2.cmml"><mi id="S2.p4.1.m1.1.2.2.2" xref="S2.p4.1.m1.1.2.2.2.cmml">U</mi><mo id="S2.p4.1.m1.1.2.2.1" xref="S2.p4.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S2.p4.1.m1.1.2.2.3.2" xref="S2.p4.1.m1.1.2.2.3.1.cmml"><mo id="S2.p4.1.m1.1.2.2.3.2.1" xref="S2.p4.1.m1.1.2.2.3.1.1.cmml">|</mo><mi id="S2.p4.1.m1.1.1" xref="S2.p4.1.m1.1.1.cmml">S</mi><mo id="S2.p4.1.m1.1.2.2.3.2.2" xref="S2.p4.1.m1.1.2.2.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S2.p4.1.m1.1.2.1" xref="S2.p4.1.m1.1.2.1.cmml">=</mo><mn id="S2.p4.1.m1.1.2.3" xref="S2.p4.1.m1.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.1.m1.1b"><apply id="S2.p4.1.m1.1.2.cmml" xref="S2.p4.1.m1.1.2"><eq id="S2.p4.1.m1.1.2.1.cmml" xref="S2.p4.1.m1.1.2.1"></eq><apply id="S2.p4.1.m1.1.2.2.cmml" xref="S2.p4.1.m1.1.2.2"><times id="S2.p4.1.m1.1.2.2.1.cmml" xref="S2.p4.1.m1.1.2.2.1"></times><ci id="S2.p4.1.m1.1.2.2.2.cmml" xref="S2.p4.1.m1.1.2.2.2">𝑈</ci><apply id="S2.p4.1.m1.1.2.2.3.1.cmml" xref="S2.p4.1.m1.1.2.2.3.2"><csymbol cd="latexml" id="S2.p4.1.m1.1.2.2.3.1.1.cmml" xref="S2.p4.1.m1.1.2.2.3.2.1">ket</csymbol><ci id="S2.p4.1.m1.1.1.cmml" xref="S2.p4.1.m1.1.1">𝑆</ci></apply></apply><cn id="S2.p4.1.m1.1.2.3.cmml" type="integer" xref="S2.p4.1.m1.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.1.m1.1c">U\left|S\right>=0</annotation><annotation encoding="application/x-llamapun" id="S2.p4.1.m1.1d">italic_U | italic_S ⟩ = 0</annotation></semantics></math>, <math alttext="U^{\dagger}\left|-S\right>=0" class="ltx_Math" display="inline" id="S2.p4.2.m2.1"><semantics id="S2.p4.2.m2.1a"><mrow id="S2.p4.2.m2.1.1" xref="S2.p4.2.m2.1.1.cmml"><mrow id="S2.p4.2.m2.1.1.1" xref="S2.p4.2.m2.1.1.1.cmml"><msup id="S2.p4.2.m2.1.1.1.3" xref="S2.p4.2.m2.1.1.1.3.cmml"><mi id="S2.p4.2.m2.1.1.1.3.2" xref="S2.p4.2.m2.1.1.1.3.2.cmml">U</mi><mo id="S2.p4.2.m2.1.1.1.3.3" xref="S2.p4.2.m2.1.1.1.3.3.cmml">†</mo></msup><mo id="S2.p4.2.m2.1.1.1.2" xref="S2.p4.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S2.p4.2.m2.1.1.1.1.1" xref="S2.p4.2.m2.1.1.1.1.2.cmml"><mo id="S2.p4.2.m2.1.1.1.1.1.2" xref="S2.p4.2.m2.1.1.1.1.2.1.cmml">|</mo><mrow id="S2.p4.2.m2.1.1.1.1.1.1" xref="S2.p4.2.m2.1.1.1.1.1.1.cmml"><mo id="S2.p4.2.m2.1.1.1.1.1.1a" xref="S2.p4.2.m2.1.1.1.1.1.1.cmml">−</mo><mi id="S2.p4.2.m2.1.1.1.1.1.1.2" xref="S2.p4.2.m2.1.1.1.1.1.1.2.cmml">S</mi></mrow><mo id="S2.p4.2.m2.1.1.1.1.1.3" xref="S2.p4.2.m2.1.1.1.1.2.1.cmml">⟩</mo></mrow></mrow><mo id="S2.p4.2.m2.1.1.2" xref="S2.p4.2.m2.1.1.2.cmml">=</mo><mn id="S2.p4.2.m2.1.1.3" xref="S2.p4.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.2.m2.1b"><apply id="S2.p4.2.m2.1.1.cmml" xref="S2.p4.2.m2.1.1"><eq id="S2.p4.2.m2.1.1.2.cmml" xref="S2.p4.2.m2.1.1.2"></eq><apply id="S2.p4.2.m2.1.1.1.cmml" xref="S2.p4.2.m2.1.1.1"><times id="S2.p4.2.m2.1.1.1.2.cmml" xref="S2.p4.2.m2.1.1.1.2"></times><apply id="S2.p4.2.m2.1.1.1.3.cmml" xref="S2.p4.2.m2.1.1.1.3"><csymbol cd="ambiguous" id="S2.p4.2.m2.1.1.1.3.1.cmml" xref="S2.p4.2.m2.1.1.1.3">superscript</csymbol><ci id="S2.p4.2.m2.1.1.1.3.2.cmml" xref="S2.p4.2.m2.1.1.1.3.2">𝑈</ci><ci id="S2.p4.2.m2.1.1.1.3.3.cmml" xref="S2.p4.2.m2.1.1.1.3.3">†</ci></apply><apply id="S2.p4.2.m2.1.1.1.1.2.cmml" xref="S2.p4.2.m2.1.1.1.1.1"><csymbol cd="latexml" id="S2.p4.2.m2.1.1.1.1.2.1.cmml" xref="S2.p4.2.m2.1.1.1.1.1.2">ket</csymbol><apply id="S2.p4.2.m2.1.1.1.1.1.1.cmml" xref="S2.p4.2.m2.1.1.1.1.1.1"><minus id="S2.p4.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.p4.2.m2.1.1.1.1.1.1"></minus><ci id="S2.p4.2.m2.1.1.1.1.1.1.2.cmml" xref="S2.p4.2.m2.1.1.1.1.1.1.2">𝑆</ci></apply></apply></apply><cn id="S2.p4.2.m2.1.1.3.cmml" type="integer" xref="S2.p4.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.2.m2.1c">U^{\dagger}\left|-S\right>=0</annotation><annotation encoding="application/x-llamapun" id="S2.p4.2.m2.1d">italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT | - italic_S ⟩ = 0</annotation></semantics></math> and <math alttext="U\left|n\right>=\ket{n+1}" class="ltx_Math" display="inline" id="S2.p4.3.m3.2"><semantics id="S2.p4.3.m3.2a"><mrow id="S2.p4.3.m3.2.3" xref="S2.p4.3.m3.2.3.cmml"><mrow id="S2.p4.3.m3.2.3.2" xref="S2.p4.3.m3.2.3.2.cmml"><mi id="S2.p4.3.m3.2.3.2.2" xref="S2.p4.3.m3.2.3.2.2.cmml">U</mi><mo id="S2.p4.3.m3.2.3.2.1" xref="S2.p4.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S2.p4.3.m3.2.3.2.3.2" xref="S2.p4.3.m3.2.3.2.3.1.cmml"><mo id="S2.p4.3.m3.2.3.2.3.2.1" xref="S2.p4.3.m3.2.3.2.3.1.1.cmml">|</mo><mi id="S2.p4.3.m3.2.2" xref="S2.p4.3.m3.2.2.cmml">n</mi><mo id="S2.p4.3.m3.2.3.2.3.2.2" xref="S2.p4.3.m3.2.3.2.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S2.p4.3.m3.2.3.1" xref="S2.p4.3.m3.2.3.1.cmml">=</mo><mrow id="S2.p4.3.m3.1.1.3" xref="S2.p4.3.m3.1.1.2.cmml"><mo id="S2.p4.3.m3.1.1.3.1" stretchy="false" xref="S2.p4.3.m3.1.1.2.1.cmml">|</mo><mrow id="S2.p4.3.m3.1.1.1.1" xref="S2.p4.3.m3.1.1.1.1.cmml"><mi id="S2.p4.3.m3.1.1.1.1.2" xref="S2.p4.3.m3.1.1.1.1.2.cmml">n</mi><mo id="S2.p4.3.m3.1.1.1.1.1" xref="S2.p4.3.m3.1.1.1.1.1.cmml">+</mo><mn id="S2.p4.3.m3.1.1.1.1.3" xref="S2.p4.3.m3.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.p4.3.m3.1.1.3.2" stretchy="false" xref="S2.p4.3.m3.1.1.2.1.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.3.m3.2b"><apply id="S2.p4.3.m3.2.3.cmml" xref="S2.p4.3.m3.2.3"><eq id="S2.p4.3.m3.2.3.1.cmml" xref="S2.p4.3.m3.2.3.1"></eq><apply id="S2.p4.3.m3.2.3.2.cmml" xref="S2.p4.3.m3.2.3.2"><times id="S2.p4.3.m3.2.3.2.1.cmml" xref="S2.p4.3.m3.2.3.2.1"></times><ci id="S2.p4.3.m3.2.3.2.2.cmml" xref="S2.p4.3.m3.2.3.2.2">𝑈</ci><apply id="S2.p4.3.m3.2.3.2.3.1.cmml" xref="S2.p4.3.m3.2.3.2.3.2"><csymbol cd="latexml" id="S2.p4.3.m3.2.3.2.3.1.1.cmml" xref="S2.p4.3.m3.2.3.2.3.2.1">ket</csymbol><ci id="S2.p4.3.m3.2.2.cmml" xref="S2.p4.3.m3.2.2">𝑛</ci></apply></apply><apply id="S2.p4.3.m3.1.1.2.cmml" xref="S2.p4.3.m3.1.1.3"><csymbol cd="latexml" id="S2.p4.3.m3.1.1.2.1.cmml" xref="S2.p4.3.m3.1.1.3.1">ket</csymbol><apply id="S2.p4.3.m3.1.1.1.1.cmml" xref="S2.p4.3.m3.1.1.1.1"><plus id="S2.p4.3.m3.1.1.1.1.1.cmml" xref="S2.p4.3.m3.1.1.1.1.1"></plus><ci id="S2.p4.3.m3.1.1.1.1.2.cmml" xref="S2.p4.3.m3.1.1.1.1.2">𝑛</ci><cn id="S2.p4.3.m3.1.1.1.1.3.cmml" type="integer" xref="S2.p4.3.m3.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.3.m3.2c">U\left|n\right>=\ket{n+1}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.3.m3.2d">italic_U | italic_n ⟩ = | start_ARG italic_n + 1 end_ARG ⟩</annotation></semantics></math> otherwise. <math alttext="U_{ni}" class="ltx_Math" display="inline" id="S2.p4.4.m4.1"><semantics id="S2.p4.4.m4.1a"><msub id="S2.p4.4.m4.1.1" xref="S2.p4.4.m4.1.1.cmml"><mi id="S2.p4.4.m4.1.1.2" xref="S2.p4.4.m4.1.1.2.cmml">U</mi><mrow id="S2.p4.4.m4.1.1.3" xref="S2.p4.4.m4.1.1.3.cmml"><mi id="S2.p4.4.m4.1.1.3.2" xref="S2.p4.4.m4.1.1.3.2.cmml">n</mi><mo id="S2.p4.4.m4.1.1.3.1" xref="S2.p4.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S2.p4.4.m4.1.1.3.3" xref="S2.p4.4.m4.1.1.3.3.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.p4.4.m4.1b"><apply id="S2.p4.4.m4.1.1.cmml" xref="S2.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S2.p4.4.m4.1.1.1.cmml" xref="S2.p4.4.m4.1.1">subscript</csymbol><ci id="S2.p4.4.m4.1.1.2.cmml" xref="S2.p4.4.m4.1.1.2">𝑈</ci><apply id="S2.p4.4.m4.1.1.3.cmml" xref="S2.p4.4.m4.1.1.3"><times id="S2.p4.4.m4.1.1.3.1.cmml" xref="S2.p4.4.m4.1.1.3.1"></times><ci id="S2.p4.4.m4.1.1.3.2.cmml" xref="S2.p4.4.m4.1.1.3.2">𝑛</ci><ci id="S2.p4.4.m4.1.1.3.3.cmml" xref="S2.p4.4.m4.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.4.m4.1c">U_{ni}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.4.m4.1d">italic_U start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="U^{\dagger}_{ni}" class="ltx_Math" display="inline" id="S2.p4.5.m5.1"><semantics id="S2.p4.5.m5.1a"><msubsup id="S2.p4.5.m5.1.1" xref="S2.p4.5.m5.1.1.cmml"><mi id="S2.p4.5.m5.1.1.2.2" xref="S2.p4.5.m5.1.1.2.2.cmml">U</mi><mrow id="S2.p4.5.m5.1.1.3" xref="S2.p4.5.m5.1.1.3.cmml"><mi id="S2.p4.5.m5.1.1.3.2" xref="S2.p4.5.m5.1.1.3.2.cmml">n</mi><mo id="S2.p4.5.m5.1.1.3.1" xref="S2.p4.5.m5.1.1.3.1.cmml">⁢</mo><mi id="S2.p4.5.m5.1.1.3.3" xref="S2.p4.5.m5.1.1.3.3.cmml">i</mi></mrow><mo id="S2.p4.5.m5.1.1.2.3" xref="S2.p4.5.m5.1.1.2.3.cmml">†</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.p4.5.m5.1b"><apply id="S2.p4.5.m5.1.1.cmml" xref="S2.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S2.p4.5.m5.1.1.1.cmml" xref="S2.p4.5.m5.1.1">subscript</csymbol><apply id="S2.p4.5.m5.1.1.2.cmml" xref="S2.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S2.p4.5.m5.1.1.2.1.cmml" xref="S2.p4.5.m5.1.1">superscript</csymbol><ci id="S2.p4.5.m5.1.1.2.2.cmml" xref="S2.p4.5.m5.1.1.2.2">𝑈</ci><ci id="S2.p4.5.m5.1.1.2.3.cmml" xref="S2.p4.5.m5.1.1.2.3">†</ci></apply><apply id="S2.p4.5.m5.1.1.3.cmml" xref="S2.p4.5.m5.1.1.3"><times id="S2.p4.5.m5.1.1.3.1.cmml" xref="S2.p4.5.m5.1.1.3.1"></times><ci id="S2.p4.5.m5.1.1.3.2.cmml" xref="S2.p4.5.m5.1.1.3.2">𝑛</ci><ci id="S2.p4.5.m5.1.1.3.3.cmml" xref="S2.p4.5.m5.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.5.m5.1c">U^{\dagger}_{ni}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.5.m5.1d">italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT</annotation></semantics></math> still do not commute, though the non-zero commutation relation is removed to the edge of the local Hilbert space (to the states <math alttext="\ket{-S}" class="ltx_Math" display="inline" id="S2.p4.6.m6.1"><semantics id="S2.p4.6.m6.1a"><mrow id="S2.p4.6.m6.1.1.3" xref="S2.p4.6.m6.1.1.2.cmml"><mo id="S2.p4.6.m6.1.1.3.1" stretchy="false" xref="S2.p4.6.m6.1.1.2.1.cmml">|</mo><mrow id="S2.p4.6.m6.1.1.1.1" xref="S2.p4.6.m6.1.1.1.1.cmml"><mo id="S2.p4.6.m6.1.1.1.1a" xref="S2.p4.6.m6.1.1.1.1.cmml">−</mo><mi id="S2.p4.6.m6.1.1.1.1.2" xref="S2.p4.6.m6.1.1.1.1.2.cmml">S</mi></mrow><mo id="S2.p4.6.m6.1.1.3.2" stretchy="false" xref="S2.p4.6.m6.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.6.m6.1b"><apply id="S2.p4.6.m6.1.1.2.cmml" xref="S2.p4.6.m6.1.1.3"><csymbol cd="latexml" id="S2.p4.6.m6.1.1.2.1.cmml" xref="S2.p4.6.m6.1.1.3.1">ket</csymbol><apply id="S2.p4.6.m6.1.1.1.1.cmml" xref="S2.p4.6.m6.1.1.1.1"><minus id="S2.p4.6.m6.1.1.1.1.1.cmml" xref="S2.p4.6.m6.1.1.1.1"></minus><ci id="S2.p4.6.m6.1.1.1.1.2.cmml" xref="S2.p4.6.m6.1.1.1.1.2">𝑆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.6.m6.1c">\ket{-S}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.6.m6.1d">| start_ARG - italic_S end_ARG ⟩</annotation></semantics></math> and <math alttext="\ket{S}" class="ltx_Math" display="inline" id="S2.p4.7.m7.1"><semantics id="S2.p4.7.m7.1a"><mrow id="S2.p4.7.m7.1.1.3" xref="S2.p4.7.m7.1.1.2.cmml"><mo id="S2.p4.7.m7.1.1.3.1" stretchy="false" xref="S2.p4.7.m7.1.1.2.1.cmml">|</mo><mi id="S2.p4.7.m7.1.1.1.1" xref="S2.p4.7.m7.1.1.1.1.cmml">S</mi><mo id="S2.p4.7.m7.1.1.3.2" stretchy="false" xref="S2.p4.7.m7.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.7.m7.1b"><apply id="S2.p4.7.m7.1.1.2.cmml" xref="S2.p4.7.m7.1.1.3"><csymbol cd="latexml" id="S2.p4.7.m7.1.1.2.1.cmml" xref="S2.p4.7.m7.1.1.3.1">ket</csymbol><ci id="S2.p4.7.m7.1.1.1.1.cmml" xref="S2.p4.7.m7.1.1.1.1">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.7.m7.1c">\ket{S}</annotation><annotation encoding="application/x-llamapun" id="S2.p4.7.m7.1d">| start_ARG italic_S end_ARG ⟩</annotation></semantics></math>). With this construction, there will be <math alttext="2S+1" class="ltx_Math" display="inline" id="S2.p4.8.m8.1"><semantics id="S2.p4.8.m8.1a"><mrow id="S2.p4.8.m8.1.1" xref="S2.p4.8.m8.1.1.cmml"><mrow id="S2.p4.8.m8.1.1.2" xref="S2.p4.8.m8.1.1.2.cmml"><mn id="S2.p4.8.m8.1.1.2.2" xref="S2.p4.8.m8.1.1.2.2.cmml">2</mn><mo id="S2.p4.8.m8.1.1.2.1" xref="S2.p4.8.m8.1.1.2.1.cmml">⁢</mo><mi id="S2.p4.8.m8.1.1.2.3" xref="S2.p4.8.m8.1.1.2.3.cmml">S</mi></mrow><mo id="S2.p4.8.m8.1.1.1" xref="S2.p4.8.m8.1.1.1.cmml">+</mo><mn id="S2.p4.8.m8.1.1.3" xref="S2.p4.8.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.8.m8.1b"><apply id="S2.p4.8.m8.1.1.cmml" xref="S2.p4.8.m8.1.1"><plus id="S2.p4.8.m8.1.1.1.cmml" xref="S2.p4.8.m8.1.1.1"></plus><apply id="S2.p4.8.m8.1.1.2.cmml" xref="S2.p4.8.m8.1.1.2"><times id="S2.p4.8.m8.1.1.2.1.cmml" xref="S2.p4.8.m8.1.1.2.1"></times><cn id="S2.p4.8.m8.1.1.2.2.cmml" type="integer" xref="S2.p4.8.m8.1.1.2.2">2</cn><ci id="S2.p4.8.m8.1.1.2.3.cmml" xref="S2.p4.8.m8.1.1.2.3">𝑆</ci></apply><cn id="S2.p4.8.m8.1.1.3.cmml" type="integer" xref="S2.p4.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.8.m8.1c">2S+1</annotation><annotation encoding="application/x-llamapun" id="S2.p4.8.m8.1d">2 italic_S + 1</annotation></semantics></math> states per link. We generically refer to <math alttext="S" class="ltx_Math" display="inline" id="S2.p4.9.m9.1"><semantics id="S2.p4.9.m9.1a"><mi id="S2.p4.9.m9.1.1" xref="S2.p4.9.m9.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p4.9.m9.1b"><ci id="S2.p4.9.m9.1.1.cmml" xref="S2.p4.9.m9.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.9.m9.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p4.9.m9.1d">italic_S</annotation></semantics></math> as the total spin, irrespective of the formulation we are using. The <math alttext="S=1" class="ltx_Math" display="inline" id="S2.p4.10.m10.1"><semantics id="S2.p4.10.m10.1a"><mrow id="S2.p4.10.m10.1.1" xref="S2.p4.10.m10.1.1.cmml"><mi id="S2.p4.10.m10.1.1.2" xref="S2.p4.10.m10.1.1.2.cmml">S</mi><mo id="S2.p4.10.m10.1.1.1" xref="S2.p4.10.m10.1.1.1.cmml">=</mo><mn id="S2.p4.10.m10.1.1.3" xref="S2.p4.10.m10.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.10.m10.1b"><apply id="S2.p4.10.m10.1.1.cmml" xref="S2.p4.10.m10.1.1"><eq id="S2.p4.10.m10.1.1.1.cmml" xref="S2.p4.10.m10.1.1.1"></eq><ci id="S2.p4.10.m10.1.1.2.cmml" xref="S2.p4.10.m10.1.1.2">𝑆</ci><cn id="S2.p4.10.m10.1.1.3.cmml" type="integer" xref="S2.p4.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.10.m10.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="S2.p4.10.m10.1d">italic_S = 1</annotation></semantics></math> TLM is equivalent to the <math alttext="S=1" class="ltx_Math" display="inline" id="S2.p4.11.m11.1"><semantics id="S2.p4.11.m11.1a"><mrow id="S2.p4.11.m11.1.1" xref="S2.p4.11.m11.1.1.cmml"><mi id="S2.p4.11.m11.1.1.2" xref="S2.p4.11.m11.1.1.2.cmml">S</mi><mo id="S2.p4.11.m11.1.1.1" xref="S2.p4.11.m11.1.1.1.cmml">=</mo><mn id="S2.p4.11.m11.1.1.3" xref="S2.p4.11.m11.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p4.11.m11.1b"><apply id="S2.p4.11.m11.1.1.cmml" xref="S2.p4.11.m11.1.1"><eq id="S2.p4.11.m11.1.1.1.cmml" xref="S2.p4.11.m11.1.1.1"></eq><ci id="S2.p4.11.m11.1.1.2.cmml" xref="S2.p4.11.m11.1.1.2">𝑆</ci><cn id="S2.p4.11.m11.1.1.3.cmml" type="integer" xref="S2.p4.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p4.11.m11.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="S2.p4.11.m11.1d">italic_S = 1</annotation></semantics></math> QLM, up to normalization of Hamiltonian parameters.</p> </div> <div class="ltx_para" id="S2.p5"> <p class="ltx_p" id="S2.p5.7">We will also write the Hamiltonian (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S2.E1" title="In II Link models in 2+1D ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">1</span></a>) as</p> <table class="ltx_equation ltx_eqn_table" id="S2.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H=K+V,\quad K=H^{+}+H^{-},\quad H^{\pm}=\sum_{n}H^{\pm}_{n}," class="ltx_Math" display="block" id="S2.E3.m1.1"><semantics id="S2.E3.m1.1a"><mrow id="S2.E3.m1.1.1.1"><mrow id="S2.E3.m1.1.1.1.1.2" xref="S2.E3.m1.1.1.1.1.3.cmml"><mrow id="S2.E3.m1.1.1.1.1.1.1" xref="S2.E3.m1.1.1.1.1.1.1.cmml"><mi id="S2.E3.m1.1.1.1.1.1.1.2" xref="S2.E3.m1.1.1.1.1.1.1.2.cmml">H</mi><mo id="S2.E3.m1.1.1.1.1.1.1.1" xref="S2.E3.m1.1.1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E3.m1.1.1.1.1.1.1.3" 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xref="S2.E3.m1.1.1.1.1.2.2.1.1.3.2.3.cmml">+</mo></msup><mo id="S2.E3.m1.1.1.1.1.2.2.1.1.3.1" xref="S2.E3.m1.1.1.1.1.2.2.1.1.3.1.cmml">+</mo><msup id="S2.E3.m1.1.1.1.1.2.2.1.1.3.3" xref="S2.E3.m1.1.1.1.1.2.2.1.1.3.3.cmml"><mi id="S2.E3.m1.1.1.1.1.2.2.1.1.3.3.2" xref="S2.E3.m1.1.1.1.1.2.2.1.1.3.3.2.cmml">H</mi><mo id="S2.E3.m1.1.1.1.1.2.2.1.1.3.3.3" xref="S2.E3.m1.1.1.1.1.2.2.1.1.3.3.3.cmml">−</mo></msup></mrow></mrow><mo id="S2.E3.m1.1.1.1.1.2.2.2.3" rspace="1.167em" xref="S2.E3.m1.1.1.1.1.2.2.3a.cmml">,</mo><mrow id="S2.E3.m1.1.1.1.1.2.2.2.2" xref="S2.E3.m1.1.1.1.1.2.2.2.2.cmml"><msup id="S2.E3.m1.1.1.1.1.2.2.2.2.2" xref="S2.E3.m1.1.1.1.1.2.2.2.2.2.cmml"><mi id="S2.E3.m1.1.1.1.1.2.2.2.2.2.2" xref="S2.E3.m1.1.1.1.1.2.2.2.2.2.2.cmml">H</mi><mo id="S2.E3.m1.1.1.1.1.2.2.2.2.2.3" xref="S2.E3.m1.1.1.1.1.2.2.2.2.2.3.cmml">±</mo></msup><mo id="S2.E3.m1.1.1.1.1.2.2.2.2.1" rspace="0.111em" xref="S2.E3.m1.1.1.1.1.2.2.2.2.1.cmml">=</mo><mrow id="S2.E3.m1.1.1.1.1.2.2.2.2.3" 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xref="S2.E3.m1.1.1.1.1.2.2.2.2.3.1.2"></sum><ci id="S2.E3.m1.1.1.1.1.2.2.2.2.3.1.3.cmml" xref="S2.E3.m1.1.1.1.1.2.2.2.2.3.1.3">𝑛</ci></apply><apply id="S2.E3.m1.1.1.1.1.2.2.2.2.3.2.cmml" xref="S2.E3.m1.1.1.1.1.2.2.2.2.3.2"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.1.1.2.2.2.2.3.2.1.cmml" xref="S2.E3.m1.1.1.1.1.2.2.2.2.3.2">subscript</csymbol><apply id="S2.E3.m1.1.1.1.1.2.2.2.2.3.2.2.cmml" xref="S2.E3.m1.1.1.1.1.2.2.2.2.3.2"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.1.1.2.2.2.2.3.2.2.1.cmml" xref="S2.E3.m1.1.1.1.1.2.2.2.2.3.2">superscript</csymbol><ci id="S2.E3.m1.1.1.1.1.2.2.2.2.3.2.2.2.cmml" xref="S2.E3.m1.1.1.1.1.2.2.2.2.3.2.2.2">𝐻</ci><csymbol cd="latexml" id="S2.E3.m1.1.1.1.1.2.2.2.2.3.2.2.3.cmml" xref="S2.E3.m1.1.1.1.1.2.2.2.2.3.2.2.3">plus-or-minus</csymbol></apply><ci id="S2.E3.m1.1.1.1.1.2.2.2.2.3.2.3.cmml" xref="S2.E3.m1.1.1.1.1.2.2.2.2.3.2.3">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.1c">H=K+V,\quad K=H^{+}+H^{-},\quad H^{\pm}=\sum_{n}H^{\pm}_{n},</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.1d">italic_H = italic_K + italic_V , italic_K = italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT + italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p5.6">where <math alttext="H^{-}_{n}=U_{n1}^{\dagger}U_{n+\hat{1}2}^{\dagger}U_{n2}U_{n+\hat{2}1}" class="ltx_Math" display="inline" id="S2.p5.1.m1.1"><semantics id="S2.p5.1.m1.1a"><mrow id="S2.p5.1.m1.1.1" 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id="S2.p5.1.m1.1.1.3.5.cmml" xref="S2.p5.1.m1.1.1.3.5"><csymbol cd="ambiguous" id="S2.p5.1.m1.1.1.3.5.1.cmml" xref="S2.p5.1.m1.1.1.3.5">subscript</csymbol><ci id="S2.p5.1.m1.1.1.3.5.2.cmml" xref="S2.p5.1.m1.1.1.3.5.2">𝑈</ci><apply id="S2.p5.1.m1.1.1.3.5.3.cmml" xref="S2.p5.1.m1.1.1.3.5.3"><plus id="S2.p5.1.m1.1.1.3.5.3.1.cmml" xref="S2.p5.1.m1.1.1.3.5.3.1"></plus><ci id="S2.p5.1.m1.1.1.3.5.3.2.cmml" xref="S2.p5.1.m1.1.1.3.5.3.2">𝑛</ci><apply id="S2.p5.1.m1.1.1.3.5.3.3.cmml" xref="S2.p5.1.m1.1.1.3.5.3.3"><times id="S2.p5.1.m1.1.1.3.5.3.3.1.cmml" xref="S2.p5.1.m1.1.1.3.5.3.3.1"></times><apply id="S2.p5.1.m1.1.1.3.5.3.3.2.cmml" xref="S2.p5.1.m1.1.1.3.5.3.3.2"><ci id="S2.p5.1.m1.1.1.3.5.3.3.2.1.cmml" xref="S2.p5.1.m1.1.1.3.5.3.3.2.1">^</ci><cn id="S2.p5.1.m1.1.1.3.5.3.3.2.2.cmml" type="integer" xref="S2.p5.1.m1.1.1.3.5.3.3.2.2">2</cn></apply><cn id="S2.p5.1.m1.1.1.3.5.3.3.3.cmml" type="integer" xref="S2.p5.1.m1.1.1.3.5.3.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.1.m1.1c">H^{-}_{n}=U_{n1}^{\dagger}U_{n+\hat{1}2}^{\dagger}U_{n2}U_{n+\hat{2}1}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.1.m1.1d">italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_U start_POSTSUBSCRIPT italic_n 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_U start_POSTSUBSCRIPT italic_n + over^ start_ARG 1 end_ARG 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_U start_POSTSUBSCRIPT italic_n 2 end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_n + over^ start_ARG 2 end_ARG 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="H^{+}=\left(H^{-}\right)^{\dagger}" class="ltx_Math" display="inline" id="S2.p5.2.m2.1"><semantics id="S2.p5.2.m2.1a"><mrow id="S2.p5.2.m2.1.1" xref="S2.p5.2.m2.1.1.cmml"><msup id="S2.p5.2.m2.1.1.3" xref="S2.p5.2.m2.1.1.3.cmml"><mi id="S2.p5.2.m2.1.1.3.2" xref="S2.p5.2.m2.1.1.3.2.cmml">H</mi><mo id="S2.p5.2.m2.1.1.3.3" xref="S2.p5.2.m2.1.1.3.3.cmml">+</mo></msup><mo id="S2.p5.2.m2.1.1.2" xref="S2.p5.2.m2.1.1.2.cmml">=</mo><msup id="S2.p5.2.m2.1.1.1" xref="S2.p5.2.m2.1.1.1.cmml"><mrow id="S2.p5.2.m2.1.1.1.1.1" xref="S2.p5.2.m2.1.1.1.1.1.1.cmml"><mo id="S2.p5.2.m2.1.1.1.1.1.2" xref="S2.p5.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S2.p5.2.m2.1.1.1.1.1.1" xref="S2.p5.2.m2.1.1.1.1.1.1.cmml"><mi id="S2.p5.2.m2.1.1.1.1.1.1.2" xref="S2.p5.2.m2.1.1.1.1.1.1.2.cmml">H</mi><mo id="S2.p5.2.m2.1.1.1.1.1.1.3" xref="S2.p5.2.m2.1.1.1.1.1.1.3.cmml">−</mo></msup><mo id="S2.p5.2.m2.1.1.1.1.1.3" xref="S2.p5.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.p5.2.m2.1.1.1.3" xref="S2.p5.2.m2.1.1.1.3.cmml">†</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.2.m2.1b"><apply id="S2.p5.2.m2.1.1.cmml" xref="S2.p5.2.m2.1.1"><eq 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id="S2.p5.2.m2.1c">H^{+}=\left(H^{-}\right)^{\dagger}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.2.m2.1d">italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT = ( italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT</annotation></semantics></math> are <em class="ltx_emph ltx_font_italic" id="S2.p5.6.1">kinetic operators</em>. This notation follows the convention that we will adopt throughout the paper. “Plaquette at site <math alttext="n" class="ltx_Math" display="inline" id="S2.p5.3.m3.1"><semantics id="S2.p5.3.m3.1a"><mi id="S2.p5.3.m3.1.1" xref="S2.p5.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.p5.3.m3.1b"><ci id="S2.p5.3.m3.1.1.cmml" xref="S2.p5.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.p5.3.m3.1d">italic_n</annotation></semantics></math>” refers to the plaquette for which <math alttext="n" class="ltx_Math" display="inline" id="S2.p5.4.m4.1"><semantics id="S2.p5.4.m4.1a"><mi id="S2.p5.4.m4.1.1" xref="S2.p5.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.p5.4.m4.1b"><ci id="S2.p5.4.m4.1.1.cmml" xref="S2.p5.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.p5.4.m4.1d">italic_n</annotation></semantics></math> is its lower left vertex. In order to simplify the formulas we will also adopt the convention <math alttext="\left(H^{+}\right)^{-m}\equiv\left(H^{-}\right)^{m}" class="ltx_Math" display="inline" id="S2.p5.5.m5.2"><semantics id="S2.p5.5.m5.2a"><mrow id="S2.p5.5.m5.2.2" xref="S2.p5.5.m5.2.2.cmml"><msup id="S2.p5.5.m5.1.1.1" xref="S2.p5.5.m5.1.1.1.cmml"><mrow id="S2.p5.5.m5.1.1.1.1.1" xref="S2.p5.5.m5.1.1.1.1.1.1.cmml"><mo id="S2.p5.5.m5.1.1.1.1.1.2" xref="S2.p5.5.m5.1.1.1.1.1.1.cmml">(</mo><msup id="S2.p5.5.m5.1.1.1.1.1.1" xref="S2.p5.5.m5.1.1.1.1.1.1.cmml"><mi id="S2.p5.5.m5.1.1.1.1.1.1.2" xref="S2.p5.5.m5.1.1.1.1.1.1.2.cmml">H</mi><mo id="S2.p5.5.m5.1.1.1.1.1.1.3" xref="S2.p5.5.m5.1.1.1.1.1.1.3.cmml">+</mo></msup><mo id="S2.p5.5.m5.1.1.1.1.1.3" xref="S2.p5.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.p5.5.m5.1.1.1.3" xref="S2.p5.5.m5.1.1.1.3.cmml"><mo id="S2.p5.5.m5.1.1.1.3a" xref="S2.p5.5.m5.1.1.1.3.cmml">−</mo><mi id="S2.p5.5.m5.1.1.1.3.2" xref="S2.p5.5.m5.1.1.1.3.2.cmml">m</mi></mrow></msup><mo id="S2.p5.5.m5.2.2.3" xref="S2.p5.5.m5.2.2.3.cmml">≡</mo><msup id="S2.p5.5.m5.2.2.2" xref="S2.p5.5.m5.2.2.2.cmml"><mrow id="S2.p5.5.m5.2.2.2.1.1" xref="S2.p5.5.m5.2.2.2.1.1.1.cmml"><mo id="S2.p5.5.m5.2.2.2.1.1.2" xref="S2.p5.5.m5.2.2.2.1.1.1.cmml">(</mo><msup id="S2.p5.5.m5.2.2.2.1.1.1" xref="S2.p5.5.m5.2.2.2.1.1.1.cmml"><mi id="S2.p5.5.m5.2.2.2.1.1.1.2" xref="S2.p5.5.m5.2.2.2.1.1.1.2.cmml">H</mi><mo id="S2.p5.5.m5.2.2.2.1.1.1.3" xref="S2.p5.5.m5.2.2.2.1.1.1.3.cmml">−</mo></msup><mo id="S2.p5.5.m5.2.2.2.1.1.3" xref="S2.p5.5.m5.2.2.2.1.1.1.cmml">)</mo></mrow><mi id="S2.p5.5.m5.2.2.2.3" xref="S2.p5.5.m5.2.2.2.3.cmml">m</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.5.m5.2b"><apply id="S2.p5.5.m5.2.2.cmml" xref="S2.p5.5.m5.2.2"><equivalent id="S2.p5.5.m5.2.2.3.cmml" xref="S2.p5.5.m5.2.2.3"></equivalent><apply id="S2.p5.5.m5.1.1.1.cmml" xref="S2.p5.5.m5.1.1.1"><csymbol cd="ambiguous" id="S2.p5.5.m5.1.1.1.2.cmml" xref="S2.p5.5.m5.1.1.1">superscript</csymbol><apply id="S2.p5.5.m5.1.1.1.1.1.1.cmml" xref="S2.p5.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.p5.5.m5.1.1.1.1.1.1.1.cmml" xref="S2.p5.5.m5.1.1.1.1.1">superscript</csymbol><ci id="S2.p5.5.m5.1.1.1.1.1.1.2.cmml" xref="S2.p5.5.m5.1.1.1.1.1.1.2">𝐻</ci><plus id="S2.p5.5.m5.1.1.1.1.1.1.3.cmml" xref="S2.p5.5.m5.1.1.1.1.1.1.3"></plus></apply><apply id="S2.p5.5.m5.1.1.1.3.cmml" xref="S2.p5.5.m5.1.1.1.3"><minus id="S2.p5.5.m5.1.1.1.3.1.cmml" xref="S2.p5.5.m5.1.1.1.3"></minus><ci id="S2.p5.5.m5.1.1.1.3.2.cmml" xref="S2.p5.5.m5.1.1.1.3.2">𝑚</ci></apply></apply><apply id="S2.p5.5.m5.2.2.2.cmml" xref="S2.p5.5.m5.2.2.2"><csymbol cd="ambiguous" id="S2.p5.5.m5.2.2.2.2.cmml" xref="S2.p5.5.m5.2.2.2">superscript</csymbol><apply id="S2.p5.5.m5.2.2.2.1.1.1.cmml" xref="S2.p5.5.m5.2.2.2.1.1"><csymbol cd="ambiguous" id="S2.p5.5.m5.2.2.2.1.1.1.1.cmml" xref="S2.p5.5.m5.2.2.2.1.1">superscript</csymbol><ci id="S2.p5.5.m5.2.2.2.1.1.1.2.cmml" xref="S2.p5.5.m5.2.2.2.1.1.1.2">𝐻</ci><minus id="S2.p5.5.m5.2.2.2.1.1.1.3.cmml" xref="S2.p5.5.m5.2.2.2.1.1.1.3"></minus></apply><ci id="S2.p5.5.m5.2.2.2.3.cmml" xref="S2.p5.5.m5.2.2.2.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.5.m5.2c">\left(H^{+}\right)^{-m}\equiv\left(H^{-}\right)^{m}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.5.m5.2d">( italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - italic_m end_POSTSUPERSCRIPT ≡ ( italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT</annotation></semantics></math>. This should not be confused with the inverse of <math alttext="\left(H^{+}\right)^{m}" class="ltx_Math" display="inline" id="S2.p5.6.m6.1"><semantics id="S2.p5.6.m6.1a"><msup id="S2.p5.6.m6.1.1" xref="S2.p5.6.m6.1.1.cmml"><mrow id="S2.p5.6.m6.1.1.1.1" xref="S2.p5.6.m6.1.1.1.1.1.cmml"><mo id="S2.p5.6.m6.1.1.1.1.2" xref="S2.p5.6.m6.1.1.1.1.1.cmml">(</mo><msup id="S2.p5.6.m6.1.1.1.1.1" xref="S2.p5.6.m6.1.1.1.1.1.cmml"><mi id="S2.p5.6.m6.1.1.1.1.1.2" xref="S2.p5.6.m6.1.1.1.1.1.2.cmml">H</mi><mo id="S2.p5.6.m6.1.1.1.1.1.3" xref="S2.p5.6.m6.1.1.1.1.1.3.cmml">+</mo></msup><mo id="S2.p5.6.m6.1.1.1.1.3" xref="S2.p5.6.m6.1.1.1.1.1.cmml">)</mo></mrow><mi id="S2.p5.6.m6.1.1.3" xref="S2.p5.6.m6.1.1.3.cmml">m</mi></msup><annotation-xml encoding="MathML-Content" id="S2.p5.6.m6.1b"><apply id="S2.p5.6.m6.1.1.cmml" xref="S2.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S2.p5.6.m6.1.1.2.cmml" xref="S2.p5.6.m6.1.1">superscript</csymbol><apply id="S2.p5.6.m6.1.1.1.1.1.cmml" xref="S2.p5.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S2.p5.6.m6.1.1.1.1.1.1.cmml" xref="S2.p5.6.m6.1.1.1.1">superscript</csymbol><ci id="S2.p5.6.m6.1.1.1.1.1.2.cmml" xref="S2.p5.6.m6.1.1.1.1.1.2">𝐻</ci><plus id="S2.p5.6.m6.1.1.1.1.1.3.cmml" xref="S2.p5.6.m6.1.1.1.1.1.3"></plus></apply><ci id="S2.p5.6.m6.1.1.3.cmml" xref="S2.p5.6.m6.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.6.m6.1c">\left(H^{+}\right)^{m}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.6.m6.1d">( italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT</annotation></semantics></math>, since these operators are not invertible.</p> </div> <div class="ltx_para" id="S2.p6"> <p class="ltx_p" id="S2.p6.2">To demonstrate the existence of scars in these models, one should show that they are non-integrable. It is expected that this is the case for pure gauge theories in <math alttext="2+1" class="ltx_Math" display="inline" id="S2.p6.1.m1.1"><semantics id="S2.p6.1.m1.1a"><mrow id="S2.p6.1.m1.1.1" xref="S2.p6.1.m1.1.1.cmml"><mn id="S2.p6.1.m1.1.1.2" xref="S2.p6.1.m1.1.1.2.cmml">2</mn><mo id="S2.p6.1.m1.1.1.1" xref="S2.p6.1.m1.1.1.1.cmml">+</mo><mn id="S2.p6.1.m1.1.1.3" xref="S2.p6.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.1.m1.1b"><apply id="S2.p6.1.m1.1.1.cmml" xref="S2.p6.1.m1.1.1"><plus id="S2.p6.1.m1.1.1.1.cmml" xref="S2.p6.1.m1.1.1.1"></plus><cn id="S2.p6.1.m1.1.1.2.cmml" type="integer" xref="S2.p6.1.m1.1.1.2">2</cn><cn id="S2.p6.1.m1.1.1.3.cmml" type="integer" xref="S2.p6.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.1.m1.1c">2+1</annotation><annotation encoding="application/x-llamapun" id="S2.p6.1.m1.1d">2 + 1</annotation></semantics></math>D, and we will further demonstrate it here for a single-leg ladder with <math alttext="S=1" class="ltx_Math" display="inline" id="S2.p6.2.m2.1"><semantics id="S2.p6.2.m2.1a"><mrow id="S2.p6.2.m2.1.1" xref="S2.p6.2.m2.1.1.cmml"><mi id="S2.p6.2.m2.1.1.2" xref="S2.p6.2.m2.1.1.2.cmml">S</mi><mo id="S2.p6.2.m2.1.1.1" xref="S2.p6.2.m2.1.1.1.cmml">=</mo><mn id="S2.p6.2.m2.1.1.3" xref="S2.p6.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.2.m2.1b"><apply id="S2.p6.2.m2.1.1.cmml" xref="S2.p6.2.m2.1.1"><eq id="S2.p6.2.m2.1.1.1.cmml" xref="S2.p6.2.m2.1.1.1"></eq><ci id="S2.p6.2.m2.1.1.2.cmml" xref="S2.p6.2.m2.1.1.2">𝑆</ci><cn id="S2.p6.2.m2.1.1.3.cmml" type="integer" xref="S2.p6.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.2.m2.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="S2.p6.2.m2.1d">italic_S = 1</annotation></semantics></math>.</p> </div> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">II.1 </span>Non-Integrability of the Spin-1 Ladder</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.2">We consider a single row of plaquettes with spin <math alttext="S=1" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.1"><semantics id="S2.SS1.p1.1.m1.1a"><mrow id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml"><mi id="S2.SS1.p1.1.m1.1.1.2" xref="S2.SS1.p1.1.m1.1.1.2.cmml">S</mi><mo id="S2.SS1.p1.1.m1.1.1.1" xref="S2.SS1.p1.1.m1.1.1.1.cmml">=</mo><mn id="S2.SS1.p1.1.m1.1.1.3" xref="S2.SS1.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><apply id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1"><eq id="S2.SS1.p1.1.m1.1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1.1"></eq><ci id="S2.SS1.p1.1.m1.1.1.2.cmml" xref="S2.SS1.p1.1.m1.1.1.2">𝑆</ci><cn id="S2.SS1.p1.1.m1.1.1.3.cmml" type="integer" xref="S2.SS1.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">italic_S = 1</annotation></semantics></math> links, open boundary conditions in both directions and the potential <math alttext="V=\lambda\sum_{n_{1}=0}E_{\left(n_{1},1\right)1}" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.2"><semantics id="S2.SS1.p1.2.m2.2a"><mrow id="S2.SS1.p1.2.m2.2.3" xref="S2.SS1.p1.2.m2.2.3.cmml"><mi id="S2.SS1.p1.2.m2.2.3.2" xref="S2.SS1.p1.2.m2.2.3.2.cmml">V</mi><mo id="S2.SS1.p1.2.m2.2.3.1" xref="S2.SS1.p1.2.m2.2.3.1.cmml">=</mo><mrow id="S2.SS1.p1.2.m2.2.3.3" xref="S2.SS1.p1.2.m2.2.3.3.cmml"><mi id="S2.SS1.p1.2.m2.2.3.3.2" xref="S2.SS1.p1.2.m2.2.3.3.2.cmml">λ</mi><mo id="S2.SS1.p1.2.m2.2.3.3.1" xref="S2.SS1.p1.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS1.p1.2.m2.2.3.3.3" xref="S2.SS1.p1.2.m2.2.3.3.3.cmml"><msub id="S2.SS1.p1.2.m2.2.3.3.3.1" xref="S2.SS1.p1.2.m2.2.3.3.3.1.cmml"><mo id="S2.SS1.p1.2.m2.2.3.3.3.1.2" xref="S2.SS1.p1.2.m2.2.3.3.3.1.2.cmml">∑</mo><mrow id="S2.SS1.p1.2.m2.2.3.3.3.1.3" xref="S2.SS1.p1.2.m2.2.3.3.3.1.3.cmml"><msub id="S2.SS1.p1.2.m2.2.3.3.3.1.3.2" xref="S2.SS1.p1.2.m2.2.3.3.3.1.3.2.cmml"><mi id="S2.SS1.p1.2.m2.2.3.3.3.1.3.2.2" xref="S2.SS1.p1.2.m2.2.3.3.3.1.3.2.2.cmml">n</mi><mn id="S2.SS1.p1.2.m2.2.3.3.3.1.3.2.3" xref="S2.SS1.p1.2.m2.2.3.3.3.1.3.2.3.cmml">1</mn></msub><mo id="S2.SS1.p1.2.m2.2.3.3.3.1.3.1" xref="S2.SS1.p1.2.m2.2.3.3.3.1.3.1.cmml">=</mo><mn id="S2.SS1.p1.2.m2.2.3.3.3.1.3.3" xref="S2.SS1.p1.2.m2.2.3.3.3.1.3.3.cmml">0</mn></mrow></msub><msub id="S2.SS1.p1.2.m2.2.3.3.3.2" xref="S2.SS1.p1.2.m2.2.3.3.3.2.cmml"><mi id="S2.SS1.p1.2.m2.2.3.3.3.2.2" xref="S2.SS1.p1.2.m2.2.3.3.3.2.2.cmml">E</mi><mrow id="S2.SS1.p1.2.m2.2.2.2" xref="S2.SS1.p1.2.m2.2.2.2.cmml"><mrow id="S2.SS1.p1.2.m2.2.2.2.2.1" xref="S2.SS1.p1.2.m2.2.2.2.2.2.cmml"><mo id="S2.SS1.p1.2.m2.2.2.2.2.1.2" xref="S2.SS1.p1.2.m2.2.2.2.2.2.cmml">(</mo><msub id="S2.SS1.p1.2.m2.2.2.2.2.1.1" 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type="integer" xref="S2.SS1.p1.2.m2.2.2.2.2.1.1.3">1</cn></apply><cn id="S2.SS1.p1.2.m2.1.1.1.1.cmml" type="integer" xref="S2.SS1.p1.2.m2.1.1.1.1">1</cn></interval><cn id="S2.SS1.p1.2.m2.2.2.2.4.cmml" type="integer" xref="S2.SS1.p1.2.m2.2.2.2.4">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m2.2c">V=\lambda\sum_{n_{1}=0}E_{\left(n_{1},1\right)1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m2.2d">italic_V = italic_λ ∑ start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , 1 ) 1 end_POSTSUBSCRIPT</annotation></semantics></math>. We show that his model is non-integrable by studying the statistics of the spectrum between adjacent energy levels. We follow the method introduced in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib61" title="">61</a>]</cite>, which was also applied to QLMs in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib30" title="">30</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib31" title="">31</a>]</cite>. This will be one of the many cases in which we demonstrate the existence of mid-spectrum low entropy states, establishing the presence of quantum many-body scars. While we do not demonstrate the non-integrability of all models considered, the result is expected to extend beyond this case.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.2">We start by resolving the symmetries of the system. Translation symmetry is broken due to the open boundaries. The potential breaks charge conjugation symmetry. Reflection across the horizontal axis, as defined above, is identical to charge conjugation and is therefore also broken by the potential. Reflection symmetry with respect to the vertical axis needs to be resolved. This results in two sectors, labeled with the eigenvalue <math alttext="\pm 1" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><mrow id="S2.SS1.p2.1.m1.1.1" xref="S2.SS1.p2.1.m1.1.1.cmml"><mo id="S2.SS1.p2.1.m1.1.1a" xref="S2.SS1.p2.1.m1.1.1.cmml">±</mo><mn id="S2.SS1.p2.1.m1.1.1.2" xref="S2.SS1.p2.1.m1.1.1.2.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.1b"><apply id="S2.SS1.p2.1.m1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1"><csymbol cd="latexml" id="S2.SS1.p2.1.m1.1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1">plus-or-minus</csymbol><cn id="S2.SS1.p2.1.m1.1.1.2.cmml" type="integer" xref="S2.SS1.p2.1.m1.1.1.2">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.1c">\pm 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.1d">± 1</annotation></semantics></math>. The distribution <math alttext="p\left(r\right)" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.1"><semantics id="S2.SS1.p2.2.m2.1a"><mrow id="S2.SS1.p2.2.m2.1.2" xref="S2.SS1.p2.2.m2.1.2.cmml"><mi id="S2.SS1.p2.2.m2.1.2.2" xref="S2.SS1.p2.2.m2.1.2.2.cmml">p</mi><mo id="S2.SS1.p2.2.m2.1.2.1" xref="S2.SS1.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.2.m2.1.2.3.2" xref="S2.SS1.p2.2.m2.1.2.cmml"><mo id="S2.SS1.p2.2.m2.1.2.3.2.1" xref="S2.SS1.p2.2.m2.1.2.cmml">(</mo><mi id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml">r</mi><mo id="S2.SS1.p2.2.m2.1.2.3.2.2" xref="S2.SS1.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.1b"><apply id="S2.SS1.p2.2.m2.1.2.cmml" xref="S2.SS1.p2.2.m2.1.2"><times id="S2.SS1.p2.2.m2.1.2.1.cmml" xref="S2.SS1.p2.2.m2.1.2.1"></times><ci id="S2.SS1.p2.2.m2.1.2.2.cmml" xref="S2.SS1.p2.2.m2.1.2.2">𝑝</ci><ci id="S2.SS1.p2.2.m2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.1c">p\left(r\right)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.1d">italic_p ( italic_r )</annotation></semantics></math> of consecutive level spacing ratios</p> <table class="ltx_equation ltx_eqn_table" id="S2.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="r_{n}=\min\left\{\frac{E_{n+1}-E_{n}}{E_{n}-E_{n-1}},\frac{E_{n}-E_{n-1}}{E_{n% +1}-E_{n}}\right\}" class="ltx_Math" display="block" id="S2.E4.m1.3"><semantics id="S2.E4.m1.3a"><mrow id="S2.E4.m1.3.4" xref="S2.E4.m1.3.4.cmml"><msub id="S2.E4.m1.3.4.2" xref="S2.E4.m1.3.4.2.cmml"><mi id="S2.E4.m1.3.4.2.2" xref="S2.E4.m1.3.4.2.2.cmml">r</mi><mi id="S2.E4.m1.3.4.2.3" xref="S2.E4.m1.3.4.2.3.cmml">n</mi></msub><mo id="S2.E4.m1.3.4.1" xref="S2.E4.m1.3.4.1.cmml">=</mo><mrow id="S2.E4.m1.3.4.3.2" xref="S2.E4.m1.3.4.3.1.cmml"><mi 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end_POSTSUBSCRIPT end_ARG }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p2.3">of both sectors match the Gaussian Orthogonal Ensemble (GOE)</p> <table class="ltx_equation ltx_eqn_table" id="S2.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="p_{GOE}(r)=\frac{27}{4}\frac{r+r^{2}}{(1+r+r^{2})^{5/2}}" class="ltx_Math" display="block" id="S2.E5.m1.2"><semantics id="S2.E5.m1.2a"><mrow id="S2.E5.m1.2.3" xref="S2.E5.m1.2.3.cmml"><mrow id="S2.E5.m1.2.3.2" xref="S2.E5.m1.2.3.2.cmml"><msub id="S2.E5.m1.2.3.2.2" xref="S2.E5.m1.2.3.2.2.cmml"><mi id="S2.E5.m1.2.3.2.2.2" xref="S2.E5.m1.2.3.2.2.2.cmml">p</mi><mrow 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id="S2.E5.m1.2c">p_{GOE}(r)=\frac{27}{4}\frac{r+r^{2}}{(1+r+r^{2})^{5/2}}</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.2d">italic_p start_POSTSUBSCRIPT italic_G italic_O italic_E end_POSTSUBSCRIPT ( italic_r ) = divide start_ARG 27 end_ARG start_ARG 4 end_ARG divide start_ARG italic_r + italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 + italic_r + italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 5 / 2 end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p2.4">as seen in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S2.F1" title="Figure 1 ‣ II.1 Non-Integrability of the Spin-1 Ladder ‣ II Link models in 2+1D ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">1</span></a>. This is in contrast to the Poisson distribution expected for integrable models</p> <table class="ltx_equation ltx_eqn_table" id="S2.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="p_{P}(r)=\frac{2}{(1+r)^{2}}," class="ltx_Math" display="block" id="S2.E6.m1.3"><semantics id="S2.E6.m1.3a"><mrow id="S2.E6.m1.3.3.1" xref="S2.E6.m1.3.3.1.1.cmml"><mrow id="S2.E6.m1.3.3.1.1" xref="S2.E6.m1.3.3.1.1.cmml"><mrow id="S2.E6.m1.3.3.1.1.2" xref="S2.E6.m1.3.3.1.1.2.cmml"><msub id="S2.E6.m1.3.3.1.1.2.2" xref="S2.E6.m1.3.3.1.1.2.2.cmml"><mi id="S2.E6.m1.3.3.1.1.2.2.2" xref="S2.E6.m1.3.3.1.1.2.2.2.cmml">p</mi><mi id="S2.E6.m1.3.3.1.1.2.2.3" xref="S2.E6.m1.3.3.1.1.2.2.3.cmml">P</mi></msub><mo id="S2.E6.m1.3.3.1.1.2.1" xref="S2.E6.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S2.E6.m1.3.3.1.1.2.3.2" xref="S2.E6.m1.3.3.1.1.2.cmml"><mo id="S2.E6.m1.3.3.1.1.2.3.2.1" stretchy="false" xref="S2.E6.m1.3.3.1.1.2.cmml">(</mo><mi id="S2.E6.m1.2.2" xref="S2.E6.m1.2.2.cmml">r</mi><mo id="S2.E6.m1.3.3.1.1.2.3.2.2" stretchy="false" xref="S2.E6.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E6.m1.3.3.1.1.1" xref="S2.E6.m1.3.3.1.1.1.cmml">=</mo><mfrac id="S2.E6.m1.1.1" xref="S2.E6.m1.1.1.cmml"><mn id="S2.E6.m1.1.1.3" xref="S2.E6.m1.1.1.3.cmml">2</mn><msup id="S2.E6.m1.1.1.1" xref="S2.E6.m1.1.1.1.cmml"><mrow id="S2.E6.m1.1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.1.1.cmml"><mo id="S2.E6.m1.1.1.1.1.1.2" stretchy="false" xref="S2.E6.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E6.m1.1.1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.1.1.cmml"><mn id="S2.E6.m1.1.1.1.1.1.1.2" xref="S2.E6.m1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S2.E6.m1.1.1.1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S2.E6.m1.1.1.1.1.1.1.3" xref="S2.E6.m1.1.1.1.1.1.1.3.cmml">r</mi></mrow><mo id="S2.E6.m1.1.1.1.1.1.3" stretchy="false" xref="S2.E6.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S2.E6.m1.1.1.1.3" xref="S2.E6.m1.1.1.1.3.cmml">2</mn></msup></mfrac></mrow><mo id="S2.E6.m1.3.3.1.2" xref="S2.E6.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E6.m1.3b"><apply id="S2.E6.m1.3.3.1.1.cmml" xref="S2.E6.m1.3.3.1"><eq id="S2.E6.m1.3.3.1.1.1.cmml" xref="S2.E6.m1.3.3.1.1.1"></eq><apply id="S2.E6.m1.3.3.1.1.2.cmml" xref="S2.E6.m1.3.3.1.1.2"><times id="S2.E6.m1.3.3.1.1.2.1.cmml" xref="S2.E6.m1.3.3.1.1.2.1"></times><apply id="S2.E6.m1.3.3.1.1.2.2.cmml" xref="S2.E6.m1.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S2.E6.m1.3.3.1.1.2.2.1.cmml" xref="S2.E6.m1.3.3.1.1.2.2">subscript</csymbol><ci id="S2.E6.m1.3.3.1.1.2.2.2.cmml" xref="S2.E6.m1.3.3.1.1.2.2.2">𝑝</ci><ci id="S2.E6.m1.3.3.1.1.2.2.3.cmml" xref="S2.E6.m1.3.3.1.1.2.2.3">𝑃</ci></apply><ci id="S2.E6.m1.2.2.cmml" xref="S2.E6.m1.2.2">𝑟</ci></apply><apply id="S2.E6.m1.1.1.cmml" xref="S2.E6.m1.1.1"><divide id="S2.E6.m1.1.1.2.cmml" xref="S2.E6.m1.1.1"></divide><cn id="S2.E6.m1.1.1.3.cmml" type="integer" xref="S2.E6.m1.1.1.3">2</cn><apply id="S2.E6.m1.1.1.1.cmml" xref="S2.E6.m1.1.1.1"><csymbol cd="ambiguous" id="S2.E6.m1.1.1.1.2.cmml" xref="S2.E6.m1.1.1.1">superscript</csymbol><apply id="S2.E6.m1.1.1.1.1.1.1.cmml" xref="S2.E6.m1.1.1.1.1.1"><plus id="S2.E6.m1.1.1.1.1.1.1.1.cmml" xref="S2.E6.m1.1.1.1.1.1.1.1"></plus><cn id="S2.E6.m1.1.1.1.1.1.1.2.cmml" type="integer" xref="S2.E6.m1.1.1.1.1.1.1.2">1</cn><ci id="S2.E6.m1.1.1.1.1.1.1.3.cmml" xref="S2.E6.m1.1.1.1.1.1.1.3">𝑟</ci></apply><cn id="S2.E6.m1.1.1.1.3.cmml" type="integer" xref="S2.E6.m1.1.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m1.3c">p_{P}(r)=\frac{2}{(1+r)^{2}},</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m1.3d">italic_p start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_r ) = divide start_ARG 2 end_ARG start_ARG ( 1 + italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p2.5">which is also plotted for reference.</p> </div> <figure class="ltx_figure" id="S2.F1"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="220" id="S2.F1.g1" src="extracted/5828746/images/distribution_L12_plus.png" width="293"/></div> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="220" id="S2.F1.g2" src="extracted/5828746/images/distribution_L12_minus.png" width="293"/></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Level spacing distribution for a <math alttext="12\times 1" class="ltx_Math" display="inline" id="S2.F1.3.m1.1"><semantics id="S2.F1.3.m1.1b"><mrow id="S2.F1.3.m1.1.1" xref="S2.F1.3.m1.1.1.cmml"><mn id="S2.F1.3.m1.1.1.2" xref="S2.F1.3.m1.1.1.2.cmml">12</mn><mo id="S2.F1.3.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.F1.3.m1.1.1.1.cmml">×</mo><mn id="S2.F1.3.m1.1.1.3" xref="S2.F1.3.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F1.3.m1.1c"><apply id="S2.F1.3.m1.1.1.cmml" xref="S2.F1.3.m1.1.1"><times id="S2.F1.3.m1.1.1.1.cmml" xref="S2.F1.3.m1.1.1.1"></times><cn id="S2.F1.3.m1.1.1.2.cmml" type="integer" xref="S2.F1.3.m1.1.1.2">12</cn><cn id="S2.F1.3.m1.1.1.3.cmml" type="integer" xref="S2.F1.3.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.3.m1.1d">12\times 1</annotation><annotation encoding="application/x-llamapun" id="S2.F1.3.m1.1e">12 × 1</annotation></semantics></math> ladder with the height potential at varying <math alttext="\lambda" class="ltx_Math" display="inline" id="S2.F1.4.m2.1"><semantics id="S2.F1.4.m2.1b"><mi id="S2.F1.4.m2.1.1" xref="S2.F1.4.m2.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S2.F1.4.m2.1c"><ci id="S2.F1.4.m2.1.1.cmml" xref="S2.F1.4.m2.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.4.m2.1d">\lambda</annotation><annotation encoding="application/x-llamapun" id="S2.F1.4.m2.1e">italic_λ</annotation></semantics></math>. Left: +1 reflection symmetry sector. Right: -1 reflection symmetry sector. Both match the GOE distribution well, indicating that this system is not integrable.</figcaption> </figure> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span>Zero-mode scars</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.2">We refer to zero-modes as eigenstates of the plaquette term of the Hamiltonian with zero eigenvalues. In our notation, these states satisfy</p> <table class="ltx_equation ltx_eqn_table" id="S3.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="K\ket{\psi_{z}}=\sum_{n}(H^{+}_{n}+H^{-}_{n})\left|\psi_{z}\right>=0." class="ltx_Math" display="block" id="S3.E7.m1.2"><semantics id="S3.E7.m1.2a"><mrow id="S3.E7.m1.2.2.1" xref="S3.E7.m1.2.2.1.1.cmml"><mrow id="S3.E7.m1.2.2.1.1" xref="S3.E7.m1.2.2.1.1.cmml"><mrow id="S3.E7.m1.2.2.1.1.4" xref="S3.E7.m1.2.2.1.1.4.cmml"><mi id="S3.E7.m1.2.2.1.1.4.2" xref="S3.E7.m1.2.2.1.1.4.2.cmml">K</mi><mo id="S3.E7.m1.2.2.1.1.4.1" xref="S3.E7.m1.2.2.1.1.4.1.cmml">⁢</mo><mrow id="S3.E7.m1.1.1.3" xref="S3.E7.m1.1.1.2.cmml"><mo id="S3.E7.m1.1.1.3.1" stretchy="false" xref="S3.E7.m1.1.1.2.1.cmml">|</mo><msub id="S3.E7.m1.1.1.1.1" xref="S3.E7.m1.1.1.1.1.cmml"><mi id="S3.E7.m1.1.1.1.1.2" xref="S3.E7.m1.1.1.1.1.2.cmml">ψ</mi><mi 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end_POSTSUBSCRIPT end_ARG ⟩ = ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) | italic_ψ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ⟩ = 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p1.1">Zero-modes can play a crucial role in forming quantum many-body scars. Systems with a spectral symmetry, together with point-group symmetries, can exhibit an exponential number of zero-modes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib62" title="">62</a>]</cite>. Their existence follows from an index theorem, which we review and adapt to our models of interest in the Supplementary Material <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib56" title="">56</a>]</cite>. While one might expect these eigenstates to be thermal, it has been shown that states with low entanglement can be constructed within this subspace for specific cases. It was further conjectured that this is a generic property of local Hamiltonians with an exponential number of zero-modes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib63" title="">63</a>]</cite>. This property was observed explicitly in the case of gauge theories in the spin-<math alttext="1/2" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mrow id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml"><mn id="S3.p1.1.m1.1.1.2" xref="S3.p1.1.m1.1.1.2.cmml">1</mn><mo id="S3.p1.1.m1.1.1.1" xref="S3.p1.1.m1.1.1.1.cmml">/</mo><mn id="S3.p1.1.m1.1.1.3" xref="S3.p1.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><apply id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1"><divide id="S3.p1.1.m1.1.1.1.cmml" xref="S3.p1.1.m1.1.1.1"></divide><cn id="S3.p1.1.m1.1.1.2.cmml" type="integer" xref="S3.p1.1.m1.1.1.2">1</cn><cn id="S3.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.p1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">1/2</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">1 / 2</annotation></semantics></math> QLM, even if it is possible to find scars that are not constructed exclusively from zero-modes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib32" title="">32</a>]</cite>.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.1">We show that it is also possible to construct such states for arbitrary spin truncated link models. The number of these low entropy states grows exponentially with the volume. By choosing the potential term of the Hamiltonian appropriately, different linear combinations can be isolated as low entropy eigenstates.</p> </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">III.1 </span>QMBS in TLM With Arbitrary Integer Spin</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.4">Next, we show that zero-modes with area-law entanglement entropy exist in truncated link models of all integer spins. We start by partitioning the 2D lattice into <math alttext="2\times 1" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mrow id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml"><mn id="S3.SS1.p1.1.m1.1.1.2" xref="S3.SS1.p1.1.m1.1.1.2.cmml">2</mn><mo id="S3.SS1.p1.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p1.1.m1.1.1.1.cmml">×</mo><mn id="S3.SS1.p1.1.m1.1.1.3" xref="S3.SS1.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><apply id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1"><times id="S3.SS1.p1.1.m1.1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1.1"></times><cn id="S3.SS1.p1.1.m1.1.1.2.cmml" type="integer" xref="S3.SS1.p1.1.m1.1.1.2">2</cn><cn id="S3.SS1.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.SS1.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">2\times 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">2 × 1</annotation></semantics></math> and <math alttext="1\times 2" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><mrow id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml"><mn id="S3.SS1.p1.2.m2.1.1.2" xref="S3.SS1.p1.2.m2.1.1.2.cmml">1</mn><mo id="S3.SS1.p1.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p1.2.m2.1.1.1.cmml">×</mo><mn id="S3.SS1.p1.2.m2.1.1.3" xref="S3.SS1.p1.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><apply id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1"><times id="S3.SS1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1.1"></times><cn id="S3.SS1.p1.2.m2.1.1.2.cmml" type="integer" xref="S3.SS1.p1.2.m2.1.1.2">1</cn><cn id="S3.SS1.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.SS1.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">1\times 2</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">1 × 2</annotation></semantics></math> tiles, as depicted on the left of Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.F2" title="Figure 2 ‣ III.1 QMBS in TLM With Arbitrary Integer Spin ‣ III Zero-mode scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">2</span></a>. A given partition will be called tiling and will be denoted by <math alttext="T" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m3.1"><semantics id="S3.SS1.p1.3.m3.1a"><mi id="S3.SS1.p1.3.m3.1.1" xref="S3.SS1.p1.3.m3.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m3.1b"><ci id="S3.SS1.p1.3.m3.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">italic_T</annotation></semantics></math>. It can be represented by a set of tuples <math alttext="\left(n,n^{\prime}\right)" class="ltx_Math" display="inline" id="S3.SS1.p1.4.m4.2"><semantics id="S3.SS1.p1.4.m4.2a"><mrow id="S3.SS1.p1.4.m4.2.2.1" xref="S3.SS1.p1.4.m4.2.2.2.cmml"><mo id="S3.SS1.p1.4.m4.2.2.1.2" xref="S3.SS1.p1.4.m4.2.2.2.cmml">(</mo><mi id="S3.SS1.p1.4.m4.1.1" xref="S3.SS1.p1.4.m4.1.1.cmml">n</mi><mo id="S3.SS1.p1.4.m4.2.2.1.3" xref="S3.SS1.p1.4.m4.2.2.2.cmml">,</mo><msup id="S3.SS1.p1.4.m4.2.2.1.1" xref="S3.SS1.p1.4.m4.2.2.1.1.cmml"><mi id="S3.SS1.p1.4.m4.2.2.1.1.2" xref="S3.SS1.p1.4.m4.2.2.1.1.2.cmml">n</mi><mo id="S3.SS1.p1.4.m4.2.2.1.1.3" xref="S3.SS1.p1.4.m4.2.2.1.1.3.cmml">′</mo></msup><mo id="S3.SS1.p1.4.m4.2.2.1.4" xref="S3.SS1.p1.4.m4.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.4.m4.2b"><interval closure="open" id="S3.SS1.p1.4.m4.2.2.2.cmml" xref="S3.SS1.p1.4.m4.2.2.1"><ci id="S3.SS1.p1.4.m4.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1">𝑛</ci><apply id="S3.SS1.p1.4.m4.2.2.1.1.cmml" xref="S3.SS1.p1.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m4.2.2.1.1.1.cmml" xref="S3.SS1.p1.4.m4.2.2.1.1">superscript</csymbol><ci id="S3.SS1.p1.4.m4.2.2.1.1.2.cmml" xref="S3.SS1.p1.4.m4.2.2.1.1.2">𝑛</ci><ci id="S3.SS1.p1.4.m4.2.2.1.1.3.cmml" xref="S3.SS1.p1.4.m4.2.2.1.1.3">′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m4.2c">\left(n,n^{\prime}\right)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m4.2d">( italic_n , italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> where the two entries indicate the two plaquettes making up each tile. 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xref="S3.Ex1.m2.4.4.2.2.1.1.3">′</ci></apply></interval><ci id="S3.Ex1.m2.4.4.2.4.cmml" xref="S3.Ex1.m2.4.4.2.4">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex1.m2.4c">\displaystyle\frac{1}{\left(S+1\right)^{\left|T\right|/2}}\prod_{(n,n^{\prime}% )\in T}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m2.4d">divide start_ARG 1 end_ARG start_ARG ( italic_S + 1 ) start_POSTSUPERSCRIPT | italic_T | / 2 end_POSTSUPERSCRIPT end_ARG ∏ start_POSTSUBSCRIPT ( italic_n , italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ italic_T end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.E8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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xref="S3.E8.m1.2.2.1.1.1.1.1.2.2.3.2.1"></minus><ci id="S3.E8.m1.2.2.1.1.1.1.1.2.2.3.2.2.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.2.2.3.2.2">𝑖</ci><ci id="S3.E8.m1.2.2.1.1.1.1.1.2.2.3.2.3.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.2.2.3.2.3">𝑆</ci></apply><ci id="S3.E8.m1.2.2.1.1.1.1.1.2.2.3.3.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.2.2.3.3">𝑘</ci></apply></apply><apply id="S3.E8.m1.2.2.1.1.1.1.1.3.3.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.E8.m1.2.2.1.1.1.1.1.3.3.2.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3">superscript</csymbol><apply id="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1"><csymbol cd="ambiguous" id="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.1.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1">subscript</csymbol><apply id="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.2.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1"><csymbol cd="ambiguous" id="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.2.1.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1">superscript</csymbol><ci id="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.2.2.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.2.2">𝐻</ci><plus id="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.2.3.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.2.3"></plus></apply><apply id="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.3.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.3.1.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.3">superscript</csymbol><ci id="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.3.2.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.3.2">𝑛</ci><ci id="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.3.3.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3.1.1.1.3.3">′</ci></apply></apply><apply id="S3.E8.m1.2.2.1.1.1.1.1.3.3.3.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3.3"><minus id="S3.E8.m1.2.2.1.1.1.1.1.3.3.3.1.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3.3.1"></minus><ci id="S3.E8.m1.2.2.1.1.1.1.1.3.3.3.2.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3.3.2">𝑖</ci><ci id="S3.E8.m1.2.2.1.1.1.1.1.3.3.3.3.cmml" xref="S3.E8.m1.2.2.1.1.1.1.1.3.3.3.3">𝑘</ci></apply></apply></apply></apply><apply id="S3.E8.m1.1.1a.2.cmml" xref="S3.E8.m1.1.1a.3"><csymbol cd="latexml" id="S3.E8.m1.1.1a.2.1.cmml" xref="S3.E8.m1.1.1a.3.1">ket</csymbol><cn id="S3.E8.m1.1.1.1.1.cmml" type="integer" xref="S3.E8.m1.1.1.1.1">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E8.m1.2c">\displaystyle\left(\sum_{k=0}^{S}(-1)^{k}(H^{+}_{n})^{i-S+k}(H^{+}_{n^{\prime}% })^{i-k}\right)\ket{\mathbf{0}},</annotation><annotation encoding="application/x-llamapun" id="S3.E8.m1.2d">( ∑ start_POSTSUBSCRIPT italic_k = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S end_POSTSUPERSCRIPT ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_i - italic_S + italic_k end_POSTSUPERSCRIPT ( italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_i - italic_k end_POSTSUPERSCRIPT ) | start_ARG bold_0 end_ARG ⟩ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p1.7">where <math alttext="\ket{\mathbf{0}}" class="ltx_Math" display="inline" id="S3.SS1.p1.5.m1.1"><semantics id="S3.SS1.p1.5.m1.1a"><mrow id="S3.SS1.p1.5.m1.1.1.3" xref="S3.SS1.p1.5.m1.1.1.2.cmml"><mo id="S3.SS1.p1.5.m1.1.1.3.1" stretchy="false" xref="S3.SS1.p1.5.m1.1.1.2.1.cmml">|</mo><mn id="S3.SS1.p1.5.m1.1.1.1.1" xref="S3.SS1.p1.5.m1.1.1.1.1.cmml">𝟎</mn><mo id="S3.SS1.p1.5.m1.1.1.3.2" stretchy="false" xref="S3.SS1.p1.5.m1.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.5.m1.1b"><apply id="S3.SS1.p1.5.m1.1.1.2.cmml" xref="S3.SS1.p1.5.m1.1.1.3"><csymbol cd="latexml" id="S3.SS1.p1.5.m1.1.1.2.1.cmml" xref="S3.SS1.p1.5.m1.1.1.3.1">ket</csymbol><cn id="S3.SS1.p1.5.m1.1.1.1.1.cmml" type="integer" xref="S3.SS1.p1.5.m1.1.1.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m1.1c">\ket{\mathbf{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.5.m1.1d">| start_ARG bold_0 end_ARG ⟩</annotation></semantics></math> is the state where all links have value zero, and <math alttext="0\leq i\leq S" class="ltx_Math" display="inline" id="S3.SS1.p1.6.m2.1"><semantics id="S3.SS1.p1.6.m2.1a"><mrow id="S3.SS1.p1.6.m2.1.1" xref="S3.SS1.p1.6.m2.1.1.cmml"><mn id="S3.SS1.p1.6.m2.1.1.2" xref="S3.SS1.p1.6.m2.1.1.2.cmml">0</mn><mo id="S3.SS1.p1.6.m2.1.1.3" xref="S3.SS1.p1.6.m2.1.1.3.cmml">≤</mo><mi id="S3.SS1.p1.6.m2.1.1.4" xref="S3.SS1.p1.6.m2.1.1.4.cmml">i</mi><mo id="S3.SS1.p1.6.m2.1.1.5" xref="S3.SS1.p1.6.m2.1.1.5.cmml">≤</mo><mi id="S3.SS1.p1.6.m2.1.1.6" xref="S3.SS1.p1.6.m2.1.1.6.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.6.m2.1b"><apply id="S3.SS1.p1.6.m2.1.1.cmml" xref="S3.SS1.p1.6.m2.1.1"><and id="S3.SS1.p1.6.m2.1.1a.cmml" xref="S3.SS1.p1.6.m2.1.1"></and><apply id="S3.SS1.p1.6.m2.1.1b.cmml" xref="S3.SS1.p1.6.m2.1.1"><leq id="S3.SS1.p1.6.m2.1.1.3.cmml" xref="S3.SS1.p1.6.m2.1.1.3"></leq><cn id="S3.SS1.p1.6.m2.1.1.2.cmml" type="integer" xref="S3.SS1.p1.6.m2.1.1.2">0</cn><ci id="S3.SS1.p1.6.m2.1.1.4.cmml" xref="S3.SS1.p1.6.m2.1.1.4">𝑖</ci></apply><apply id="S3.SS1.p1.6.m2.1.1c.cmml" xref="S3.SS1.p1.6.m2.1.1"><leq id="S3.SS1.p1.6.m2.1.1.5.cmml" xref="S3.SS1.p1.6.m2.1.1.5"></leq><share href="https://arxiv.org/html/2403.08892v3#S3.SS1.p1.6.m2.1.1.4.cmml" id="S3.SS1.p1.6.m2.1.1d.cmml" xref="S3.SS1.p1.6.m2.1.1"></share><ci id="S3.SS1.p1.6.m2.1.1.6.cmml" xref="S3.SS1.p1.6.m2.1.1.6">𝑆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.6.m2.1c">0\leq i\leq S</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.6.m2.1d">0 ≤ italic_i ≤ italic_S</annotation></semantics></math>. The states <math alttext="\ket{\psi_{s}^{(i,T)}}" class="ltx_Math" display="inline" id="S3.SS1.p1.7.m3.1"><semantics id="S3.SS1.p1.7.m3.1a"><mrow id="S3.SS1.p1.7.m3.1.1.3" xref="S3.SS1.p1.7.m3.1.1.2.cmml"><mo id="S3.SS1.p1.7.m3.1.1.3.1" stretchy="false" xref="S3.SS1.p1.7.m3.1.1.2.1.cmml">|</mo><msubsup id="S3.SS1.p1.7.m3.1.1.1.1" xref="S3.SS1.p1.7.m3.1.1.1.1.cmml"><mi id="S3.SS1.p1.7.m3.1.1.1.1.4.2" xref="S3.SS1.p1.7.m3.1.1.1.1.4.2.cmml">ψ</mi><mi id="S3.SS1.p1.7.m3.1.1.1.1.4.3" xref="S3.SS1.p1.7.m3.1.1.1.1.4.3.cmml">s</mi><mrow id="S3.SS1.p1.7.m3.1.1.1.1.2.2.4" xref="S3.SS1.p1.7.m3.1.1.1.1.2.2.3.cmml"><mo id="S3.SS1.p1.7.m3.1.1.1.1.2.2.4.1" stretchy="false" xref="S3.SS1.p1.7.m3.1.1.1.1.2.2.3.cmml">(</mo><mi id="S3.SS1.p1.7.m3.1.1.1.1.1.1.1" xref="S3.SS1.p1.7.m3.1.1.1.1.1.1.1.cmml">i</mi><mo id="S3.SS1.p1.7.m3.1.1.1.1.2.2.4.2" xref="S3.SS1.p1.7.m3.1.1.1.1.2.2.3.cmml">,</mo><mi id="S3.SS1.p1.7.m3.1.1.1.1.2.2.2" xref="S3.SS1.p1.7.m3.1.1.1.1.2.2.2.cmml">T</mi><mo id="S3.SS1.p1.7.m3.1.1.1.1.2.2.4.3" stretchy="false" xref="S3.SS1.p1.7.m3.1.1.1.1.2.2.3.cmml">)</mo></mrow></msubsup><mo id="S3.SS1.p1.7.m3.1.1.3.2" stretchy="false" xref="S3.SS1.p1.7.m3.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.7.m3.1b"><apply id="S3.SS1.p1.7.m3.1.1.2.cmml" xref="S3.SS1.p1.7.m3.1.1.3"><csymbol cd="latexml" id="S3.SS1.p1.7.m3.1.1.2.1.cmml" xref="S3.SS1.p1.7.m3.1.1.3.1">ket</csymbol><apply id="S3.SS1.p1.7.m3.1.1.1.1.cmml" xref="S3.SS1.p1.7.m3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.7.m3.1.1.1.1.3.cmml" xref="S3.SS1.p1.7.m3.1.1.1.1">superscript</csymbol><apply id="S3.SS1.p1.7.m3.1.1.1.1.4.cmml" xref="S3.SS1.p1.7.m3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.7.m3.1.1.1.1.4.1.cmml" xref="S3.SS1.p1.7.m3.1.1.1.1">subscript</csymbol><ci id="S3.SS1.p1.7.m3.1.1.1.1.4.2.cmml" xref="S3.SS1.p1.7.m3.1.1.1.1.4.2">𝜓</ci><ci id="S3.SS1.p1.7.m3.1.1.1.1.4.3.cmml" xref="S3.SS1.p1.7.m3.1.1.1.1.4.3">𝑠</ci></apply><interval closure="open" id="S3.SS1.p1.7.m3.1.1.1.1.2.2.3.cmml" xref="S3.SS1.p1.7.m3.1.1.1.1.2.2.4"><ci id="S3.SS1.p1.7.m3.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p1.7.m3.1.1.1.1.1.1.1">𝑖</ci><ci id="S3.SS1.p1.7.m3.1.1.1.1.2.2.2.cmml" xref="S3.SS1.p1.7.m3.1.1.1.1.2.2.2">𝑇</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.7.m3.1c">\ket{\psi_{s}^{(i,T)}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.7.m3.1d">| start_ARG italic_ψ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i , italic_T ) end_POSTSUPERSCRIPT end_ARG ⟩</annotation></semantics></math> have the property</p> <table class="ltx_equation ltx_eqn_table" id="S3.E9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="(H^{-}_{m}+H^{+}_{m^{\prime}})\ket{\psi_{s}^{(i,T)}}=(H^{+}_{m}+H^{-}_{m^{% \prime}})\ket{\psi_{s}^{(i,T)}}=0" 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\prime}})\ket{\psi_{s}^{(i,T)}}=0</annotation><annotation encoding="application/x-llamapun" id="S3.E9.m1.4d">( italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT + italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) | start_ARG italic_ψ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i , italic_T ) end_POSTSUPERSCRIPT end_ARG ⟩ = ( italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT + italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) | start_ARG italic_ψ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i , italic_T ) end_POSTSUPERSCRIPT end_ARG ⟩ = 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p1.9">if <math alttext="(m,m^{\prime})" class="ltx_Math" display="inline" id="S3.SS1.p1.8.m1.2"><semantics id="S3.SS1.p1.8.m1.2a"><mrow id="S3.SS1.p1.8.m1.2.2.1" xref="S3.SS1.p1.8.m1.2.2.2.cmml"><mo id="S3.SS1.p1.8.m1.2.2.1.2" stretchy="false" xref="S3.SS1.p1.8.m1.2.2.2.cmml">(</mo><mi id="S3.SS1.p1.8.m1.1.1" xref="S3.SS1.p1.8.m1.1.1.cmml">m</mi><mo id="S3.SS1.p1.8.m1.2.2.1.3" xref="S3.SS1.p1.8.m1.2.2.2.cmml">,</mo><msup id="S3.SS1.p1.8.m1.2.2.1.1" xref="S3.SS1.p1.8.m1.2.2.1.1.cmml"><mi id="S3.SS1.p1.8.m1.2.2.1.1.2" xref="S3.SS1.p1.8.m1.2.2.1.1.2.cmml">m</mi><mo id="S3.SS1.p1.8.m1.2.2.1.1.3" xref="S3.SS1.p1.8.m1.2.2.1.1.3.cmml">′</mo></msup><mo id="S3.SS1.p1.8.m1.2.2.1.4" stretchy="false" xref="S3.SS1.p1.8.m1.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.8.m1.2b"><interval closure="open" id="S3.SS1.p1.8.m1.2.2.2.cmml" xref="S3.SS1.p1.8.m1.2.2.1"><ci id="S3.SS1.p1.8.m1.1.1.cmml" xref="S3.SS1.p1.8.m1.1.1">𝑚</ci><apply id="S3.SS1.p1.8.m1.2.2.1.1.cmml" xref="S3.SS1.p1.8.m1.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.8.m1.2.2.1.1.1.cmml" xref="S3.SS1.p1.8.m1.2.2.1.1">superscript</csymbol><ci id="S3.SS1.p1.8.m1.2.2.1.1.2.cmml" xref="S3.SS1.p1.8.m1.2.2.1.1.2">𝑚</ci><ci id="S3.SS1.p1.8.m1.2.2.1.1.3.cmml" xref="S3.SS1.p1.8.m1.2.2.1.1.3">′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.8.m1.2c">(m,m^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.8.m1.2d">( italic_m , italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> is a tile in <math alttext="T" class="ltx_Math" display="inline" id="S3.SS1.p1.9.m2.1"><semantics id="S3.SS1.p1.9.m2.1a"><mi id="S3.SS1.p1.9.m2.1.1" xref="S3.SS1.p1.9.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.9.m2.1b"><ci id="S3.SS1.p1.9.m2.1.1.cmml" xref="S3.SS1.p1.9.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.9.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.9.m2.1d">italic_T</annotation></semantics></math>. Therefore,</p> <table class="ltx_equation ltx_eqn_table" id="S3.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="K\ket{\psi_{s}^{(i,T)}}=\sum_{(n,n^{\prime})\in T}(H^{+}_{n}+H^{-}_{n^{\prime}% }+H^{-}_{n}+H^{+}_{n^{\prime}})\ket{\psi_{s}^{(i,T)}}=0," class="ltx_Math" display="block" id="S3.E10.m1.5"><semantics id="S3.E10.m1.5a"><mrow id="S3.E10.m1.5.5.1" xref="S3.E10.m1.5.5.1.1.cmml"><mrow id="S3.E10.m1.5.5.1.1" xref="S3.E10.m1.5.5.1.1.cmml"><mrow id="S3.E10.m1.5.5.1.1.3" xref="S3.E10.m1.5.5.1.1.3.cmml"><mi id="S3.E10.m1.5.5.1.1.3.2" xref="S3.E10.m1.5.5.1.1.3.2.cmml">K</mi><mo id="S3.E10.m1.5.5.1.1.3.1" xref="S3.E10.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S3.E10.m1.1.1.3" xref="S3.E10.m1.1.1.2.cmml"><mo id="S3.E10.m1.1.1.3.1" stretchy="false" xref="S3.E10.m1.1.1.2.1.cmml">|</mo><msubsup id="S3.E10.m1.1.1.1.1" xref="S3.E10.m1.1.1.1.1.cmml"><mi 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end_POSTSUBSCRIPT ( italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT + italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) | start_ARG italic_ψ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i , italic_T ) end_POSTSUPERSCRIPT end_ARG ⟩ = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.p1.10">making them zero-modes for truncated link models of arbitrary spin. For quantum link models with spin <math alttext="S>1" class="ltx_Math" display="inline" id="S3.SS1.p1.10.m1.1"><semantics id="S3.SS1.p1.10.m1.1a"><mrow id="S3.SS1.p1.10.m1.1.1" xref="S3.SS1.p1.10.m1.1.1.cmml"><mi id="S3.SS1.p1.10.m1.1.1.2" xref="S3.SS1.p1.10.m1.1.1.2.cmml">S</mi><mo id="S3.SS1.p1.10.m1.1.1.1" xref="S3.SS1.p1.10.m1.1.1.1.cmml">></mo><mn id="S3.SS1.p1.10.m1.1.1.3" xref="S3.SS1.p1.10.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.10.m1.1b"><apply id="S3.SS1.p1.10.m1.1.1.cmml" xref="S3.SS1.p1.10.m1.1.1"><gt id="S3.SS1.p1.10.m1.1.1.1.cmml" xref="S3.SS1.p1.10.m1.1.1.1"></gt><ci id="S3.SS1.p1.10.m1.1.1.2.cmml" xref="S3.SS1.p1.10.m1.1.1.2">𝑆</ci><cn id="S3.SS1.p1.10.m1.1.1.3.cmml" type="integer" xref="S3.SS1.p1.10.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.10.m1.1c">S>1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.10.m1.1d">italic_S > 1</annotation></semantics></math>, this would no longer be the case due to the electric field-dependent pre-factors when raising/lowering the spins.</p> </div> <figure class="ltx_figure" id="S3.F2"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_square" height="210" id="S3.F2.g1" src="extracted/5828746/images/tiling.png" width="210"/></div> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_square" height="210" id="S3.F2.g2" src="extracted/5828746/images/scar4.png" width="210"/></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Left: Example of a tiling <math alttext="T" class="ltx_Math" display="inline" id="S3.F2.6.m1.1"><semantics id="S3.F2.6.m1.1b"><mi id="S3.F2.6.m1.1.1" xref="S3.F2.6.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.F2.6.m1.1c"><ci id="S3.F2.6.m1.1.1.cmml" xref="S3.F2.6.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.6.m1.1d">T</annotation><annotation encoding="application/x-llamapun" id="S3.F2.6.m1.1e">italic_T</annotation></semantics></math> of the 2D lattice, consisting of partitioning it into <math alttext="2\times 1" class="ltx_Math" display="inline" id="S3.F2.7.m2.1"><semantics id="S3.F2.7.m2.1b"><mrow id="S3.F2.7.m2.1.1" xref="S3.F2.7.m2.1.1.cmml"><mn id="S3.F2.7.m2.1.1.2" xref="S3.F2.7.m2.1.1.2.cmml">2</mn><mo id="S3.F2.7.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.F2.7.m2.1.1.1.cmml">×</mo><mn id="S3.F2.7.m2.1.1.3" xref="S3.F2.7.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.7.m2.1c"><apply id="S3.F2.7.m2.1.1.cmml" xref="S3.F2.7.m2.1.1"><times id="S3.F2.7.m2.1.1.1.cmml" xref="S3.F2.7.m2.1.1.1"></times><cn id="S3.F2.7.m2.1.1.2.cmml" type="integer" xref="S3.F2.7.m2.1.1.2">2</cn><cn id="S3.F2.7.m2.1.1.3.cmml" type="integer" xref="S3.F2.7.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.7.m2.1d">2\times 1</annotation><annotation encoding="application/x-llamapun" id="S3.F2.7.m2.1e">2 × 1</annotation></semantics></math> and <math alttext="1\times 2" class="ltx_Math" display="inline" id="S3.F2.8.m3.1"><semantics id="S3.F2.8.m3.1b"><mrow id="S3.F2.8.m3.1.1" xref="S3.F2.8.m3.1.1.cmml"><mn id="S3.F2.8.m3.1.1.2" xref="S3.F2.8.m3.1.1.2.cmml">1</mn><mo id="S3.F2.8.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.F2.8.m3.1.1.1.cmml">×</mo><mn id="S3.F2.8.m3.1.1.3" xref="S3.F2.8.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.8.m3.1c"><apply id="S3.F2.8.m3.1.1.cmml" xref="S3.F2.8.m3.1.1"><times id="S3.F2.8.m3.1.1.1.cmml" xref="S3.F2.8.m3.1.1.1"></times><cn id="S3.F2.8.m3.1.1.2.cmml" type="integer" xref="S3.F2.8.m3.1.1.2">1</cn><cn id="S3.F2.8.m3.1.1.3.cmml" type="integer" xref="S3.F2.8.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.8.m3.1d">1\times 2</annotation><annotation encoding="application/x-llamapun" id="S3.F2.8.m3.1e">1 × 2</annotation></semantics></math> tiles, to a total of 8 tiles. Right: Representation of a scar state based on an alternative tiling and in the dual representation. An independent sum over every <math alttext="k_{i}" class="ltx_Math" display="inline" id="S3.F2.9.m4.1"><semantics id="S3.F2.9.m4.1b"><msub id="S3.F2.9.m4.1.1" xref="S3.F2.9.m4.1.1.cmml"><mi id="S3.F2.9.m4.1.1.2" xref="S3.F2.9.m4.1.1.2.cmml">k</mi><mi id="S3.F2.9.m4.1.1.3" xref="S3.F2.9.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.F2.9.m4.1c"><apply id="S3.F2.9.m4.1.1.cmml" xref="S3.F2.9.m4.1.1"><csymbol cd="ambiguous" id="S3.F2.9.m4.1.1.1.cmml" xref="S3.F2.9.m4.1.1">subscript</csymbol><ci id="S3.F2.9.m4.1.1.2.cmml" xref="S3.F2.9.m4.1.1.2">𝑘</ci><ci id="S3.F2.9.m4.1.1.3.cmml" xref="S3.F2.9.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.9.m4.1d">k_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.9.m4.1e">italic_k start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> with a weight of <math alttext="\prod_{i}\left(-1\right)^{k_{i}}" class="ltx_Math" display="inline" id="S3.F2.10.m5.1"><semantics id="S3.F2.10.m5.1b"><mrow id="S3.F2.10.m5.1.1" xref="S3.F2.10.m5.1.1.cmml"><msub id="S3.F2.10.m5.1.1.2" xref="S3.F2.10.m5.1.1.2.cmml"><mo id="S3.F2.10.m5.1.1.2.2" xref="S3.F2.10.m5.1.1.2.2.cmml">∏</mo><mi id="S3.F2.10.m5.1.1.2.3" xref="S3.F2.10.m5.1.1.2.3.cmml">i</mi></msub><msup id="S3.F2.10.m5.1.1.1" xref="S3.F2.10.m5.1.1.1.cmml"><mrow id="S3.F2.10.m5.1.1.1.1.1" xref="S3.F2.10.m5.1.1.1.1.1.1.cmml"><mo id="S3.F2.10.m5.1.1.1.1.1.2" lspace="0em" xref="S3.F2.10.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.F2.10.m5.1.1.1.1.1.1" xref="S3.F2.10.m5.1.1.1.1.1.1.cmml"><mo id="S3.F2.10.m5.1.1.1.1.1.1b" xref="S3.F2.10.m5.1.1.1.1.1.1.cmml">−</mo><mn id="S3.F2.10.m5.1.1.1.1.1.1.2" xref="S3.F2.10.m5.1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S3.F2.10.m5.1.1.1.1.1.3" xref="S3.F2.10.m5.1.1.1.1.1.1.cmml">)</mo></mrow><msub id="S3.F2.10.m5.1.1.1.3" xref="S3.F2.10.m5.1.1.1.3.cmml"><mi id="S3.F2.10.m5.1.1.1.3.2" xref="S3.F2.10.m5.1.1.1.3.2.cmml">k</mi><mi id="S3.F2.10.m5.1.1.1.3.3" xref="S3.F2.10.m5.1.1.1.3.3.cmml">i</mi></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.10.m5.1c"><apply id="S3.F2.10.m5.1.1.cmml" xref="S3.F2.10.m5.1.1"><apply id="S3.F2.10.m5.1.1.2.cmml" xref="S3.F2.10.m5.1.1.2"><csymbol cd="ambiguous" id="S3.F2.10.m5.1.1.2.1.cmml" xref="S3.F2.10.m5.1.1.2">subscript</csymbol><csymbol cd="latexml" id="S3.F2.10.m5.1.1.2.2.cmml" xref="S3.F2.10.m5.1.1.2.2">product</csymbol><ci id="S3.F2.10.m5.1.1.2.3.cmml" xref="S3.F2.10.m5.1.1.2.3">𝑖</ci></apply><apply id="S3.F2.10.m5.1.1.1.cmml" xref="S3.F2.10.m5.1.1.1"><csymbol cd="ambiguous" id="S3.F2.10.m5.1.1.1.2.cmml" xref="S3.F2.10.m5.1.1.1">superscript</csymbol><apply id="S3.F2.10.m5.1.1.1.1.1.1.cmml" xref="S3.F2.10.m5.1.1.1.1.1"><minus id="S3.F2.10.m5.1.1.1.1.1.1.1.cmml" xref="S3.F2.10.m5.1.1.1.1.1"></minus><cn id="S3.F2.10.m5.1.1.1.1.1.1.2.cmml" type="integer" xref="S3.F2.10.m5.1.1.1.1.1.1.2">1</cn></apply><apply id="S3.F2.10.m5.1.1.1.3.cmml" xref="S3.F2.10.m5.1.1.1.3"><csymbol cd="ambiguous" id="S3.F2.10.m5.1.1.1.3.1.cmml" xref="S3.F2.10.m5.1.1.1.3">subscript</csymbol><ci id="S3.F2.10.m5.1.1.1.3.2.cmml" xref="S3.F2.10.m5.1.1.1.3.2">𝑘</ci><ci id="S3.F2.10.m5.1.1.1.3.3.cmml" xref="S3.F2.10.m5.1.1.1.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.10.m5.1d">\prod_{i}\left(-1\right)^{k_{i}}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.10.m5.1e">∏ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( - 1 ) start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> is assumed. See Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S5.E24" title="In V A 4×4 Scar for the 𝐸² potential ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">24</span></a>). </figcaption> </figure> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.4">These states are zero-modes in all lattices where <math alttext="L_{1}L_{2}" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><mrow id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml"><msub id="S3.SS1.p2.1.m1.1.1.2" xref="S3.SS1.p2.1.m1.1.1.2.cmml"><mi id="S3.SS1.p2.1.m1.1.1.2.2" xref="S3.SS1.p2.1.m1.1.1.2.2.cmml">L</mi><mn id="S3.SS1.p2.1.m1.1.1.2.3" xref="S3.SS1.p2.1.m1.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS1.p2.1.m1.1.1.1" xref="S3.SS1.p2.1.m1.1.1.1.cmml">⁢</mo><msub id="S3.SS1.p2.1.m1.1.1.3" xref="S3.SS1.p2.1.m1.1.1.3.cmml"><mi id="S3.SS1.p2.1.m1.1.1.3.2" xref="S3.SS1.p2.1.m1.1.1.3.2.cmml">L</mi><mn id="S3.SS1.p2.1.m1.1.1.3.3" xref="S3.SS1.p2.1.m1.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><apply id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1"><times id="S3.SS1.p2.1.m1.1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1.1"></times><apply id="S3.SS1.p2.1.m1.1.1.2.cmml" xref="S3.SS1.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p2.1.m1.1.1.2.1.cmml" xref="S3.SS1.p2.1.m1.1.1.2">subscript</csymbol><ci id="S3.SS1.p2.1.m1.1.1.2.2.cmml" xref="S3.SS1.p2.1.m1.1.1.2.2">𝐿</ci><cn id="S3.SS1.p2.1.m1.1.1.2.3.cmml" type="integer" xref="S3.SS1.p2.1.m1.1.1.2.3">1</cn></apply><apply id="S3.SS1.p2.1.m1.1.1.3.cmml" xref="S3.SS1.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p2.1.m1.1.1.3.1.cmml" xref="S3.SS1.p2.1.m1.1.1.3">subscript</csymbol><ci id="S3.SS1.p2.1.m1.1.1.3.2.cmml" xref="S3.SS1.p2.1.m1.1.1.3.2">𝐿</ci><cn id="S3.SS1.p2.1.m1.1.1.3.3.cmml" type="integer" xref="S3.SS1.p2.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">L_{1}L_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is even. The number of possible tilings grows exponentially with <math alttext="L_{1}L_{2}" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.1"><semantics id="S3.SS1.p2.2.m2.1a"><mrow id="S3.SS1.p2.2.m2.1.1" xref="S3.SS1.p2.2.m2.1.1.cmml"><msub id="S3.SS1.p2.2.m2.1.1.2" xref="S3.SS1.p2.2.m2.1.1.2.cmml"><mi id="S3.SS1.p2.2.m2.1.1.2.2" xref="S3.SS1.p2.2.m2.1.1.2.2.cmml">L</mi><mn id="S3.SS1.p2.2.m2.1.1.2.3" xref="S3.SS1.p2.2.m2.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS1.p2.2.m2.1.1.1" xref="S3.SS1.p2.2.m2.1.1.1.cmml">⁢</mo><msub id="S3.SS1.p2.2.m2.1.1.3" xref="S3.SS1.p2.2.m2.1.1.3.cmml"><mi id="S3.SS1.p2.2.m2.1.1.3.2" xref="S3.SS1.p2.2.m2.1.1.3.2.cmml">L</mi><mn id="S3.SS1.p2.2.m2.1.1.3.3" xref="S3.SS1.p2.2.m2.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.1b"><apply id="S3.SS1.p2.2.m2.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1"><times id="S3.SS1.p2.2.m2.1.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1.1"></times><apply id="S3.SS1.p2.2.m2.1.1.2.cmml" xref="S3.SS1.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p2.2.m2.1.1.2.1.cmml" xref="S3.SS1.p2.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS1.p2.2.m2.1.1.2.2.cmml" xref="S3.SS1.p2.2.m2.1.1.2.2">𝐿</ci><cn id="S3.SS1.p2.2.m2.1.1.2.3.cmml" type="integer" xref="S3.SS1.p2.2.m2.1.1.2.3">1</cn></apply><apply id="S3.SS1.p2.2.m2.1.1.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p2.2.m2.1.1.3.1.cmml" xref="S3.SS1.p2.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS1.p2.2.m2.1.1.3.2.cmml" xref="S3.SS1.p2.2.m2.1.1.3.2">𝐿</ci><cn id="S3.SS1.p2.2.m2.1.1.3.3.cmml" type="integer" xref="S3.SS1.p2.2.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.1c">L_{1}L_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.1d">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> if both <math alttext="L_{1}" class="ltx_Math" display="inline" id="S3.SS1.p2.3.m3.1"><semantics id="S3.SS1.p2.3.m3.1a"><msub id="S3.SS1.p2.3.m3.1.1" xref="S3.SS1.p2.3.m3.1.1.cmml"><mi id="S3.SS1.p2.3.m3.1.1.2" xref="S3.SS1.p2.3.m3.1.1.2.cmml">L</mi><mn id="S3.SS1.p2.3.m3.1.1.3" xref="S3.SS1.p2.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.3.m3.1b"><apply id="S3.SS1.p2.3.m3.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.3.m3.1.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1">subscript</csymbol><ci id="S3.SS1.p2.3.m3.1.1.2.cmml" xref="S3.SS1.p2.3.m3.1.1.2">𝐿</ci><cn id="S3.SS1.p2.3.m3.1.1.3.cmml" type="integer" xref="S3.SS1.p2.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.3.m3.1c">L_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.3.m3.1d">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="L_{2}" class="ltx_Math" display="inline" id="S3.SS1.p2.4.m4.1"><semantics id="S3.SS1.p2.4.m4.1a"><msub id="S3.SS1.p2.4.m4.1.1" xref="S3.SS1.p2.4.m4.1.1.cmml"><mi id="S3.SS1.p2.4.m4.1.1.2" xref="S3.SS1.p2.4.m4.1.1.2.cmml">L</mi><mn id="S3.SS1.p2.4.m4.1.1.3" xref="S3.SS1.p2.4.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.4.m4.1b"><apply id="S3.SS1.p2.4.m4.1.1.cmml" xref="S3.SS1.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.4.m4.1.1.1.cmml" xref="S3.SS1.p2.4.m4.1.1">subscript</csymbol><ci id="S3.SS1.p2.4.m4.1.1.2.cmml" xref="S3.SS1.p2.4.m4.1.1.2">𝐿</ci><cn id="S3.SS1.p2.4.m4.1.1.3.cmml" type="integer" xref="S3.SS1.p2.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.4.m4.1c">L_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.4.m4.1d">italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are larger than 1.</p> </div> <div class="ltx_para" id="S3.SS1.p3"> <p class="ltx_p" id="S3.SS1.p3.1">By partitioning the systems into two regions, we see that the entanglement between them is generated by tiles that touch both regions resulting in an area law. The states <math alttext="\ket{\psi_{s}^{(i,T)}}" class="ltx_Math" display="inline" id="S3.SS1.p3.1.m1.1"><semantics id="S3.SS1.p3.1.m1.1a"><mrow id="S3.SS1.p3.1.m1.1.1.3" xref="S3.SS1.p3.1.m1.1.1.2.cmml"><mo id="S3.SS1.p3.1.m1.1.1.3.1" stretchy="false" xref="S3.SS1.p3.1.m1.1.1.2.1.cmml">|</mo><msubsup id="S3.SS1.p3.1.m1.1.1.1.1" xref="S3.SS1.p3.1.m1.1.1.1.1.cmml"><mi id="S3.SS1.p3.1.m1.1.1.1.1.4.2" xref="S3.SS1.p3.1.m1.1.1.1.1.4.2.cmml">ψ</mi><mi id="S3.SS1.p3.1.m1.1.1.1.1.4.3" xref="S3.SS1.p3.1.m1.1.1.1.1.4.3.cmml">s</mi><mrow id="S3.SS1.p3.1.m1.1.1.1.1.2.2.4" xref="S3.SS1.p3.1.m1.1.1.1.1.2.2.3.cmml"><mo id="S3.SS1.p3.1.m1.1.1.1.1.2.2.4.1" stretchy="false" xref="S3.SS1.p3.1.m1.1.1.1.1.2.2.3.cmml">(</mo><mi id="S3.SS1.p3.1.m1.1.1.1.1.1.1.1" xref="S3.SS1.p3.1.m1.1.1.1.1.1.1.1.cmml">i</mi><mo id="S3.SS1.p3.1.m1.1.1.1.1.2.2.4.2" xref="S3.SS1.p3.1.m1.1.1.1.1.2.2.3.cmml">,</mo><mi id="S3.SS1.p3.1.m1.1.1.1.1.2.2.2" xref="S3.SS1.p3.1.m1.1.1.1.1.2.2.2.cmml">T</mi><mo id="S3.SS1.p3.1.m1.1.1.1.1.2.2.4.3" stretchy="false" xref="S3.SS1.p3.1.m1.1.1.1.1.2.2.3.cmml">)</mo></mrow></msubsup><mo id="S3.SS1.p3.1.m1.1.1.3.2" stretchy="false" xref="S3.SS1.p3.1.m1.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.1.m1.1b"><apply id="S3.SS1.p3.1.m1.1.1.2.cmml" xref="S3.SS1.p3.1.m1.1.1.3"><csymbol cd="latexml" id="S3.SS1.p3.1.m1.1.1.2.1.cmml" xref="S3.SS1.p3.1.m1.1.1.3.1">ket</csymbol><apply id="S3.SS1.p3.1.m1.1.1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.1.m1.1.1.1.1.3.cmml" xref="S3.SS1.p3.1.m1.1.1.1.1">superscript</csymbol><apply id="S3.SS1.p3.1.m1.1.1.1.1.4.cmml" xref="S3.SS1.p3.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.1.m1.1.1.1.1.4.1.cmml" xref="S3.SS1.p3.1.m1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.p3.1.m1.1.1.1.1.4.2.cmml" xref="S3.SS1.p3.1.m1.1.1.1.1.4.2">𝜓</ci><ci id="S3.SS1.p3.1.m1.1.1.1.1.4.3.cmml" xref="S3.SS1.p3.1.m1.1.1.1.1.4.3">𝑠</ci></apply><interval closure="open" id="S3.SS1.p3.1.m1.1.1.1.1.2.2.3.cmml" xref="S3.SS1.p3.1.m1.1.1.1.1.2.2.4"><ci id="S3.SS1.p3.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1.1.1.1.1.1">𝑖</ci><ci id="S3.SS1.p3.1.m1.1.1.1.1.2.2.2.cmml" xref="S3.SS1.p3.1.m1.1.1.1.1.2.2.2">𝑇</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.1.m1.1c">\ket{\psi_{s}^{(i,T)}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.1.m1.1d">| start_ARG italic_ψ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i , italic_T ) end_POSTSUPERSCRIPT end_ARG ⟩</annotation></semantics></math> are then mid-spectrum states with area law entanglement entropy, making them quantum scars.</p> </div> <div class="ltx_para" id="S3.SS1.p4"> <p class="ltx_p" id="S3.SS1.p4.8">To get a deeper understanding of the structure of these scars, and to connect them to lego scars that have been constructed in the <math alttext="S=1/2" class="ltx_Math" display="inline" id="S3.SS1.p4.1.m1.1"><semantics id="S3.SS1.p4.1.m1.1a"><mrow id="S3.SS1.p4.1.m1.1.1" xref="S3.SS1.p4.1.m1.1.1.cmml"><mi id="S3.SS1.p4.1.m1.1.1.2" xref="S3.SS1.p4.1.m1.1.1.2.cmml">S</mi><mo id="S3.SS1.p4.1.m1.1.1.1" xref="S3.SS1.p4.1.m1.1.1.1.cmml">=</mo><mrow id="S3.SS1.p4.1.m1.1.1.3" xref="S3.SS1.p4.1.m1.1.1.3.cmml"><mn id="S3.SS1.p4.1.m1.1.1.3.2" xref="S3.SS1.p4.1.m1.1.1.3.2.cmml">1</mn><mo id="S3.SS1.p4.1.m1.1.1.3.1" xref="S3.SS1.p4.1.m1.1.1.3.1.cmml">/</mo><mn id="S3.SS1.p4.1.m1.1.1.3.3" xref="S3.SS1.p4.1.m1.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.1.m1.1b"><apply id="S3.SS1.p4.1.m1.1.1.cmml" xref="S3.SS1.p4.1.m1.1.1"><eq id="S3.SS1.p4.1.m1.1.1.1.cmml" xref="S3.SS1.p4.1.m1.1.1.1"></eq><ci id="S3.SS1.p4.1.m1.1.1.2.cmml" xref="S3.SS1.p4.1.m1.1.1.2">𝑆</ci><apply id="S3.SS1.p4.1.m1.1.1.3.cmml" xref="S3.SS1.p4.1.m1.1.1.3"><divide id="S3.SS1.p4.1.m1.1.1.3.1.cmml" xref="S3.SS1.p4.1.m1.1.1.3.1"></divide><cn id="S3.SS1.p4.1.m1.1.1.3.2.cmml" type="integer" xref="S3.SS1.p4.1.m1.1.1.3.2">1</cn><cn id="S3.SS1.p4.1.m1.1.1.3.3.cmml" type="integer" xref="S3.SS1.p4.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.1.m1.1c">S=1/2</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.1.m1.1d">italic_S = 1 / 2</annotation></semantics></math> QLM, it is useful to define the <em class="ltx_emph ltx_font_italic" id="S3.SS1.p4.8.1">dual basis</em> for integer spin. Height variables <math alttext="h_{n}\in\mathbb{Z}" class="ltx_Math" display="inline" id="S3.SS1.p4.2.m2.1"><semantics id="S3.SS1.p4.2.m2.1a"><mrow id="S3.SS1.p4.2.m2.1.1" xref="S3.SS1.p4.2.m2.1.1.cmml"><msub id="S3.SS1.p4.2.m2.1.1.2" xref="S3.SS1.p4.2.m2.1.1.2.cmml"><mi id="S3.SS1.p4.2.m2.1.1.2.2" xref="S3.SS1.p4.2.m2.1.1.2.2.cmml">h</mi><mi id="S3.SS1.p4.2.m2.1.1.2.3" xref="S3.SS1.p4.2.m2.1.1.2.3.cmml">n</mi></msub><mo id="S3.SS1.p4.2.m2.1.1.1" xref="S3.SS1.p4.2.m2.1.1.1.cmml">∈</mo><mi id="S3.SS1.p4.2.m2.1.1.3" xref="S3.SS1.p4.2.m2.1.1.3.cmml">ℤ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.2.m2.1b"><apply id="S3.SS1.p4.2.m2.1.1.cmml" xref="S3.SS1.p4.2.m2.1.1"><in id="S3.SS1.p4.2.m2.1.1.1.cmml" xref="S3.SS1.p4.2.m2.1.1.1"></in><apply id="S3.SS1.p4.2.m2.1.1.2.cmml" xref="S3.SS1.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p4.2.m2.1.1.2.1.cmml" xref="S3.SS1.p4.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS1.p4.2.m2.1.1.2.2.cmml" xref="S3.SS1.p4.2.m2.1.1.2.2">ℎ</ci><ci id="S3.SS1.p4.2.m2.1.1.2.3.cmml" xref="S3.SS1.p4.2.m2.1.1.2.3">𝑛</ci></apply><ci id="S3.SS1.p4.2.m2.1.1.3.cmml" xref="S3.SS1.p4.2.m2.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.2.m2.1c">h_{n}\in\mathbb{Z}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.2.m2.1d">italic_h start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ blackboard_Z</annotation></semantics></math> live at the center of plaquettes. The values of the vertical links are given by the difference of the height variables of the plaquettes to their right and left, while the values of the horizontal links are given by the difference of the height variables of the plaquettes to their bottom and top, i.e. <math alttext="E_{n1}=h_{n-\hat{2}}-h_{n}" class="ltx_Math" display="inline" id="S3.SS1.p4.3.m3.1"><semantics id="S3.SS1.p4.3.m3.1a"><mrow id="S3.SS1.p4.3.m3.1.1" xref="S3.SS1.p4.3.m3.1.1.cmml"><msub id="S3.SS1.p4.3.m3.1.1.2" xref="S3.SS1.p4.3.m3.1.1.2.cmml"><mi id="S3.SS1.p4.3.m3.1.1.2.2" xref="S3.SS1.p4.3.m3.1.1.2.2.cmml">E</mi><mrow id="S3.SS1.p4.3.m3.1.1.2.3" xref="S3.SS1.p4.3.m3.1.1.2.3.cmml"><mi id="S3.SS1.p4.3.m3.1.1.2.3.2" xref="S3.SS1.p4.3.m3.1.1.2.3.2.cmml">n</mi><mo id="S3.SS1.p4.3.m3.1.1.2.3.1" xref="S3.SS1.p4.3.m3.1.1.2.3.1.cmml">⁢</mo><mn id="S3.SS1.p4.3.m3.1.1.2.3.3" xref="S3.SS1.p4.3.m3.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.SS1.p4.3.m3.1.1.1" xref="S3.SS1.p4.3.m3.1.1.1.cmml">=</mo><mrow id="S3.SS1.p4.3.m3.1.1.3" xref="S3.SS1.p4.3.m3.1.1.3.cmml"><msub id="S3.SS1.p4.3.m3.1.1.3.2" xref="S3.SS1.p4.3.m3.1.1.3.2.cmml"><mi id="S3.SS1.p4.3.m3.1.1.3.2.2" xref="S3.SS1.p4.3.m3.1.1.3.2.2.cmml">h</mi><mrow id="S3.SS1.p4.3.m3.1.1.3.2.3" xref="S3.SS1.p4.3.m3.1.1.3.2.3.cmml"><mi id="S3.SS1.p4.3.m3.1.1.3.2.3.2" xref="S3.SS1.p4.3.m3.1.1.3.2.3.2.cmml">n</mi><mo id="S3.SS1.p4.3.m3.1.1.3.2.3.1" xref="S3.SS1.p4.3.m3.1.1.3.2.3.1.cmml">−</mo><mover accent="true" id="S3.SS1.p4.3.m3.1.1.3.2.3.3" xref="S3.SS1.p4.3.m3.1.1.3.2.3.3.cmml"><mn id="S3.SS1.p4.3.m3.1.1.3.2.3.3.2" xref="S3.SS1.p4.3.m3.1.1.3.2.3.3.2.cmml">2</mn><mo id="S3.SS1.p4.3.m3.1.1.3.2.3.3.1" xref="S3.SS1.p4.3.m3.1.1.3.2.3.3.1.cmml">^</mo></mover></mrow></msub><mo id="S3.SS1.p4.3.m3.1.1.3.1" xref="S3.SS1.p4.3.m3.1.1.3.1.cmml">−</mo><msub id="S3.SS1.p4.3.m3.1.1.3.3" xref="S3.SS1.p4.3.m3.1.1.3.3.cmml"><mi id="S3.SS1.p4.3.m3.1.1.3.3.2" xref="S3.SS1.p4.3.m3.1.1.3.3.2.cmml">h</mi><mi id="S3.SS1.p4.3.m3.1.1.3.3.3" xref="S3.SS1.p4.3.m3.1.1.3.3.3.cmml">n</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.3.m3.1b"><apply id="S3.SS1.p4.3.m3.1.1.cmml" xref="S3.SS1.p4.3.m3.1.1"><eq id="S3.SS1.p4.3.m3.1.1.1.cmml" xref="S3.SS1.p4.3.m3.1.1.1"></eq><apply id="S3.SS1.p4.3.m3.1.1.2.cmml" xref="S3.SS1.p4.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p4.3.m3.1.1.2.1.cmml" xref="S3.SS1.p4.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS1.p4.3.m3.1.1.2.2.cmml" xref="S3.SS1.p4.3.m3.1.1.2.2">𝐸</ci><apply id="S3.SS1.p4.3.m3.1.1.2.3.cmml" xref="S3.SS1.p4.3.m3.1.1.2.3"><times id="S3.SS1.p4.3.m3.1.1.2.3.1.cmml" xref="S3.SS1.p4.3.m3.1.1.2.3.1"></times><ci id="S3.SS1.p4.3.m3.1.1.2.3.2.cmml" xref="S3.SS1.p4.3.m3.1.1.2.3.2">𝑛</ci><cn id="S3.SS1.p4.3.m3.1.1.2.3.3.cmml" type="integer" xref="S3.SS1.p4.3.m3.1.1.2.3.3">1</cn></apply></apply><apply id="S3.SS1.p4.3.m3.1.1.3.cmml" xref="S3.SS1.p4.3.m3.1.1.3"><minus id="S3.SS1.p4.3.m3.1.1.3.1.cmml" xref="S3.SS1.p4.3.m3.1.1.3.1"></minus><apply id="S3.SS1.p4.3.m3.1.1.3.2.cmml" xref="S3.SS1.p4.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.p4.3.m3.1.1.3.2.1.cmml" xref="S3.SS1.p4.3.m3.1.1.3.2">subscript</csymbol><ci id="S3.SS1.p4.3.m3.1.1.3.2.2.cmml" xref="S3.SS1.p4.3.m3.1.1.3.2.2">ℎ</ci><apply id="S3.SS1.p4.3.m3.1.1.3.2.3.cmml" xref="S3.SS1.p4.3.m3.1.1.3.2.3"><minus id="S3.SS1.p4.3.m3.1.1.3.2.3.1.cmml" xref="S3.SS1.p4.3.m3.1.1.3.2.3.1"></minus><ci id="S3.SS1.p4.3.m3.1.1.3.2.3.2.cmml" xref="S3.SS1.p4.3.m3.1.1.3.2.3.2">𝑛</ci><apply id="S3.SS1.p4.3.m3.1.1.3.2.3.3.cmml" xref="S3.SS1.p4.3.m3.1.1.3.2.3.3"><ci id="S3.SS1.p4.3.m3.1.1.3.2.3.3.1.cmml" xref="S3.SS1.p4.3.m3.1.1.3.2.3.3.1">^</ci><cn id="S3.SS1.p4.3.m3.1.1.3.2.3.3.2.cmml" type="integer" xref="S3.SS1.p4.3.m3.1.1.3.2.3.3.2">2</cn></apply></apply></apply><apply id="S3.SS1.p4.3.m3.1.1.3.3.cmml" xref="S3.SS1.p4.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p4.3.m3.1.1.3.3.1.cmml" xref="S3.SS1.p4.3.m3.1.1.3.3">subscript</csymbol><ci id="S3.SS1.p4.3.m3.1.1.3.3.2.cmml" xref="S3.SS1.p4.3.m3.1.1.3.3.2">ℎ</ci><ci id="S3.SS1.p4.3.m3.1.1.3.3.3.cmml" xref="S3.SS1.p4.3.m3.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.3.m3.1c">E_{n1}=h_{n-\hat{2}}-h_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.3.m3.1d">italic_E start_POSTSUBSCRIPT italic_n 1 end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT italic_n - over^ start_ARG 2 end_ARG end_POSTSUBSCRIPT - italic_h start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="E_{n2}=h_{n}-h_{n-\hat{1}}" class="ltx_Math" display="inline" id="S3.SS1.p4.4.m4.1"><semantics id="S3.SS1.p4.4.m4.1a"><mrow id="S3.SS1.p4.4.m4.1.1" xref="S3.SS1.p4.4.m4.1.1.cmml"><msub id="S3.SS1.p4.4.m4.1.1.2" xref="S3.SS1.p4.4.m4.1.1.2.cmml"><mi id="S3.SS1.p4.4.m4.1.1.2.2" xref="S3.SS1.p4.4.m4.1.1.2.2.cmml">E</mi><mrow id="S3.SS1.p4.4.m4.1.1.2.3" xref="S3.SS1.p4.4.m4.1.1.2.3.cmml"><mi id="S3.SS1.p4.4.m4.1.1.2.3.2" xref="S3.SS1.p4.4.m4.1.1.2.3.2.cmml">n</mi><mo id="S3.SS1.p4.4.m4.1.1.2.3.1" xref="S3.SS1.p4.4.m4.1.1.2.3.1.cmml">⁢</mo><mn id="S3.SS1.p4.4.m4.1.1.2.3.3" xref="S3.SS1.p4.4.m4.1.1.2.3.3.cmml">2</mn></mrow></msub><mo id="S3.SS1.p4.4.m4.1.1.1" xref="S3.SS1.p4.4.m4.1.1.1.cmml">=</mo><mrow id="S3.SS1.p4.4.m4.1.1.3" xref="S3.SS1.p4.4.m4.1.1.3.cmml"><msub id="S3.SS1.p4.4.m4.1.1.3.2" xref="S3.SS1.p4.4.m4.1.1.3.2.cmml"><mi id="S3.SS1.p4.4.m4.1.1.3.2.2" xref="S3.SS1.p4.4.m4.1.1.3.2.2.cmml">h</mi><mi id="S3.SS1.p4.4.m4.1.1.3.2.3" xref="S3.SS1.p4.4.m4.1.1.3.2.3.cmml">n</mi></msub><mo id="S3.SS1.p4.4.m4.1.1.3.1" xref="S3.SS1.p4.4.m4.1.1.3.1.cmml">−</mo><msub id="S3.SS1.p4.4.m4.1.1.3.3" xref="S3.SS1.p4.4.m4.1.1.3.3.cmml"><mi id="S3.SS1.p4.4.m4.1.1.3.3.2" xref="S3.SS1.p4.4.m4.1.1.3.3.2.cmml">h</mi><mrow id="S3.SS1.p4.4.m4.1.1.3.3.3" xref="S3.SS1.p4.4.m4.1.1.3.3.3.cmml"><mi id="S3.SS1.p4.4.m4.1.1.3.3.3.2" xref="S3.SS1.p4.4.m4.1.1.3.3.3.2.cmml">n</mi><mo id="S3.SS1.p4.4.m4.1.1.3.3.3.1" xref="S3.SS1.p4.4.m4.1.1.3.3.3.1.cmml">−</mo><mover accent="true" id="S3.SS1.p4.4.m4.1.1.3.3.3.3" xref="S3.SS1.p4.4.m4.1.1.3.3.3.3.cmml"><mn id="S3.SS1.p4.4.m4.1.1.3.3.3.3.2" xref="S3.SS1.p4.4.m4.1.1.3.3.3.3.2.cmml">1</mn><mo id="S3.SS1.p4.4.m4.1.1.3.3.3.3.1" xref="S3.SS1.p4.4.m4.1.1.3.3.3.3.1.cmml">^</mo></mover></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.4.m4.1b"><apply id="S3.SS1.p4.4.m4.1.1.cmml" xref="S3.SS1.p4.4.m4.1.1"><eq id="S3.SS1.p4.4.m4.1.1.1.cmml" xref="S3.SS1.p4.4.m4.1.1.1"></eq><apply id="S3.SS1.p4.4.m4.1.1.2.cmml" xref="S3.SS1.p4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p4.4.m4.1.1.2.1.cmml" xref="S3.SS1.p4.4.m4.1.1.2">subscript</csymbol><ci id="S3.SS1.p4.4.m4.1.1.2.2.cmml" xref="S3.SS1.p4.4.m4.1.1.2.2">𝐸</ci><apply id="S3.SS1.p4.4.m4.1.1.2.3.cmml" xref="S3.SS1.p4.4.m4.1.1.2.3"><times id="S3.SS1.p4.4.m4.1.1.2.3.1.cmml" xref="S3.SS1.p4.4.m4.1.1.2.3.1"></times><ci id="S3.SS1.p4.4.m4.1.1.2.3.2.cmml" xref="S3.SS1.p4.4.m4.1.1.2.3.2">𝑛</ci><cn id="S3.SS1.p4.4.m4.1.1.2.3.3.cmml" type="integer" xref="S3.SS1.p4.4.m4.1.1.2.3.3">2</cn></apply></apply><apply id="S3.SS1.p4.4.m4.1.1.3.cmml" xref="S3.SS1.p4.4.m4.1.1.3"><minus id="S3.SS1.p4.4.m4.1.1.3.1.cmml" xref="S3.SS1.p4.4.m4.1.1.3.1"></minus><apply id="S3.SS1.p4.4.m4.1.1.3.2.cmml" xref="S3.SS1.p4.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.p4.4.m4.1.1.3.2.1.cmml" xref="S3.SS1.p4.4.m4.1.1.3.2">subscript</csymbol><ci id="S3.SS1.p4.4.m4.1.1.3.2.2.cmml" xref="S3.SS1.p4.4.m4.1.1.3.2.2">ℎ</ci><ci id="S3.SS1.p4.4.m4.1.1.3.2.3.cmml" xref="S3.SS1.p4.4.m4.1.1.3.2.3">𝑛</ci></apply><apply id="S3.SS1.p4.4.m4.1.1.3.3.cmml" xref="S3.SS1.p4.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p4.4.m4.1.1.3.3.1.cmml" xref="S3.SS1.p4.4.m4.1.1.3.3">subscript</csymbol><ci id="S3.SS1.p4.4.m4.1.1.3.3.2.cmml" xref="S3.SS1.p4.4.m4.1.1.3.3.2">ℎ</ci><apply id="S3.SS1.p4.4.m4.1.1.3.3.3.cmml" xref="S3.SS1.p4.4.m4.1.1.3.3.3"><minus id="S3.SS1.p4.4.m4.1.1.3.3.3.1.cmml" xref="S3.SS1.p4.4.m4.1.1.3.3.3.1"></minus><ci id="S3.SS1.p4.4.m4.1.1.3.3.3.2.cmml" xref="S3.SS1.p4.4.m4.1.1.3.3.3.2">𝑛</ci><apply id="S3.SS1.p4.4.m4.1.1.3.3.3.3.cmml" xref="S3.SS1.p4.4.m4.1.1.3.3.3.3"><ci id="S3.SS1.p4.4.m4.1.1.3.3.3.3.1.cmml" xref="S3.SS1.p4.4.m4.1.1.3.3.3.3.1">^</ci><cn id="S3.SS1.p4.4.m4.1.1.3.3.3.3.2.cmml" type="integer" xref="S3.SS1.p4.4.m4.1.1.3.3.3.3.2">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.4.m4.1c">E_{n2}=h_{n}-h_{n-\hat{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.4.m4.1d">italic_E start_POSTSUBSCRIPT italic_n 2 end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_h start_POSTSUBSCRIPT italic_n - over^ start_ARG 1 end_ARG end_POSTSUBSCRIPT</annotation></semantics></math>. Height variable configurations with <math alttext="|h_{n}-h_{n^{\prime}}|\leq S" class="ltx_Math" display="inline" id="S3.SS1.p4.5.m5.1"><semantics id="S3.SS1.p4.5.m5.1a"><mrow id="S3.SS1.p4.5.m5.1.1" xref="S3.SS1.p4.5.m5.1.1.cmml"><mrow id="S3.SS1.p4.5.m5.1.1.1.1" xref="S3.SS1.p4.5.m5.1.1.1.2.cmml"><mo id="S3.SS1.p4.5.m5.1.1.1.1.2" stretchy="false" xref="S3.SS1.p4.5.m5.1.1.1.2.1.cmml">|</mo><mrow id="S3.SS1.p4.5.m5.1.1.1.1.1" xref="S3.SS1.p4.5.m5.1.1.1.1.1.cmml"><msub id="S3.SS1.p4.5.m5.1.1.1.1.1.2" xref="S3.SS1.p4.5.m5.1.1.1.1.1.2.cmml"><mi id="S3.SS1.p4.5.m5.1.1.1.1.1.2.2" xref="S3.SS1.p4.5.m5.1.1.1.1.1.2.2.cmml">h</mi><mi id="S3.SS1.p4.5.m5.1.1.1.1.1.2.3" xref="S3.SS1.p4.5.m5.1.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S3.SS1.p4.5.m5.1.1.1.1.1.1" xref="S3.SS1.p4.5.m5.1.1.1.1.1.1.cmml">−</mo><msub id="S3.SS1.p4.5.m5.1.1.1.1.1.3" xref="S3.SS1.p4.5.m5.1.1.1.1.1.3.cmml"><mi id="S3.SS1.p4.5.m5.1.1.1.1.1.3.2" xref="S3.SS1.p4.5.m5.1.1.1.1.1.3.2.cmml">h</mi><msup id="S3.SS1.p4.5.m5.1.1.1.1.1.3.3" xref="S3.SS1.p4.5.m5.1.1.1.1.1.3.3.cmml"><mi id="S3.SS1.p4.5.m5.1.1.1.1.1.3.3.2" xref="S3.SS1.p4.5.m5.1.1.1.1.1.3.3.2.cmml">n</mi><mo id="S3.SS1.p4.5.m5.1.1.1.1.1.3.3.3" xref="S3.SS1.p4.5.m5.1.1.1.1.1.3.3.3.cmml">′</mo></msup></msub></mrow><mo id="S3.SS1.p4.5.m5.1.1.1.1.3" stretchy="false" xref="S3.SS1.p4.5.m5.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.SS1.p4.5.m5.1.1.2" xref="S3.SS1.p4.5.m5.1.1.2.cmml">≤</mo><mi id="S3.SS1.p4.5.m5.1.1.3" xref="S3.SS1.p4.5.m5.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.5.m5.1b"><apply id="S3.SS1.p4.5.m5.1.1.cmml" xref="S3.SS1.p4.5.m5.1.1"><leq id="S3.SS1.p4.5.m5.1.1.2.cmml" xref="S3.SS1.p4.5.m5.1.1.2"></leq><apply id="S3.SS1.p4.5.m5.1.1.1.2.cmml" xref="S3.SS1.p4.5.m5.1.1.1.1"><abs id="S3.SS1.p4.5.m5.1.1.1.2.1.cmml" xref="S3.SS1.p4.5.m5.1.1.1.1.2"></abs><apply id="S3.SS1.p4.5.m5.1.1.1.1.1.cmml" xref="S3.SS1.p4.5.m5.1.1.1.1.1"><minus id="S3.SS1.p4.5.m5.1.1.1.1.1.1.cmml" xref="S3.SS1.p4.5.m5.1.1.1.1.1.1"></minus><apply id="S3.SS1.p4.5.m5.1.1.1.1.1.2.cmml" xref="S3.SS1.p4.5.m5.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p4.5.m5.1.1.1.1.1.2.1.cmml" xref="S3.SS1.p4.5.m5.1.1.1.1.1.2">subscript</csymbol><ci id="S3.SS1.p4.5.m5.1.1.1.1.1.2.2.cmml" xref="S3.SS1.p4.5.m5.1.1.1.1.1.2.2">ℎ</ci><ci id="S3.SS1.p4.5.m5.1.1.1.1.1.2.3.cmml" xref="S3.SS1.p4.5.m5.1.1.1.1.1.2.3">𝑛</ci></apply><apply id="S3.SS1.p4.5.m5.1.1.1.1.1.3.cmml" xref="S3.SS1.p4.5.m5.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p4.5.m5.1.1.1.1.1.3.1.cmml" xref="S3.SS1.p4.5.m5.1.1.1.1.1.3">subscript</csymbol><ci id="S3.SS1.p4.5.m5.1.1.1.1.1.3.2.cmml" xref="S3.SS1.p4.5.m5.1.1.1.1.1.3.2">ℎ</ci><apply id="S3.SS1.p4.5.m5.1.1.1.1.1.3.3.cmml" xref="S3.SS1.p4.5.m5.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p4.5.m5.1.1.1.1.1.3.3.1.cmml" xref="S3.SS1.p4.5.m5.1.1.1.1.1.3.3">superscript</csymbol><ci id="S3.SS1.p4.5.m5.1.1.1.1.1.3.3.2.cmml" xref="S3.SS1.p4.5.m5.1.1.1.1.1.3.3.2">𝑛</ci><ci id="S3.SS1.p4.5.m5.1.1.1.1.1.3.3.3.cmml" xref="S3.SS1.p4.5.m5.1.1.1.1.1.3.3.3">′</ci></apply></apply></apply></apply><ci id="S3.SS1.p4.5.m5.1.1.3.cmml" xref="S3.SS1.p4.5.m5.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.5.m5.1c">|h_{n}-h_{n^{\prime}}|\leq S</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.5.m5.1d">| italic_h start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_h start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | ≤ italic_S</annotation></semantics></math> for all neighboring plaquettes <math alttext="n" class="ltx_Math" display="inline" id="S3.SS1.p4.6.m6.1"><semantics id="S3.SS1.p4.6.m6.1a"><mi id="S3.SS1.p4.6.m6.1.1" xref="S3.SS1.p4.6.m6.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.6.m6.1b"><ci id="S3.SS1.p4.6.m6.1.1.cmml" xref="S3.SS1.p4.6.m6.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.6.m6.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.6.m6.1d">italic_n</annotation></semantics></math>, <math alttext="n^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.p4.7.m7.1"><semantics id="S3.SS1.p4.7.m7.1a"><msup id="S3.SS1.p4.7.m7.1.1" xref="S3.SS1.p4.7.m7.1.1.cmml"><mi id="S3.SS1.p4.7.m7.1.1.2" xref="S3.SS1.p4.7.m7.1.1.2.cmml">n</mi><mo id="S3.SS1.p4.7.m7.1.1.3" xref="S3.SS1.p4.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.7.m7.1b"><apply id="S3.SS1.p4.7.m7.1.1.cmml" xref="S3.SS1.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS1.p4.7.m7.1.1.1.cmml" xref="S3.SS1.p4.7.m7.1.1">superscript</csymbol><ci id="S3.SS1.p4.7.m7.1.1.2.cmml" xref="S3.SS1.p4.7.m7.1.1.2">𝑛</ci><ci id="S3.SS1.p4.7.m7.1.1.3.cmml" xref="S3.SS1.p4.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.7.m7.1c">n^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.7.m7.1d">italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> represent a valid spin-<math alttext="S" class="ltx_Math" display="inline" id="S3.SS1.p4.8.m8.1"><semantics id="S3.SS1.p4.8.m8.1a"><mi id="S3.SS1.p4.8.m8.1.1" xref="S3.SS1.p4.8.m8.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.8.m8.1b"><ci id="S3.SS1.p4.8.m8.1.1.cmml" xref="S3.SS1.p4.8.m8.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.8.m8.1c">S</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.8.m8.1d">italic_S</annotation></semantics></math> link configuration.</p> </div> <div class="ltx_para" id="S3.SS1.p5"> <p class="ltx_p" id="S3.SS1.p5.3">The height variable representation with periodic boundary conditions is not unique, since adding a constant to all height variables gives the same state. If there is an open boundary condition in at least one direction, we set all height variables at one open boundary to have the value of the neighboring boundary link. In our construction, we always choose the top boundary, which makes the representation unique. The kinetic operators <math alttext="H^{+}_{n}" class="ltx_Math" display="inline" id="S3.SS1.p5.1.m1.1"><semantics id="S3.SS1.p5.1.m1.1a"><msubsup id="S3.SS1.p5.1.m1.1.1" xref="S3.SS1.p5.1.m1.1.1.cmml"><mi id="S3.SS1.p5.1.m1.1.1.2.2" xref="S3.SS1.p5.1.m1.1.1.2.2.cmml">H</mi><mi id="S3.SS1.p5.1.m1.1.1.3" xref="S3.SS1.p5.1.m1.1.1.3.cmml">n</mi><mo id="S3.SS1.p5.1.m1.1.1.2.3" xref="S3.SS1.p5.1.m1.1.1.2.3.cmml">+</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.1.m1.1b"><apply id="S3.SS1.p5.1.m1.1.1.cmml" xref="S3.SS1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p5.1.m1.1.1.1.cmml" xref="S3.SS1.p5.1.m1.1.1">subscript</csymbol><apply id="S3.SS1.p5.1.m1.1.1.2.cmml" xref="S3.SS1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p5.1.m1.1.1.2.1.cmml" xref="S3.SS1.p5.1.m1.1.1">superscript</csymbol><ci id="S3.SS1.p5.1.m1.1.1.2.2.cmml" xref="S3.SS1.p5.1.m1.1.1.2.2">𝐻</ci><plus id="S3.SS1.p5.1.m1.1.1.2.3.cmml" xref="S3.SS1.p5.1.m1.1.1.2.3"></plus></apply><ci id="S3.SS1.p5.1.m1.1.1.3.cmml" xref="S3.SS1.p5.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.1.m1.1c">H^{+}_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.1.m1.1d">italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="H^{-}_{n}" class="ltx_Math" display="inline" id="S3.SS1.p5.2.m2.1"><semantics id="S3.SS1.p5.2.m2.1a"><msubsup id="S3.SS1.p5.2.m2.1.1" xref="S3.SS1.p5.2.m2.1.1.cmml"><mi id="S3.SS1.p5.2.m2.1.1.2.2" xref="S3.SS1.p5.2.m2.1.1.2.2.cmml">H</mi><mi id="S3.SS1.p5.2.m2.1.1.3" xref="S3.SS1.p5.2.m2.1.1.3.cmml">n</mi><mo id="S3.SS1.p5.2.m2.1.1.2.3" xref="S3.SS1.p5.2.m2.1.1.2.3.cmml">−</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.2.m2.1b"><apply id="S3.SS1.p5.2.m2.1.1.cmml" xref="S3.SS1.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.p5.2.m2.1.1.1.cmml" xref="S3.SS1.p5.2.m2.1.1">subscript</csymbol><apply id="S3.SS1.p5.2.m2.1.1.2.cmml" xref="S3.SS1.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.p5.2.m2.1.1.2.1.cmml" xref="S3.SS1.p5.2.m2.1.1">superscript</csymbol><ci id="S3.SS1.p5.2.m2.1.1.2.2.cmml" xref="S3.SS1.p5.2.m2.1.1.2.2">𝐻</ci><minus id="S3.SS1.p5.2.m2.1.1.2.3.cmml" xref="S3.SS1.p5.2.m2.1.1.2.3"></minus></apply><ci id="S3.SS1.p5.2.m2.1.1.3.cmml" xref="S3.SS1.p5.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.2.m2.1c">H^{-}_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.2.m2.1d">italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, are raising/lowering operators for the height variable <math alttext="h_{n}" class="ltx_Math" display="inline" id="S3.SS1.p5.3.m3.1"><semantics id="S3.SS1.p5.3.m3.1a"><msub id="S3.SS1.p5.3.m3.1.1" xref="S3.SS1.p5.3.m3.1.1.cmml"><mi id="S3.SS1.p5.3.m3.1.1.2" xref="S3.SS1.p5.3.m3.1.1.2.cmml">h</mi><mi id="S3.SS1.p5.3.m3.1.1.3" xref="S3.SS1.p5.3.m3.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.3.m3.1b"><apply id="S3.SS1.p5.3.m3.1.1.cmml" xref="S3.SS1.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p5.3.m3.1.1.1.cmml" xref="S3.SS1.p5.3.m3.1.1">subscript</csymbol><ci id="S3.SS1.p5.3.m3.1.1.2.cmml" xref="S3.SS1.p5.3.m3.1.1.2">ℎ</ci><ci id="S3.SS1.p5.3.m3.1.1.3.cmml" xref="S3.SS1.p5.3.m3.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.3.m3.1c">h_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.3.m3.1d">italic_h start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.p6"> <p class="ltx_p" id="S3.SS1.p6.1">In the dual representation, the state (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S5.E24" title="In V A 4×4 Scar for the 𝐸² potential ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">24</span></a>) in a <math alttext="2\times 1" class="ltx_Math" display="inline" id="S3.SS1.p6.1.m1.1"><semantics id="S3.SS1.p6.1.m1.1a"><mrow id="S3.SS1.p6.1.m1.1.1" xref="S3.SS1.p6.1.m1.1.1.cmml"><mn id="S3.SS1.p6.1.m1.1.1.2" xref="S3.SS1.p6.1.m1.1.1.2.cmml">2</mn><mo id="S3.SS1.p6.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.p6.1.m1.1.1.1.cmml">×</mo><mn id="S3.SS1.p6.1.m1.1.1.3" xref="S3.SS1.p6.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.1.m1.1b"><apply id="S3.SS1.p6.1.m1.1.1.cmml" xref="S3.SS1.p6.1.m1.1.1"><times id="S3.SS1.p6.1.m1.1.1.1.cmml" xref="S3.SS1.p6.1.m1.1.1.1"></times><cn id="S3.SS1.p6.1.m1.1.1.2.cmml" type="integer" xref="S3.SS1.p6.1.m1.1.1.2">2</cn><cn id="S3.SS1.p6.1.m1.1.1.3.cmml" type="integer" xref="S3.SS1.p6.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.1.m1.1c">2\times 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.1.m1.1d">2 × 1</annotation></semantics></math> plaquette system is</p> <table class="ltx_equation ltx_eqn_table" id="S3.E11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\ket{\psi_{i}}=\frac{1}{\sqrt{S+1}}\sum_{k=0}^{S}(-1)^{k}\ket{(i-S+k)\ (i-k)}" class="ltx_Math" display="block" id="S3.E11.m1.3"><semantics id="S3.E11.m1.3a"><mrow id="S3.E11.m1.3.3" xref="S3.E11.m1.3.3.cmml"><mrow id="S3.E11.m1.1.1.3" xref="S3.E11.m1.1.1.2.cmml"><mo id="S3.E11.m1.1.1.3.1" stretchy="false" xref="S3.E11.m1.1.1.2.1.cmml">|</mo><msub id="S3.E11.m1.1.1.1.1" xref="S3.E11.m1.1.1.1.1.cmml"><mi id="S3.E11.m1.1.1.1.1.2" xref="S3.E11.m1.1.1.1.1.2.cmml">ψ</mi><mi id="S3.E11.m1.1.1.1.1.3" xref="S3.E11.m1.1.1.1.1.3.cmml">i</mi></msub><mo 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xref="S3.SS1.p6.2.m1.3.3.1.1.1.1">subscript</csymbol><apply id="S3.SS1.p6.2.m1.3.3.1.1.1.1.1.2.cmml" xref="S3.SS1.p6.2.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p6.2.m1.3.3.1.1.1.1.1.2.1.cmml" xref="S3.SS1.p6.2.m1.3.3.1.1.1.1">superscript</csymbol><ci id="S3.SS1.p6.2.m1.3.3.1.1.1.1.1.2.2.cmml" xref="S3.SS1.p6.2.m1.3.3.1.1.1.1.1.2.2">𝐻</ci><plus id="S3.SS1.p6.2.m1.3.3.1.1.1.1.1.2.3.cmml" xref="S3.SS1.p6.2.m1.3.3.1.1.1.1.1.2.3"></plus></apply><cn id="S3.SS1.p6.2.m1.3.3.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.p6.2.m1.3.3.1.1.1.1.1.3">1</cn></apply><apply id="S3.SS1.p6.2.m1.3.3.1.1.3.cmml" xref="S3.SS1.p6.2.m1.3.3.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p6.2.m1.3.3.1.1.3.1.cmml" xref="S3.SS1.p6.2.m1.3.3.1.1.3">subscript</csymbol><ci id="S3.SS1.p6.2.m1.3.3.1.1.3.2.cmml" xref="S3.SS1.p6.2.m1.3.3.1.1.3.2">ℎ</ci><cn id="S3.SS1.p6.2.m1.3.3.1.1.3.3.cmml" type="integer" xref="S3.SS1.p6.2.m1.3.3.1.1.3.3">1</cn></apply></apply><apply id="S3.SS1.p6.2.m1.4.4.2.2.cmml" xref="S3.SS1.p6.2.m1.4.4.2.2"><csymbol cd="ambiguous" id="S3.SS1.p6.2.m1.4.4.2.2.2.cmml" xref="S3.SS1.p6.2.m1.4.4.2.2">superscript</csymbol><apply id="S3.SS1.p6.2.m1.4.4.2.2.1.1.1.cmml" xref="S3.SS1.p6.2.m1.4.4.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS1.p6.2.m1.4.4.2.2.1.1.1.1.cmml" xref="S3.SS1.p6.2.m1.4.4.2.2.1.1">subscript</csymbol><apply id="S3.SS1.p6.2.m1.4.4.2.2.1.1.1.2.cmml" xref="S3.SS1.p6.2.m1.4.4.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS1.p6.2.m1.4.4.2.2.1.1.1.2.1.cmml" xref="S3.SS1.p6.2.m1.4.4.2.2.1.1">superscript</csymbol><ci id="S3.SS1.p6.2.m1.4.4.2.2.1.1.1.2.2.cmml" xref="S3.SS1.p6.2.m1.4.4.2.2.1.1.1.2.2">𝐻</ci><plus id="S3.SS1.p6.2.m1.4.4.2.2.1.1.1.2.3.cmml" xref="S3.SS1.p6.2.m1.4.4.2.2.1.1.1.2.3"></plus></apply><cn id="S3.SS1.p6.2.m1.4.4.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS1.p6.2.m1.4.4.2.2.1.1.1.3">2</cn></apply><apply id="S3.SS1.p6.2.m1.4.4.2.2.3.cmml" xref="S3.SS1.p6.2.m1.4.4.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.p6.2.m1.4.4.2.2.3.1.cmml" xref="S3.SS1.p6.2.m1.4.4.2.2.3">subscript</csymbol><ci id="S3.SS1.p6.2.m1.4.4.2.2.3.2.cmml" xref="S3.SS1.p6.2.m1.4.4.2.2.3.2">ℎ</ci><cn id="S3.SS1.p6.2.m1.4.4.2.2.3.3.cmml" type="integer" xref="S3.SS1.p6.2.m1.4.4.2.2.3.3">2</cn></apply></apply><apply id="S3.SS1.p6.2.m1.2.2.2.cmml" xref="S3.SS1.p6.2.m1.2.2.3"><csymbol cd="latexml" id="S3.SS1.p6.2.m1.2.2.2.1.cmml" xref="S3.SS1.p6.2.m1.2.2.3.1">ket</csymbol><cn id="S3.SS1.p6.2.m1.2.2.1.1.cmml" type="integer" xref="S3.SS1.p6.2.m1.2.2.1.1">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.2.m1.4c">\ket{h_{1}\ h_{2}}=(H^{+}_{1})^{h_{1}}(H^{+}_{2})^{h_{2}}\ket{\mathbf{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.2.m1.4d">| start_ARG italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ⟩ = ( italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_h start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT | start_ARG bold_0 end_ARG ⟩</annotation></semantics></math>. An illustration of this tile and the terms that cancel each other in the sum <math alttext="K\ket{\psi_{i}}" class="ltx_Math" display="inline" id="S3.SS1.p6.3.m2.1"><semantics id="S3.SS1.p6.3.m2.1a"><mrow id="S3.SS1.p6.3.m2.1.2" xref="S3.SS1.p6.3.m2.1.2.cmml"><mi id="S3.SS1.p6.3.m2.1.2.2" xref="S3.SS1.p6.3.m2.1.2.2.cmml">K</mi><mo id="S3.SS1.p6.3.m2.1.2.1" xref="S3.SS1.p6.3.m2.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.p6.3.m2.1.1.3" xref="S3.SS1.p6.3.m2.1.1.2.cmml"><mo id="S3.SS1.p6.3.m2.1.1.3.1" stretchy="false" xref="S3.SS1.p6.3.m2.1.1.2.1.cmml">|</mo><msub id="S3.SS1.p6.3.m2.1.1.1.1" xref="S3.SS1.p6.3.m2.1.1.1.1.cmml"><mi id="S3.SS1.p6.3.m2.1.1.1.1.2" xref="S3.SS1.p6.3.m2.1.1.1.1.2.cmml">ψ</mi><mi id="S3.SS1.p6.3.m2.1.1.1.1.3" xref="S3.SS1.p6.3.m2.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS1.p6.3.m2.1.1.3.2" stretchy="false" xref="S3.SS1.p6.3.m2.1.1.2.1.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.3.m2.1b"><apply id="S3.SS1.p6.3.m2.1.2.cmml" xref="S3.SS1.p6.3.m2.1.2"><times id="S3.SS1.p6.3.m2.1.2.1.cmml" xref="S3.SS1.p6.3.m2.1.2.1"></times><ci id="S3.SS1.p6.3.m2.1.2.2.cmml" xref="S3.SS1.p6.3.m2.1.2.2">𝐾</ci><apply id="S3.SS1.p6.3.m2.1.1.2.cmml" xref="S3.SS1.p6.3.m2.1.1.3"><csymbol cd="latexml" id="S3.SS1.p6.3.m2.1.1.2.1.cmml" xref="S3.SS1.p6.3.m2.1.1.3.1">ket</csymbol><apply id="S3.SS1.p6.3.m2.1.1.1.1.cmml" xref="S3.SS1.p6.3.m2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p6.3.m2.1.1.1.1.1.cmml" xref="S3.SS1.p6.3.m2.1.1.1.1">subscript</csymbol><ci id="S3.SS1.p6.3.m2.1.1.1.1.2.cmml" xref="S3.SS1.p6.3.m2.1.1.1.1.2">𝜓</ci><ci id="S3.SS1.p6.3.m2.1.1.1.1.3.cmml" xref="S3.SS1.p6.3.m2.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.3.m2.1c">K\ket{\psi_{i}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.3.m2.1d">italic_K | start_ARG italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> for <math alttext="S=1" class="ltx_Math" display="inline" id="S3.SS1.p6.4.m3.1"><semantics id="S3.SS1.p6.4.m3.1a"><mrow id="S3.SS1.p6.4.m3.1.1" xref="S3.SS1.p6.4.m3.1.1.cmml"><mi id="S3.SS1.p6.4.m3.1.1.2" xref="S3.SS1.p6.4.m3.1.1.2.cmml">S</mi><mo id="S3.SS1.p6.4.m3.1.1.1" xref="S3.SS1.p6.4.m3.1.1.1.cmml">=</mo><mn id="S3.SS1.p6.4.m3.1.1.3" xref="S3.SS1.p6.4.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.4.m3.1b"><apply id="S3.SS1.p6.4.m3.1.1.cmml" xref="S3.SS1.p6.4.m3.1.1"><eq id="S3.SS1.p6.4.m3.1.1.1.cmml" xref="S3.SS1.p6.4.m3.1.1.1"></eq><ci id="S3.SS1.p6.4.m3.1.1.2.cmml" xref="S3.SS1.p6.4.m3.1.1.2">𝑆</ci><cn id="S3.SS1.p6.4.m3.1.1.3.cmml" type="integer" xref="S3.SS1.p6.4.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.4.m3.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.4.m3.1d">italic_S = 1</annotation></semantics></math> can be found in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.F3" title="Figure 3 ‣ III.1 QMBS in TLM With Arbitrary Integer Spin ‣ III Zero-mode scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">3</span></a>.</p> </div> <figure class="ltx_figure" id="S3.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="327" id="S3.F3.g1" src="extracted/5828746/images/two_pq.png" width="419"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>Schematic depiction of the scar building block for <math alttext="S=1" class="ltx_Math" display="inline" id="S3.F3.4.m1.1"><semantics id="S3.F3.4.m1.1b"><mrow id="S3.F3.4.m1.1.1" xref="S3.F3.4.m1.1.1.cmml"><mi id="S3.F3.4.m1.1.1.2" xref="S3.F3.4.m1.1.1.2.cmml">S</mi><mo id="S3.F3.4.m1.1.1.1" xref="S3.F3.4.m1.1.1.1.cmml">=</mo><mn id="S3.F3.4.m1.1.1.3" xref="S3.F3.4.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F3.4.m1.1c"><apply id="S3.F3.4.m1.1.1.cmml" xref="S3.F3.4.m1.1.1"><eq id="S3.F3.4.m1.1.1.1.cmml" xref="S3.F3.4.m1.1.1.1"></eq><ci id="S3.F3.4.m1.1.1.2.cmml" xref="S3.F3.4.m1.1.1.2">𝑆</ci><cn id="S3.F3.4.m1.1.1.3.cmml" type="integer" xref="S3.F3.4.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.4.m1.1d">S=1</annotation><annotation encoding="application/x-llamapun" id="S3.F3.4.m1.1e">italic_S = 1</annotation></semantics></math>. The tiles depicted with solid lines make up the scar state. The operators <math alttext="H^{\pm}_{n}" class="ltx_Math" display="inline" id="S3.F3.5.m2.1"><semantics id="S3.F3.5.m2.1b"><msubsup id="S3.F3.5.m2.1.1" xref="S3.F3.5.m2.1.1.cmml"><mi id="S3.F3.5.m2.1.1.2.2" xref="S3.F3.5.m2.1.1.2.2.cmml">H</mi><mi id="S3.F3.5.m2.1.1.3" xref="S3.F3.5.m2.1.1.3.cmml">n</mi><mo id="S3.F3.5.m2.1.1.2.3" xref="S3.F3.5.m2.1.1.2.3.cmml">±</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.F3.5.m2.1c"><apply id="S3.F3.5.m2.1.1.cmml" xref="S3.F3.5.m2.1.1"><csymbol cd="ambiguous" id="S3.F3.5.m2.1.1.1.cmml" xref="S3.F3.5.m2.1.1">subscript</csymbol><apply id="S3.F3.5.m2.1.1.2.cmml" xref="S3.F3.5.m2.1.1"><csymbol cd="ambiguous" id="S3.F3.5.m2.1.1.2.1.cmml" xref="S3.F3.5.m2.1.1">superscript</csymbol><ci id="S3.F3.5.m2.1.1.2.2.cmml" xref="S3.F3.5.m2.1.1.2.2">𝐻</ci><csymbol cd="latexml" id="S3.F3.5.m2.1.1.2.3.cmml" xref="S3.F3.5.m2.1.1.2.3">plus-or-minus</csymbol></apply><ci id="S3.F3.5.m2.1.1.3.cmml" xref="S3.F3.5.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.5.m2.1d">H^{\pm}_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.F3.5.m2.1e">italic_H start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> raise or lower the height variables in plaquette <math alttext="n" class="ltx_Math" display="inline" id="S3.F3.6.m3.1"><semantics id="S3.F3.6.m3.1b"><mi id="S3.F3.6.m3.1.1" xref="S3.F3.6.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.F3.6.m3.1c"><ci id="S3.F3.6.m3.1.1.cmml" xref="S3.F3.6.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F3.6.m3.1d">n</annotation><annotation encoding="application/x-llamapun" id="S3.F3.6.m3.1e">italic_n</annotation></semantics></math>. They either annihilate the state or generate another state, depicted with dashed lines. This is canceled by applying the other type of operator on the second state.</figcaption> </figure> <div class="ltx_para" id="S3.SS1.p7"> <p class="ltx_p" id="S3.SS1.p7.3">All states that contribute in (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S5.E24" title="In V A 4×4 Scar for the 𝐸² potential ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">24</span></a>) have a height variable representation where there is one state out of the sum (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.E11" title="In III.1 QMBS in TLM With Arbitrary Integer Spin ‣ III Zero-mode scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">11</span></a>) on all tiles of the tiling <math alttext="T" class="ltx_Math" display="inline" id="S3.SS1.p7.1.m1.1"><semantics id="S3.SS1.p7.1.m1.1a"><mi id="S3.SS1.p7.1.m1.1.1" xref="S3.SS1.p7.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p7.1.m1.1b"><ci id="S3.SS1.p7.1.m1.1.1.cmml" xref="S3.SS1.p7.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p7.1.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p7.1.m1.1d">italic_T</annotation></semantics></math>. The state sums over all <math alttext="(S+1)^{\left|T\right|}" class="ltx_Math" display="inline" id="S3.SS1.p7.2.m2.2"><semantics id="S3.SS1.p7.2.m2.2a"><msup id="S3.SS1.p7.2.m2.2.2" xref="S3.SS1.p7.2.m2.2.2.cmml"><mrow id="S3.SS1.p7.2.m2.2.2.1.1" xref="S3.SS1.p7.2.m2.2.2.1.1.1.cmml"><mo id="S3.SS1.p7.2.m2.2.2.1.1.2" stretchy="false" xref="S3.SS1.p7.2.m2.2.2.1.1.1.cmml">(</mo><mrow id="S3.SS1.p7.2.m2.2.2.1.1.1" xref="S3.SS1.p7.2.m2.2.2.1.1.1.cmml"><mi id="S3.SS1.p7.2.m2.2.2.1.1.1.2" xref="S3.SS1.p7.2.m2.2.2.1.1.1.2.cmml">S</mi><mo id="S3.SS1.p7.2.m2.2.2.1.1.1.1" xref="S3.SS1.p7.2.m2.2.2.1.1.1.1.cmml">+</mo><mn id="S3.SS1.p7.2.m2.2.2.1.1.1.3" xref="S3.SS1.p7.2.m2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S3.SS1.p7.2.m2.2.2.1.1.3" stretchy="false" xref="S3.SS1.p7.2.m2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS1.p7.2.m2.1.1.1.3" xref="S3.SS1.p7.2.m2.1.1.1.2.cmml"><mo id="S3.SS1.p7.2.m2.1.1.1.3.1" xref="S3.SS1.p7.2.m2.1.1.1.2.1.cmml">|</mo><mi id="S3.SS1.p7.2.m2.1.1.1.1" xref="S3.SS1.p7.2.m2.1.1.1.1.cmml">T</mi><mo id="S3.SS1.p7.2.m2.1.1.1.3.2" xref="S3.SS1.p7.2.m2.1.1.1.2.1.cmml">|</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p7.2.m2.2b"><apply id="S3.SS1.p7.2.m2.2.2.cmml" xref="S3.SS1.p7.2.m2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p7.2.m2.2.2.2.cmml" xref="S3.SS1.p7.2.m2.2.2">superscript</csymbol><apply id="S3.SS1.p7.2.m2.2.2.1.1.1.cmml" xref="S3.SS1.p7.2.m2.2.2.1.1"><plus id="S3.SS1.p7.2.m2.2.2.1.1.1.1.cmml" xref="S3.SS1.p7.2.m2.2.2.1.1.1.1"></plus><ci id="S3.SS1.p7.2.m2.2.2.1.1.1.2.cmml" xref="S3.SS1.p7.2.m2.2.2.1.1.1.2">𝑆</ci><cn id="S3.SS1.p7.2.m2.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS1.p7.2.m2.2.2.1.1.1.3">1</cn></apply><apply id="S3.SS1.p7.2.m2.1.1.1.2.cmml" xref="S3.SS1.p7.2.m2.1.1.1.3"><abs id="S3.SS1.p7.2.m2.1.1.1.2.1.cmml" xref="S3.SS1.p7.2.m2.1.1.1.3.1"></abs><ci id="S3.SS1.p7.2.m2.1.1.1.1.cmml" xref="S3.SS1.p7.2.m2.1.1.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p7.2.m2.2c">(S+1)^{\left|T\right|}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p7.2.m2.2d">( italic_S + 1 ) start_POSTSUPERSCRIPT | italic_T | end_POSTSUPERSCRIPT</annotation></semantics></math> combinations of these states. A pictorial representation of this tiling is shown on the right-hand side of Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.F2" title="Figure 2 ‣ III.1 QMBS in TLM With Arbitrary Integer Spin ‣ III Zero-mode scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">2</span></a>. This tiling structure is similar to the lego and sub-lattice scars constructed for the <math alttext="S=1/2" class="ltx_Math" display="inline" id="S3.SS1.p7.3.m3.1"><semantics id="S3.SS1.p7.3.m3.1a"><mrow id="S3.SS1.p7.3.m3.1.1" xref="S3.SS1.p7.3.m3.1.1.cmml"><mi id="S3.SS1.p7.3.m3.1.1.2" xref="S3.SS1.p7.3.m3.1.1.2.cmml">S</mi><mo id="S3.SS1.p7.3.m3.1.1.1" xref="S3.SS1.p7.3.m3.1.1.1.cmml">=</mo><mrow id="S3.SS1.p7.3.m3.1.1.3" xref="S3.SS1.p7.3.m3.1.1.3.cmml"><mn id="S3.SS1.p7.3.m3.1.1.3.2" xref="S3.SS1.p7.3.m3.1.1.3.2.cmml">1</mn><mo id="S3.SS1.p7.3.m3.1.1.3.1" xref="S3.SS1.p7.3.m3.1.1.3.1.cmml">/</mo><mn id="S3.SS1.p7.3.m3.1.1.3.3" xref="S3.SS1.p7.3.m3.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p7.3.m3.1b"><apply id="S3.SS1.p7.3.m3.1.1.cmml" xref="S3.SS1.p7.3.m3.1.1"><eq id="S3.SS1.p7.3.m3.1.1.1.cmml" xref="S3.SS1.p7.3.m3.1.1.1"></eq><ci id="S3.SS1.p7.3.m3.1.1.2.cmml" xref="S3.SS1.p7.3.m3.1.1.2">𝑆</ci><apply id="S3.SS1.p7.3.m3.1.1.3.cmml" xref="S3.SS1.p7.3.m3.1.1.3"><divide id="S3.SS1.p7.3.m3.1.1.3.1.cmml" xref="S3.SS1.p7.3.m3.1.1.3.1"></divide><cn id="S3.SS1.p7.3.m3.1.1.3.2.cmml" type="integer" xref="S3.SS1.p7.3.m3.1.1.3.2">1</cn><cn id="S3.SS1.p7.3.m3.1.1.3.3.cmml" type="integer" xref="S3.SS1.p7.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p7.3.m3.1c">S=1/2</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p7.3.m3.1d">italic_S = 1 / 2</annotation></semantics></math> case <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib32" title="">32</a>]</cite>, and allows for the construction of an exponential number of scars for arbitrary truncated link models. A formal definition of this tiling, in the form of a tiling product, can be found in the Supplementary Material <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib56" title="">56</a>]</cite>. There, we also describe a more general framework to construct more low-entropy zero-modes in TLMs.</p> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">III.2 </span>Beyond Zero-Mode Building Blocks</h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.3">The above-described mechanism follows two basic ingredients. It constructs zero-modes from a basic tile made of two plaquettes joined together in one link. The full state is obtained by tiling these building blocks together. In this section, we demonstrate that it is possible to form zero-modes with different structures. This demonstrates that even though we have constructed numerous scars for arbitrary spin, there are additional routes to obtain other types of zero-mode scars. We provide explicit constructions for the <math alttext="S=1" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><mrow id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml"><mi id="S3.SS2.p1.1.m1.1.1.2" xref="S3.SS2.p1.1.m1.1.1.2.cmml">S</mi><mo id="S3.SS2.p1.1.m1.1.1.1" xref="S3.SS2.p1.1.m1.1.1.1.cmml">=</mo><mn id="S3.SS2.p1.1.m1.1.1.3" xref="S3.SS2.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><apply id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1"><eq id="S3.SS2.p1.1.m1.1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1.1"></eq><ci id="S3.SS2.p1.1.m1.1.1.2.cmml" xref="S3.SS2.p1.1.m1.1.1.2">𝑆</ci><cn id="S3.SS2.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.SS2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">italic_S = 1</annotation></semantics></math> case. The two examples described here exhibit a sublattice structure similar to what has been found for spin-<math alttext="1/2" class="ltx_Math" display="inline" id="S3.SS2.p1.2.m2.1"><semantics id="S3.SS2.p1.2.m2.1a"><mrow id="S3.SS2.p1.2.m2.1.1" xref="S3.SS2.p1.2.m2.1.1.cmml"><mn id="S3.SS2.p1.2.m2.1.1.2" xref="S3.SS2.p1.2.m2.1.1.2.cmml">1</mn><mo id="S3.SS2.p1.2.m2.1.1.1" xref="S3.SS2.p1.2.m2.1.1.1.cmml">/</mo><mn id="S3.SS2.p1.2.m2.1.1.3" xref="S3.SS2.p1.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.2.m2.1b"><apply id="S3.SS2.p1.2.m2.1.1.cmml" xref="S3.SS2.p1.2.m2.1.1"><divide id="S3.SS2.p1.2.m2.1.1.1.cmml" xref="S3.SS2.p1.2.m2.1.1.1"></divide><cn id="S3.SS2.p1.2.m2.1.1.2.cmml" type="integer" xref="S3.SS2.p1.2.m2.1.1.2">1</cn><cn id="S3.SS2.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.SS2.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.2.m2.1c">1/2</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.2.m2.1d">1 / 2</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib32" title="">32</a>]</cite>. One can be constructed by making use of diagonal tiles, while the other does not exhibit any tiling structure. A third example, that we have found for the specific case of a <math alttext="4\times 4" class="ltx_Math" display="inline" id="S3.SS2.p1.3.m3.1"><semantics id="S3.SS2.p1.3.m3.1a"><mrow id="S3.SS2.p1.3.m3.1.1" xref="S3.SS2.p1.3.m3.1.1.cmml"><mn id="S3.SS2.p1.3.m3.1.1.2" xref="S3.SS2.p1.3.m3.1.1.2.cmml">4</mn><mo id="S3.SS2.p1.3.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p1.3.m3.1.1.1.cmml">×</mo><mn id="S3.SS2.p1.3.m3.1.1.3" xref="S3.SS2.p1.3.m3.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.3.m3.1b"><apply id="S3.SS2.p1.3.m3.1.1.cmml" xref="S3.SS2.p1.3.m3.1.1"><times id="S3.SS2.p1.3.m3.1.1.1.cmml" xref="S3.SS2.p1.3.m3.1.1.1"></times><cn id="S3.SS2.p1.3.m3.1.1.2.cmml" type="integer" xref="S3.SS2.p1.3.m3.1.1.2">4</cn><cn id="S3.SS2.p1.3.m3.1.1.3.cmml" type="integer" xref="S3.SS2.p1.3.m3.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.3.m3.1c">4\times 4</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.3.m3.1d">4 × 4</annotation></semantics></math> volume, can be found in the Supplemental Material <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib56" title="">56</a>]</cite>. To the best of our knowledge, that specific construction is not generalizable for arbitrary volumes.</p> </div> <section class="ltx_paragraph" id="S3.SS2.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Diagonal Tiling</h4> <div class="ltx_para" id="S3.SS2.SSS0.Px1.p1"> <p class="ltx_p" id="S3.SS2.SSS0.Px1.p1.1">We consider a scar with the generic form previously described in (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S5.E24" title="In V A 4×4 Scar for the 𝐸² potential ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">24</span></a>) for the specific case of spin <math alttext="S=1" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px1.p1.1.m1.1"><semantics id="S3.SS2.SSS0.Px1.p1.1.m1.1a"><mrow id="S3.SS2.SSS0.Px1.p1.1.m1.1.1" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.cmml"><mi id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.2" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.2.cmml">S</mi><mo id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.1" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.1.cmml">=</mo><mn id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.3" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px1.p1.1.m1.1b"><apply id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1"><eq id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.1"></eq><ci id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.2">𝑆</ci><cn id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px1.p1.1.m1.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px1.p1.1.m1.1d">italic_S = 1</annotation></semantics></math>, i.e.</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx2"> <tbody id="S3.E12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell 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xref="S3.E12.m2.2.2.1.1.2">0</cn><cn id="S3.E12.m2.2.2.1.1.3.cmml" type="integer" xref="S3.E12.m2.2.2.1.1.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E12.m2.6c">\displaystyle\frac{1}{2^{\left|T\right|/2}}\prod_{(n,n^{\prime})\in T}\left(% \ket{\pm 1\ 0}-\ket{0\ \pm 1}\right),</annotation><annotation encoding="application/x-llamapun" id="S3.E12.m2.6d">divide start_ARG 1 end_ARG start_ARG 2 start_POSTSUPERSCRIPT | italic_T | / 2 end_POSTSUPERSCRIPT end_ARG ∏ start_POSTSUBSCRIPT ( italic_n , italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ italic_T end_POSTSUBSCRIPT ( | start_ARG ± 1 0 end_ARG ⟩ - | start_ARG 0 ± 1 end_ARG ⟩ ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.SSS0.Px1.p1.3">in the dual representation. The two plaquettes that make up tiles no longer share a link, but neighbor each other diagonally in a pattern depicted in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.F4" title="Figure 4 ‣ Non-Tiling Scar ‣ III.2 Beyond Zero-Mode Building Blocks ‣ III Zero-mode scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">4</span></a>. Individual tiles are not zero-modes, i.e. they do not satisfy (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.E9" title="In III.1 QMBS in TLM With Arbitrary Integer Spin ‣ III Zero-mode scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">9</span></a>). However, the resulting state is a zero-mode of the full system. This follows because each plaquette neighbors both sites of another tile. One of the plaquettes of this tile will take the value <math alttext="\pm 1" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px1.p1.2.m1.1"><semantics id="S3.SS2.SSS0.Px1.p1.2.m1.1a"><mrow id="S3.SS2.SSS0.Px1.p1.2.m1.1.1" xref="S3.SS2.SSS0.Px1.p1.2.m1.1.1.cmml"><mo id="S3.SS2.SSS0.Px1.p1.2.m1.1.1a" xref="S3.SS2.SSS0.Px1.p1.2.m1.1.1.cmml">±</mo><mn id="S3.SS2.SSS0.Px1.p1.2.m1.1.1.2" xref="S3.SS2.SSS0.Px1.p1.2.m1.1.1.2.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px1.p1.2.m1.1b"><apply id="S3.SS2.SSS0.Px1.p1.2.m1.1.1.cmml" xref="S3.SS2.SSS0.Px1.p1.2.m1.1.1"><csymbol cd="latexml" id="S3.SS2.SSS0.Px1.p1.2.m1.1.1.1.cmml" xref="S3.SS2.SSS0.Px1.p1.2.m1.1.1">plus-or-minus</csymbol><cn id="S3.SS2.SSS0.Px1.p1.2.m1.1.1.2.cmml" type="integer" xref="S3.SS2.SSS0.Px1.p1.2.m1.1.1.2">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px1.p1.2.m1.1c">\pm 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px1.p1.2.m1.1d">± 1</annotation></semantics></math>. None of the plaquettes can therefore ever take height <math alttext="\mp 1" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px1.p1.3.m2.1"><semantics id="S3.SS2.SSS0.Px1.p1.3.m2.1a"><mrow id="S3.SS2.SSS0.Px1.p1.3.m2.1.1" xref="S3.SS2.SSS0.Px1.p1.3.m2.1.1.cmml"><mo id="S3.SS2.SSS0.Px1.p1.3.m2.1.1a" xref="S3.SS2.SSS0.Px1.p1.3.m2.1.1.cmml">∓</mo><mn id="S3.SS2.SSS0.Px1.p1.3.m2.1.1.2" xref="S3.SS2.SSS0.Px1.p1.3.m2.1.1.2.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px1.p1.3.m2.1b"><apply id="S3.SS2.SSS0.Px1.p1.3.m2.1.1.cmml" xref="S3.SS2.SSS0.Px1.p1.3.m2.1.1"><csymbol cd="latexml" id="S3.SS2.SSS0.Px1.p1.3.m2.1.1.1.cmml" xref="S3.SS2.SSS0.Px1.p1.3.m2.1.1">minus-or-plus</csymbol><cn id="S3.SS2.SSS0.Px1.p1.3.m2.1.1.2.cmml" type="integer" xref="S3.SS2.SSS0.Px1.p1.3.m2.1.1.2">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px1.p1.3.m2.1c">\mp 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px1.p1.3.m2.1d">∓ 1</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="S3.SS2.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Non-Tiling Scar</h4> <div class="ltx_para" id="S3.SS2.SSS0.Px2.p1"> <p class="ltx_p" id="S3.SS2.SSS0.Px2.p1.2">It is also possible to have scars that do not follow (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S5.E24" title="In V A 4×4 Scar for the 𝐸² potential ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">24</span></a>) nor exhibit a similar tiling structure. We divide the lattice into two sublattices <math alttext="A" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px2.p1.1.m1.1"><semantics id="S3.SS2.SSS0.Px2.p1.1.m1.1a"><mi id="S3.SS2.SSS0.Px2.p1.1.m1.1.1" xref="S3.SS2.SSS0.Px2.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px2.p1.1.m1.1b"><ci id="S3.SS2.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S3.SS2.SSS0.Px2.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px2.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px2.p1.1.m1.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px2.p1.2.m2.1"><semantics id="S3.SS2.SSS0.Px2.p1.2.m2.1a"><mi id="S3.SS2.SSS0.Px2.p1.2.m2.1.1" xref="S3.SS2.SSS0.Px2.p1.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px2.p1.2.m2.1b"><ci id="S3.SS2.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S3.SS2.SSS0.Px2.p1.2.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px2.p1.2.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px2.p1.2.m2.1d">italic_B</annotation></semantics></math> and construct a zero-mode according to</p> </div> <div class="ltx_para" id="S3.SS2.SSS0.Px2.p2"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx3"> <tbody id="S3.Ex2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\ket{\psi_{s}}=\frac{1}{2^{L_{1}L_{2}/2}}" class="ltx_Math" display="inline" id="S3.Ex2.m1.1"><semantics id="S3.Ex2.m1.1a"><mrow id="S3.Ex2.m1.1.2" xref="S3.Ex2.m1.1.2.cmml"><mrow id="S3.Ex2.m1.1.1a.3" xref="S3.Ex2.m1.1.1a.2.cmml"><mo id="S3.Ex2.m1.1.1a.3.1" stretchy="false" xref="S3.Ex2.m1.1.1a.2.1.cmml">|</mo><msub id="S3.Ex2.m1.1.1.1.1" xref="S3.Ex2.m1.1.1.1.1.cmml"><mi id="S3.Ex2.m1.1.1.1.1.2" xref="S3.Ex2.m1.1.1.1.1.2.cmml">ψ</mi><mi id="S3.Ex2.m1.1.1.1.1.3" xref="S3.Ex2.m1.1.1.1.1.3.cmml">s</mi></msub><mo id="S3.Ex2.m1.1.1a.3.2" stretchy="false" xref="S3.Ex2.m1.1.1a.2.1.cmml">⟩</mo></mrow><mo id="S3.Ex2.m1.1.2.1" xref="S3.Ex2.m1.1.2.1.cmml">=</mo><mstyle displaystyle="true" id="S3.Ex2.m1.1.2.2" xref="S3.Ex2.m1.1.2.2.cmml"><mfrac id="S3.Ex2.m1.1.2.2a" xref="S3.Ex2.m1.1.2.2.cmml"><mn id="S3.Ex2.m1.1.2.2.2" xref="S3.Ex2.m1.1.2.2.2.cmml">1</mn><msup id="S3.Ex2.m1.1.2.2.3" xref="S3.Ex2.m1.1.2.2.3.cmml"><mn id="S3.Ex2.m1.1.2.2.3.2" xref="S3.Ex2.m1.1.2.2.3.2.cmml">2</mn><mrow id="S3.Ex2.m1.1.2.2.3.3" xref="S3.Ex2.m1.1.2.2.3.3.cmml"><mrow id="S3.Ex2.m1.1.2.2.3.3.2" xref="S3.Ex2.m1.1.2.2.3.3.2.cmml"><msub id="S3.Ex2.m1.1.2.2.3.3.2.2" xref="S3.Ex2.m1.1.2.2.3.3.2.2.cmml"><mi id="S3.Ex2.m1.1.2.2.3.3.2.2.2" xref="S3.Ex2.m1.1.2.2.3.3.2.2.2.cmml">L</mi><mn id="S3.Ex2.m1.1.2.2.3.3.2.2.3" xref="S3.Ex2.m1.1.2.2.3.3.2.2.3.cmml">1</mn></msub><mo id="S3.Ex2.m1.1.2.2.3.3.2.1" xref="S3.Ex2.m1.1.2.2.3.3.2.1.cmml">⁢</mo><msub id="S3.Ex2.m1.1.2.2.3.3.2.3" xref="S3.Ex2.m1.1.2.2.3.3.2.3.cmml"><mi id="S3.Ex2.m1.1.2.2.3.3.2.3.2" xref="S3.Ex2.m1.1.2.2.3.3.2.3.2.cmml">L</mi><mn id="S3.Ex2.m1.1.2.2.3.3.2.3.3" xref="S3.Ex2.m1.1.2.2.3.3.2.3.3.cmml">2</mn></msub></mrow><mo id="S3.Ex2.m1.1.2.2.3.3.1" xref="S3.Ex2.m1.1.2.2.3.3.1.cmml">/</mo><mn id="S3.Ex2.m1.1.2.2.3.3.3" xref="S3.Ex2.m1.1.2.2.3.3.3.cmml">2</mn></mrow></msup></mfrac></mstyle></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex2.m1.1b"><apply id="S3.Ex2.m1.1.2.cmml" xref="S3.Ex2.m1.1.2"><eq id="S3.Ex2.m1.1.2.1.cmml" xref="S3.Ex2.m1.1.2.1"></eq><apply id="S3.Ex2.m1.1.1a.2.cmml" xref="S3.Ex2.m1.1.1a.3"><csymbol cd="latexml" id="S3.Ex2.m1.1.1a.2.1.cmml" xref="S3.Ex2.m1.1.1a.3.1">ket</csymbol><apply id="S3.Ex2.m1.1.1.1.1.cmml" xref="S3.Ex2.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex2.m1.1.1.1.1.1.cmml" 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xref="S3.Ex2.m1.1.2.2.3.3.2.2"><csymbol cd="ambiguous" id="S3.Ex2.m1.1.2.2.3.3.2.2.1.cmml" xref="S3.Ex2.m1.1.2.2.3.3.2.2">subscript</csymbol><ci id="S3.Ex2.m1.1.2.2.3.3.2.2.2.cmml" xref="S3.Ex2.m1.1.2.2.3.3.2.2.2">𝐿</ci><cn id="S3.Ex2.m1.1.2.2.3.3.2.2.3.cmml" type="integer" xref="S3.Ex2.m1.1.2.2.3.3.2.2.3">1</cn></apply><apply id="S3.Ex2.m1.1.2.2.3.3.2.3.cmml" xref="S3.Ex2.m1.1.2.2.3.3.2.3"><csymbol cd="ambiguous" id="S3.Ex2.m1.1.2.2.3.3.2.3.1.cmml" xref="S3.Ex2.m1.1.2.2.3.3.2.3">subscript</csymbol><ci id="S3.Ex2.m1.1.2.2.3.3.2.3.2.cmml" xref="S3.Ex2.m1.1.2.2.3.3.2.3.2">𝐿</ci><cn id="S3.Ex2.m1.1.2.2.3.3.2.3.3.cmml" type="integer" xref="S3.Ex2.m1.1.2.2.3.3.2.3.3">2</cn></apply></apply><cn id="S3.Ex2.m1.1.2.2.3.3.3.cmml" type="integer" xref="S3.Ex2.m1.1.2.2.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex2.m1.1c">\displaystyle\ket{\psi_{s}}=\frac{1}{2^{L_{1}L_{2}/2}}</annotation><annotation encoding="application/x-llamapun" 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id="S3.E13.m1.2.2.1.1.2.3.cmml" xref="S3.E13.m1.2.2.1.1.2.3"><in id="S3.E13.m1.2.2.1.1.2.3.1.cmml" xref="S3.E13.m1.2.2.1.1.2.3.1"></in><ci id="S3.E13.m1.2.2.1.1.2.3.2.cmml" xref="S3.E13.m1.2.2.1.1.2.3.2">𝑛</ci><ci id="S3.E13.m1.2.2.1.1.2.3.3.cmml" xref="S3.E13.m1.2.2.1.1.2.3.3">𝐵</ci></apply></apply><apply id="S3.E13.m1.2.2.1.1.1.cmml" xref="S3.E13.m1.2.2.1.1.1"><times id="S3.E13.m1.2.2.1.1.1.2.cmml" xref="S3.E13.m1.2.2.1.1.1.2"></times><apply id="S3.E13.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1"><minus id="S3.E13.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.1"></minus><apply id="S3.E13.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.E13.m1.2.2.1.1.1.1.1.1.2.1.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.2">subscript</csymbol><apply id="S3.E13.m1.2.2.1.1.1.1.1.1.2.2.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.E13.m1.2.2.1.1.1.1.1.1.2.2.1.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.2">superscript</csymbol><ci id="S3.E13.m1.2.2.1.1.1.1.1.1.2.2.2.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.2.2.2">𝐻</ci><plus id="S3.E13.m1.2.2.1.1.1.1.1.1.2.2.3.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.2.2.3"></plus></apply><ci id="S3.E13.m1.2.2.1.1.1.1.1.1.2.3.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.2.3">𝑛</ci></apply><apply id="S3.E13.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.E13.m1.2.2.1.1.1.1.1.1.3.1.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.3">subscript</csymbol><apply id="S3.E13.m1.2.2.1.1.1.1.1.1.3.2.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.E13.m1.2.2.1.1.1.1.1.1.3.2.1.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.3">superscript</csymbol><ci id="S3.E13.m1.2.2.1.1.1.1.1.1.3.2.2.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.3.2.2">𝐻</ci><minus id="S3.E13.m1.2.2.1.1.1.1.1.1.3.2.3.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.3.2.3"></minus></apply><ci id="S3.E13.m1.2.2.1.1.1.1.1.1.3.3.cmml" xref="S3.E13.m1.2.2.1.1.1.1.1.1.3.3">𝑛</ci></apply></apply><apply id="S3.E13.m1.1.1a.2.cmml" xref="S3.E13.m1.1.1a.3"><csymbol cd="latexml" id="S3.E13.m1.1.1a.2.1.cmml" xref="S3.E13.m1.1.1a.3.1">ket</csymbol><cn id="S3.E13.m1.1.1.1.1.cmml" type="integer" xref="S3.E13.m1.1.1.1.1">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E13.m1.2c">\displaystyle\prod_{n\in B}\left(H^{+}_{n}-H^{-}_{n}\right)\ket{\mathbb{0}},</annotation><annotation encoding="application/x-llamapun" id="S3.E13.m1.2d">∏ start_POSTSUBSCRIPT italic_n ∈ italic_B end_POSTSUBSCRIPT ( italic_H start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_H start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) | start_ARG blackboard_0 end_ARG ⟩ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.SSS0.Px2.p2.3">where <math alttext="P^{\pm}_{m}" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px2.p2.1.m1.1"><semantics id="S3.SS2.SSS0.Px2.p2.1.m1.1a"><msubsup id="S3.SS2.SSS0.Px2.p2.1.m1.1.1" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1.cmml"><mi id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.2.2" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1.2.2.cmml">P</mi><mi id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.3" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1.3.cmml">m</mi><mo id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.2.3" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1.2.3.cmml">±</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px2.p2.1.m1.1b"><apply id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.1.cmml" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1">subscript</csymbol><apply id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.2.cmml" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.2.1.cmml" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1">superscript</csymbol><ci id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.2.2.cmml" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1.2.2">𝑃</ci><csymbol cd="latexml" id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.2.3.cmml" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1.2.3">plus-or-minus</csymbol></apply><ci id="S3.SS2.SSS0.Px2.p2.1.m1.1.1.3.cmml" xref="S3.SS2.SSS0.Px2.p2.1.m1.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px2.p2.1.m1.1c">P^{\pm}_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px2.p2.1.m1.1d">italic_P start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> is the projector on <math alttext="\pm 1" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px2.p2.2.m2.1"><semantics id="S3.SS2.SSS0.Px2.p2.2.m2.1a"><mrow id="S3.SS2.SSS0.Px2.p2.2.m2.1.1" xref="S3.SS2.SSS0.Px2.p2.2.m2.1.1.cmml"><mo id="S3.SS2.SSS0.Px2.p2.2.m2.1.1a" xref="S3.SS2.SSS0.Px2.p2.2.m2.1.1.cmml">±</mo><mn id="S3.SS2.SSS0.Px2.p2.2.m2.1.1.2" xref="S3.SS2.SSS0.Px2.p2.2.m2.1.1.2.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px2.p2.2.m2.1b"><apply id="S3.SS2.SSS0.Px2.p2.2.m2.1.1.cmml" xref="S3.SS2.SSS0.Px2.p2.2.m2.1.1"><csymbol cd="latexml" id="S3.SS2.SSS0.Px2.p2.2.m2.1.1.1.cmml" xref="S3.SS2.SSS0.Px2.p2.2.m2.1.1">plus-or-minus</csymbol><cn id="S3.SS2.SSS0.Px2.p2.2.m2.1.1.2.cmml" type="integer" xref="S3.SS2.SSS0.Px2.p2.2.m2.1.1.2">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px2.p2.2.m2.1c">\pm 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px2.p2.2.m2.1d">± 1</annotation></semantics></math> on all plaquettes surrounding the plaquette at <math alttext="m" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px2.p2.3.m3.1"><semantics id="S3.SS2.SSS0.Px2.p2.3.m3.1a"><mi id="S3.SS2.SSS0.Px2.p2.3.m3.1.1" xref="S3.SS2.SSS0.Px2.p2.3.m3.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px2.p2.3.m3.1b"><ci id="S3.SS2.SSS0.Px2.p2.3.m3.1.1.cmml" xref="S3.SS2.SSS0.Px2.p2.3.m3.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px2.p2.3.m3.1c">m</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px2.p2.3.m3.1d">italic_m</annotation></semantics></math>. An illustration can be found in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.F4" title="Figure 4 ‣ Non-Tiling Scar ‣ III.2 Beyond Zero-Mode Building Blocks ‣ III Zero-mode scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">4</span></a>.</p> </div> <figure class="ltx_figure" id="S3.F4"> <p class="ltx_p ltx_align_center" id="S3.F4.2"><span class="ltx_text" id="S3.F4.1.1" style="position:relative; bottom:-0.5pt;"> <img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="235" id="S3.F4.1.1.g1" src="extracted/5828746/images/scars3.png" width="237"/></span> <span class="ltx_text" id="S3.F4.2.2" style="position:relative; bottom:-0.5pt;"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="210" id="S3.F4.2.2.g1" src="extracted/5828746/images/scar2.png" width="210"/></span></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>Illustrations of <math alttext="S=1" class="ltx_Math" display="inline" id="S3.F4.9.m1.1"><semantics id="S3.F4.9.m1.1b"><mrow id="S3.F4.9.m1.1.1" xref="S3.F4.9.m1.1.1.cmml"><mi id="S3.F4.9.m1.1.1.2" xref="S3.F4.9.m1.1.1.2.cmml">S</mi><mo id="S3.F4.9.m1.1.1.1" xref="S3.F4.9.m1.1.1.1.cmml">=</mo><mn id="S3.F4.9.m1.1.1.3" xref="S3.F4.9.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.9.m1.1c"><apply id="S3.F4.9.m1.1.1.cmml" xref="S3.F4.9.m1.1.1"><eq id="S3.F4.9.m1.1.1.1.cmml" xref="S3.F4.9.m1.1.1.1"></eq><ci id="S3.F4.9.m1.1.1.2.cmml" xref="S3.F4.9.m1.1.1.2">𝑆</ci><cn id="S3.F4.9.m1.1.1.3.cmml" type="integer" xref="S3.F4.9.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.9.m1.1d">S=1</annotation><annotation encoding="application/x-llamapun" id="S3.F4.9.m1.1e">italic_S = 1</annotation></semantics></math> zero-mode scars that are eigenstates of <math alttext="\sum_{n}E_{n}^{2}" class="ltx_Math" display="inline" id="S3.F4.10.m2.1"><semantics id="S3.F4.10.m2.1b"><mrow id="S3.F4.10.m2.1.1" xref="S3.F4.10.m2.1.1.cmml"><msub id="S3.F4.10.m2.1.1.1" xref="S3.F4.10.m2.1.1.1.cmml"><mo id="S3.F4.10.m2.1.1.1.2" xref="S3.F4.10.m2.1.1.1.2.cmml">∑</mo><mi id="S3.F4.10.m2.1.1.1.3" xref="S3.F4.10.m2.1.1.1.3.cmml">n</mi></msub><msubsup id="S3.F4.10.m2.1.1.2" xref="S3.F4.10.m2.1.1.2.cmml"><mi id="S3.F4.10.m2.1.1.2.2.2" xref="S3.F4.10.m2.1.1.2.2.2.cmml">E</mi><mi id="S3.F4.10.m2.1.1.2.2.3" xref="S3.F4.10.m2.1.1.2.2.3.cmml">n</mi><mn id="S3.F4.10.m2.1.1.2.3" xref="S3.F4.10.m2.1.1.2.3.cmml">2</mn></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.10.m2.1c"><apply id="S3.F4.10.m2.1.1.cmml" xref="S3.F4.10.m2.1.1"><apply id="S3.F4.10.m2.1.1.1.cmml" xref="S3.F4.10.m2.1.1.1"><csymbol cd="ambiguous" id="S3.F4.10.m2.1.1.1.1.cmml" xref="S3.F4.10.m2.1.1.1">subscript</csymbol><sum id="S3.F4.10.m2.1.1.1.2.cmml" xref="S3.F4.10.m2.1.1.1.2"></sum><ci id="S3.F4.10.m2.1.1.1.3.cmml" xref="S3.F4.10.m2.1.1.1.3">𝑛</ci></apply><apply id="S3.F4.10.m2.1.1.2.cmml" xref="S3.F4.10.m2.1.1.2"><csymbol cd="ambiguous" id="S3.F4.10.m2.1.1.2.1.cmml" xref="S3.F4.10.m2.1.1.2">superscript</csymbol><apply id="S3.F4.10.m2.1.1.2.2.cmml" xref="S3.F4.10.m2.1.1.2"><csymbol cd="ambiguous" id="S3.F4.10.m2.1.1.2.2.1.cmml" xref="S3.F4.10.m2.1.1.2">subscript</csymbol><ci id="S3.F4.10.m2.1.1.2.2.2.cmml" xref="S3.F4.10.m2.1.1.2.2.2">𝐸</ci><ci id="S3.F4.10.m2.1.1.2.2.3.cmml" xref="S3.F4.10.m2.1.1.2.2.3">𝑛</ci></apply><cn id="S3.F4.10.m2.1.1.2.3.cmml" type="integer" xref="S3.F4.10.m2.1.1.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.10.m2.1d">\sum_{n}E_{n}^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.F4.10.m2.1e">∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, exhibiting sublattice structure. Left: Scar built by singlet states of two plaquettes connected diagonally. Plaquettes connected by red or green lines form tiles. Right: Singlet states are placed in sublattice <math alttext="B" class="ltx_Math" display="inline" id="S3.F4.11.m3.1"><semantics id="S3.F4.11.m3.1b"><mi id="S3.F4.11.m3.1.1" xref="S3.F4.11.m3.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.F4.11.m3.1c"><ci id="S3.F4.11.m3.1.1.cmml" xref="S3.F4.11.m3.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.11.m3.1d">B</annotation><annotation encoding="application/x-llamapun" id="S3.F4.11.m3.1e">italic_B</annotation></semantics></math> while sublattice <math alttext="A" class="ltx_Math" display="inline" id="S3.F4.12.m4.1"><semantics id="S3.F4.12.m4.1b"><mi id="S3.F4.12.m4.1.1" xref="S3.F4.12.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.F4.12.m4.1c"><ci id="S3.F4.12.m4.1.1.cmml" xref="S3.F4.12.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.12.m4.1d">A</annotation><annotation encoding="application/x-llamapun" id="S3.F4.12.m4.1e">italic_A</annotation></semantics></math> is filled by 0’s. If all plaquettes surrounding a 0 have the same value of <math alttext="\pm 1" class="ltx_Math" display="inline" id="S3.F4.13.m5.1"><semantics id="S3.F4.13.m5.1b"><mrow id="S3.F4.13.m5.1.1" xref="S3.F4.13.m5.1.1.cmml"><mo id="S3.F4.13.m5.1.1b" xref="S3.F4.13.m5.1.1.cmml">±</mo><mn id="S3.F4.13.m5.1.1.2" xref="S3.F4.13.m5.1.1.2.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.13.m5.1c"><apply id="S3.F4.13.m5.1.1.cmml" xref="S3.F4.13.m5.1.1"><csymbol cd="latexml" id="S3.F4.13.m5.1.1.1.cmml" xref="S3.F4.13.m5.1.1">plus-or-minus</csymbol><cn id="S3.F4.13.m5.1.1.2.cmml" type="integer" xref="S3.F4.13.m5.1.1.2">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.13.m5.1d">\pm 1</annotation><annotation encoding="application/x-llamapun" id="S3.F4.13.m5.1e">± 1</annotation></semantics></math>, then that plaquette is combined with <math alttext="\pm 2" class="ltx_Math" display="inline" id="S3.F4.14.m6.1"><semantics id="S3.F4.14.m6.1b"><mrow id="S3.F4.14.m6.1.1" xref="S3.F4.14.m6.1.1.cmml"><mo id="S3.F4.14.m6.1.1b" xref="S3.F4.14.m6.1.1.cmml">±</mo><mn id="S3.F4.14.m6.1.1.2" xref="S3.F4.14.m6.1.1.2.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F4.14.m6.1c"><apply id="S3.F4.14.m6.1.1.cmml" xref="S3.F4.14.m6.1.1"><csymbol cd="latexml" id="S3.F4.14.m6.1.1.1.cmml" xref="S3.F4.14.m6.1.1">plus-or-minus</csymbol><cn id="S3.F4.14.m6.1.1.2.cmml" type="integer" xref="S3.F4.14.m6.1.1.2">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F4.14.m6.1d">\pm 2</annotation><annotation encoding="application/x-llamapun" id="S3.F4.14.m6.1e">± 2</annotation></semantics></math>.</figcaption> </figure> <div class="ltx_para" id="S3.SS2.SSS0.Px2.p3"> <p class="ltx_p" id="S3.SS2.SSS0.Px2.p3.3">This state can be understood by considering one sublattice and creating a linear superposition of heights <math alttext="\pm 1" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px2.p3.1.m1.1"><semantics id="S3.SS2.SSS0.Px2.p3.1.m1.1a"><mrow id="S3.SS2.SSS0.Px2.p3.1.m1.1.1" xref="S3.SS2.SSS0.Px2.p3.1.m1.1.1.cmml"><mo id="S3.SS2.SSS0.Px2.p3.1.m1.1.1a" xref="S3.SS2.SSS0.Px2.p3.1.m1.1.1.cmml">±</mo><mn id="S3.SS2.SSS0.Px2.p3.1.m1.1.1.2" xref="S3.SS2.SSS0.Px2.p3.1.m1.1.1.2.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px2.p3.1.m1.1b"><apply id="S3.SS2.SSS0.Px2.p3.1.m1.1.1.cmml" xref="S3.SS2.SSS0.Px2.p3.1.m1.1.1"><csymbol cd="latexml" id="S3.SS2.SSS0.Px2.p3.1.m1.1.1.1.cmml" xref="S3.SS2.SSS0.Px2.p3.1.m1.1.1">plus-or-minus</csymbol><cn id="S3.SS2.SSS0.Px2.p3.1.m1.1.1.2.cmml" type="integer" xref="S3.SS2.SSS0.Px2.p3.1.m1.1.1.2">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px2.p3.1.m1.1c">\pm 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px2.p3.1.m1.1d">± 1</annotation></semantics></math>. In our example, this is done for sublattice <math alttext="B" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px2.p3.2.m2.1"><semantics id="S3.SS2.SSS0.Px2.p3.2.m2.1a"><mi id="S3.SS2.SSS0.Px2.p3.2.m2.1.1" xref="S3.SS2.SSS0.Px2.p3.2.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px2.p3.2.m2.1b"><ci id="S3.SS2.SSS0.Px2.p3.2.m2.1.1.cmml" xref="S3.SS2.SSS0.Px2.p3.2.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px2.p3.2.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px2.p3.2.m2.1d">italic_B</annotation></semantics></math>. For plaquettes of sublattice <math alttext="A" class="ltx_Math" display="inline" id="S3.SS2.SSS0.Px2.p3.3.m3.1"><semantics id="S3.SS2.SSS0.Px2.p3.3.m3.1a"><mi id="S3.SS2.SSS0.Px2.p3.3.m3.1.1" xref="S3.SS2.SSS0.Px2.p3.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px2.p3.3.m3.1b"><ci id="S3.SS2.SSS0.Px2.p3.3.m3.1.1.cmml" xref="S3.SS2.SSS0.Px2.p3.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px2.p3.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.SSS0.Px2.p3.3.m3.1d">italic_A</annotation></semantics></math>, the height variable is stuck at zero as long as all surrounding plaquettes are not equal. Otherwise, the plaquette can either be raised or lowered depending on the value of the surrounding plaquettes being positive or negative. By combining those states with the respective lowered and raised states with the opposite sign, we obtain a zero-mode.</p> </div> </section> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span>Numerical results and discussion</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.2">While the derived scars are low entropy zero-modes, they are degenerate with all the other zero-modes and are not easily identifiable in the spectrum. In the following, we introduce potentials that have these states as eigenstates. By analyzing the Shannon entropy and the bipartite entanglement entropy of the eigenstates of the Hamiltonian, we verify the existence of these low entropy states numerically. Concretely, we calculate the Shannon entropy with respect to the electric field basis <math alttext="\{\ket{\phi_{i}}\}" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><mrow id="S4.p1.1.m1.1.2.2" xref="S4.p1.1.m1.1.2.1.cmml"><mo id="S4.p1.1.m1.1.2.2.1" stretchy="false" xref="S4.p1.1.m1.1.2.1.cmml">{</mo><mrow id="S4.p1.1.m1.1.1.3" xref="S4.p1.1.m1.1.1.2.cmml"><mo id="S4.p1.1.m1.1.1.3.1" stretchy="false" xref="S4.p1.1.m1.1.1.2.1.cmml">|</mo><msub id="S4.p1.1.m1.1.1.1.1" xref="S4.p1.1.m1.1.1.1.1.cmml"><mi id="S4.p1.1.m1.1.1.1.1.2" xref="S4.p1.1.m1.1.1.1.1.2.cmml">ϕ</mi><mi id="S4.p1.1.m1.1.1.1.1.3" xref="S4.p1.1.m1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.p1.1.m1.1.1.3.2" stretchy="false" xref="S4.p1.1.m1.1.1.2.1.cmml">⟩</mo></mrow><mo id="S4.p1.1.m1.1.2.2.2" stretchy="false" xref="S4.p1.1.m1.1.2.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><set id="S4.p1.1.m1.1.2.1.cmml" xref="S4.p1.1.m1.1.2.2"><apply id="S4.p1.1.m1.1.1.2.cmml" xref="S4.p1.1.m1.1.1.3"><csymbol cd="latexml" id="S4.p1.1.m1.1.1.2.1.cmml" xref="S4.p1.1.m1.1.1.3.1">ket</csymbol><apply id="S4.p1.1.m1.1.1.1.1.cmml" xref="S4.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.p1.1.m1.1.1.1.1.1.cmml" xref="S4.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S4.p1.1.m1.1.1.1.1.2.cmml" xref="S4.p1.1.m1.1.1.1.1.2">italic-ϕ</ci><ci id="S4.p1.1.m1.1.1.1.1.3.cmml" xref="S4.p1.1.m1.1.1.1.1.3">𝑖</ci></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">\{\ket{\phi_{i}}\}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">{ | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ }</annotation></semantics></math>. For a state <math alttext="\ket{\psi}=\sum_{i}c_{i}\ket{\phi_{i}}" class="ltx_Math" display="inline" id="S4.p1.2.m2.2"><semantics id="S4.p1.2.m2.2a"><mrow id="S4.p1.2.m2.2.3" xref="S4.p1.2.m2.2.3.cmml"><mrow id="S4.p1.2.m2.1.1.3" xref="S4.p1.2.m2.1.1.2.cmml"><mo id="S4.p1.2.m2.1.1.3.1" stretchy="false" xref="S4.p1.2.m2.1.1.2.1.cmml">|</mo><mi id="S4.p1.2.m2.1.1.1.1" xref="S4.p1.2.m2.1.1.1.1.cmml">ψ</mi><mo id="S4.p1.2.m2.1.1.3.2" stretchy="false" xref="S4.p1.2.m2.1.1.2.1.cmml">⟩</mo></mrow><mo id="S4.p1.2.m2.2.3.1" rspace="0.111em" xref="S4.p1.2.m2.2.3.1.cmml">=</mo><mrow id="S4.p1.2.m2.2.3.2" xref="S4.p1.2.m2.2.3.2.cmml"><msub id="S4.p1.2.m2.2.3.2.1" xref="S4.p1.2.m2.2.3.2.1.cmml"><mo id="S4.p1.2.m2.2.3.2.1.2" xref="S4.p1.2.m2.2.3.2.1.2.cmml">∑</mo><mi id="S4.p1.2.m2.2.3.2.1.3" xref="S4.p1.2.m2.2.3.2.1.3.cmml">i</mi></msub><mrow id="S4.p1.2.m2.2.3.2.2" xref="S4.p1.2.m2.2.3.2.2.cmml"><msub id="S4.p1.2.m2.2.3.2.2.2" xref="S4.p1.2.m2.2.3.2.2.2.cmml"><mi id="S4.p1.2.m2.2.3.2.2.2.2" xref="S4.p1.2.m2.2.3.2.2.2.2.cmml">c</mi><mi id="S4.p1.2.m2.2.3.2.2.2.3" xref="S4.p1.2.m2.2.3.2.2.2.3.cmml">i</mi></msub><mo id="S4.p1.2.m2.2.3.2.2.1" xref="S4.p1.2.m2.2.3.2.2.1.cmml">⁢</mo><mrow id="S4.p1.2.m2.2.2.3" xref="S4.p1.2.m2.2.2.2.cmml"><mo id="S4.p1.2.m2.2.2.3.1" stretchy="false" xref="S4.p1.2.m2.2.2.2.1.cmml">|</mo><msub id="S4.p1.2.m2.2.2.1.1" xref="S4.p1.2.m2.2.2.1.1.cmml"><mi id="S4.p1.2.m2.2.2.1.1.2" xref="S4.p1.2.m2.2.2.1.1.2.cmml">ϕ</mi><mi id="S4.p1.2.m2.2.2.1.1.3" xref="S4.p1.2.m2.2.2.1.1.3.cmml">i</mi></msub><mo id="S4.p1.2.m2.2.2.3.2" stretchy="false" xref="S4.p1.2.m2.2.2.2.1.cmml">⟩</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.2.m2.2b"><apply id="S4.p1.2.m2.2.3.cmml" xref="S4.p1.2.m2.2.3"><eq id="S4.p1.2.m2.2.3.1.cmml" xref="S4.p1.2.m2.2.3.1"></eq><apply id="S4.p1.2.m2.1.1.2.cmml" xref="S4.p1.2.m2.1.1.3"><csymbol cd="latexml" id="S4.p1.2.m2.1.1.2.1.cmml" xref="S4.p1.2.m2.1.1.3.1">ket</csymbol><ci id="S4.p1.2.m2.1.1.1.1.cmml" xref="S4.p1.2.m2.1.1.1.1">𝜓</ci></apply><apply id="S4.p1.2.m2.2.3.2.cmml" xref="S4.p1.2.m2.2.3.2"><apply id="S4.p1.2.m2.2.3.2.1.cmml" xref="S4.p1.2.m2.2.3.2.1"><csymbol cd="ambiguous" id="S4.p1.2.m2.2.3.2.1.1.cmml" xref="S4.p1.2.m2.2.3.2.1">subscript</csymbol><sum id="S4.p1.2.m2.2.3.2.1.2.cmml" xref="S4.p1.2.m2.2.3.2.1.2"></sum><ci id="S4.p1.2.m2.2.3.2.1.3.cmml" xref="S4.p1.2.m2.2.3.2.1.3">𝑖</ci></apply><apply id="S4.p1.2.m2.2.3.2.2.cmml" xref="S4.p1.2.m2.2.3.2.2"><times id="S4.p1.2.m2.2.3.2.2.1.cmml" xref="S4.p1.2.m2.2.3.2.2.1"></times><apply id="S4.p1.2.m2.2.3.2.2.2.cmml" xref="S4.p1.2.m2.2.3.2.2.2"><csymbol cd="ambiguous" id="S4.p1.2.m2.2.3.2.2.2.1.cmml" xref="S4.p1.2.m2.2.3.2.2.2">subscript</csymbol><ci id="S4.p1.2.m2.2.3.2.2.2.2.cmml" xref="S4.p1.2.m2.2.3.2.2.2.2">𝑐</ci><ci id="S4.p1.2.m2.2.3.2.2.2.3.cmml" xref="S4.p1.2.m2.2.3.2.2.2.3">𝑖</ci></apply><apply id="S4.p1.2.m2.2.2.2.cmml" xref="S4.p1.2.m2.2.2.3"><csymbol cd="latexml" id="S4.p1.2.m2.2.2.2.1.cmml" xref="S4.p1.2.m2.2.2.3.1">ket</csymbol><apply id="S4.p1.2.m2.2.2.1.1.cmml" xref="S4.p1.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S4.p1.2.m2.2.2.1.1.1.cmml" xref="S4.p1.2.m2.2.2.1.1">subscript</csymbol><ci id="S4.p1.2.m2.2.2.1.1.2.cmml" xref="S4.p1.2.m2.2.2.1.1.2">italic-ϕ</ci><ci id="S4.p1.2.m2.2.2.1.1.3.cmml" xref="S4.p1.2.m2.2.2.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.2.m2.2c">\ket{\psi}=\sum_{i}c_{i}\ket{\phi_{i}}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.2.m2.2d">| start_ARG italic_ψ end_ARG ⟩ = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math>, the Shannon entropy has the value</p> <table class="ltx_equation ltx_eqn_table" id="S4.E14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="S=-\sum_{i}|c_{i}|^{2}\log|c_{i}|^{2}." class="ltx_Math" display="block" id="S4.E14.m1.1"><semantics id="S4.E14.m1.1a"><mrow id="S4.E14.m1.1.1.1" xref="S4.E14.m1.1.1.1.1.cmml"><mrow id="S4.E14.m1.1.1.1.1" xref="S4.E14.m1.1.1.1.1.cmml"><mi id="S4.E14.m1.1.1.1.1.4" xref="S4.E14.m1.1.1.1.1.4.cmml">S</mi><mo id="S4.E14.m1.1.1.1.1.3" xref="S4.E14.m1.1.1.1.1.3.cmml">=</mo><mrow id="S4.E14.m1.1.1.1.1.2" xref="S4.E14.m1.1.1.1.1.2.cmml"><mo id="S4.E14.m1.1.1.1.1.2a" xref="S4.E14.m1.1.1.1.1.2.cmml">−</mo><mrow id="S4.E14.m1.1.1.1.1.2.2" xref="S4.E14.m1.1.1.1.1.2.2.cmml"><munder id="S4.E14.m1.1.1.1.1.2.2.3" xref="S4.E14.m1.1.1.1.1.2.2.3.cmml"><mo id="S4.E14.m1.1.1.1.1.2.2.3.2" movablelimits="false" rspace="0em" xref="S4.E14.m1.1.1.1.1.2.2.3.2.cmml">∑</mo><mi id="S4.E14.m1.1.1.1.1.2.2.3.3" xref="S4.E14.m1.1.1.1.1.2.2.3.3.cmml">i</mi></munder><mrow id="S4.E14.m1.1.1.1.1.2.2.2" 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xref="S4.E14.m1.1.1.1.1.2.2.2.2.1.3">2</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E14.m1.1c">S=-\sum_{i}|c_{i}|^{2}\log|c_{i}|^{2}.</annotation><annotation encoding="application/x-llamapun" id="S4.E14.m1.1d">italic_S = - ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_log | italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p1.4">For the entanglement entropy, we split the system into two subsystems <math alttext="A" class="ltx_Math" display="inline" id="S4.p1.3.m1.1"><semantics id="S4.p1.3.m1.1a"><mi id="S4.p1.3.m1.1.1" xref="S4.p1.3.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.p1.3.m1.1b"><ci id="S4.p1.3.m1.1.1.cmml" xref="S4.p1.3.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.3.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.p1.3.m1.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S4.p1.4.m2.1"><semantics id="S4.p1.4.m2.1a"><mi id="S4.p1.4.m2.1.1" xref="S4.p1.4.m2.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.p1.4.m2.1b"><ci id="S4.p1.4.m2.1.1.cmml" xref="S4.p1.4.m2.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.4.m2.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.p1.4.m2.1d">italic_B</annotation></semantics></math> of equal size. The bipartite entanglement entropy is then given by,</p> <table class="ltx_equation ltx_eqn_table" id="S4.E15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="S_{\frac{L}{2}}=-\text{Tr}\rho_{A}\log\rho_{A}=-\text{Tr}\rho_{B}\log\rho_{B}." class="ltx_Math" display="block" id="S4.E15.m1.1"><semantics id="S4.E15.m1.1a"><mrow id="S4.E15.m1.1.1.1" xref="S4.E15.m1.1.1.1.1.cmml"><mrow id="S4.E15.m1.1.1.1.1" xref="S4.E15.m1.1.1.1.1.cmml"><msub id="S4.E15.m1.1.1.1.1.2" xref="S4.E15.m1.1.1.1.1.2.cmml"><mi id="S4.E15.m1.1.1.1.1.2.2" xref="S4.E15.m1.1.1.1.1.2.2.cmml">S</mi><mfrac id="S4.E15.m1.1.1.1.1.2.3" xref="S4.E15.m1.1.1.1.1.2.3.cmml"><mi id="S4.E15.m1.1.1.1.1.2.3.2" xref="S4.E15.m1.1.1.1.1.2.3.2.cmml">L</mi><mn id="S4.E15.m1.1.1.1.1.2.3.3" xref="S4.E15.m1.1.1.1.1.2.3.3.cmml">2</mn></mfrac></msub><mo id="S4.E15.m1.1.1.1.1.3" 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xref="S4.E15.m1.1.1.1.1.6.2.4"><log id="S4.E15.m1.1.1.1.1.6.2.4.1.cmml" xref="S4.E15.m1.1.1.1.1.6.2.4.1"></log><apply id="S4.E15.m1.1.1.1.1.6.2.4.2.cmml" xref="S4.E15.m1.1.1.1.1.6.2.4.2"><csymbol cd="ambiguous" id="S4.E15.m1.1.1.1.1.6.2.4.2.1.cmml" xref="S4.E15.m1.1.1.1.1.6.2.4.2">subscript</csymbol><ci id="S4.E15.m1.1.1.1.1.6.2.4.2.2.cmml" xref="S4.E15.m1.1.1.1.1.6.2.4.2.2">𝜌</ci><ci id="S4.E15.m1.1.1.1.1.6.2.4.2.3.cmml" xref="S4.E15.m1.1.1.1.1.6.2.4.2.3">𝐵</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E15.m1.1c">S_{\frac{L}{2}}=-\text{Tr}\rho_{A}\log\rho_{A}=-\text{Tr}\rho_{B}\log\rho_{B}.</annotation><annotation encoding="application/x-llamapun" id="S4.E15.m1.1d">italic_S start_POSTSUBSCRIPT divide start_ARG italic_L end_ARG start_ARG 2 end_ARG end_POSTSUBSCRIPT = - Tr italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT roman_log italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = - Tr italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT roman_log italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p1.5">Explicitly, we always partition the system by dividing it vertically into two halves. We consider the bordering vertical links on the left as belonging to the subsystem while the ones on the right do not. See e.g. the Supplementary Material of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib30" title="">30</a>]</cite> for details on how to calculate the entanglement entropy in these systems.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.1">We start by isolating the scars of the form of (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S5.E24" title="In V A 4×4 Scar for the 𝐸² potential ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">24</span></a>). We observe that the sum of the height variables remains constant and equal to <math alttext="(2i-S)" class="ltx_Math" display="inline" id="S4.p2.1.m1.1"><semantics id="S4.p2.1.m1.1a"><mrow id="S4.p2.1.m1.1.1.1" xref="S4.p2.1.m1.1.1.1.1.cmml"><mo id="S4.p2.1.m1.1.1.1.2" stretchy="false" xref="S4.p2.1.m1.1.1.1.1.cmml">(</mo><mrow id="S4.p2.1.m1.1.1.1.1" xref="S4.p2.1.m1.1.1.1.1.cmml"><mrow id="S4.p2.1.m1.1.1.1.1.2" xref="S4.p2.1.m1.1.1.1.1.2.cmml"><mn id="S4.p2.1.m1.1.1.1.1.2.2" xref="S4.p2.1.m1.1.1.1.1.2.2.cmml">2</mn><mo id="S4.p2.1.m1.1.1.1.1.2.1" xref="S4.p2.1.m1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S4.p2.1.m1.1.1.1.1.2.3" xref="S4.p2.1.m1.1.1.1.1.2.3.cmml">i</mi></mrow><mo id="S4.p2.1.m1.1.1.1.1.1" xref="S4.p2.1.m1.1.1.1.1.1.cmml">−</mo><mi id="S4.p2.1.m1.1.1.1.1.3" xref="S4.p2.1.m1.1.1.1.1.3.cmml">S</mi></mrow><mo id="S4.p2.1.m1.1.1.1.3" stretchy="false" xref="S4.p2.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.1b"><apply id="S4.p2.1.m1.1.1.1.1.cmml" xref="S4.p2.1.m1.1.1.1"><minus id="S4.p2.1.m1.1.1.1.1.1.cmml" xref="S4.p2.1.m1.1.1.1.1.1"></minus><apply id="S4.p2.1.m1.1.1.1.1.2.cmml" xref="S4.p2.1.m1.1.1.1.1.2"><times id="S4.p2.1.m1.1.1.1.1.2.1.cmml" xref="S4.p2.1.m1.1.1.1.1.2.1"></times><cn id="S4.p2.1.m1.1.1.1.1.2.2.cmml" type="integer" xref="S4.p2.1.m1.1.1.1.1.2.2">2</cn><ci id="S4.p2.1.m1.1.1.1.1.2.3.cmml" xref="S4.p2.1.m1.1.1.1.1.2.3">𝑖</ci></apply><ci id="S4.p2.1.m1.1.1.1.1.3.cmml" xref="S4.p2.1.m1.1.1.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.1c">(2i-S)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.1d">( 2 italic_i - italic_S )</annotation></semantics></math> for each term. This does not mean that the sum over height variables is a valid potential because, in general, the height variables are not uniquely defined from the physical configuration. However, they can be made unique for open boundary conditions, as described above. For example, in the case of a single-leg ladder, the sum of all height variables can be seen as the sum of the horizontal links of the top row. In general, for open boundary conditions, we write</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx4"> <tbody id="S4.E16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle V=\lambda\sum_{n}h_{n}," class="ltx_Math" display="inline" id="S4.E16.m1.1"><semantics id="S4.E16.m1.1a"><mrow id="S4.E16.m1.1.1.1" xref="S4.E16.m1.1.1.1.1.cmml"><mrow id="S4.E16.m1.1.1.1.1" xref="S4.E16.m1.1.1.1.1.cmml"><mi id="S4.E16.m1.1.1.1.1.2" xref="S4.E16.m1.1.1.1.1.2.cmml">V</mi><mo id="S4.E16.m1.1.1.1.1.1" xref="S4.E16.m1.1.1.1.1.1.cmml">=</mo><mrow id="S4.E16.m1.1.1.1.1.3" xref="S4.E16.m1.1.1.1.1.3.cmml"><mi id="S4.E16.m1.1.1.1.1.3.2" xref="S4.E16.m1.1.1.1.1.3.2.cmml">λ</mi><mo id="S4.E16.m1.1.1.1.1.3.1" xref="S4.E16.m1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S4.E16.m1.1.1.1.1.3.3" xref="S4.E16.m1.1.1.1.1.3.3.cmml"><mstyle displaystyle="true" 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id="S4.E16.m1.1c">\displaystyle V=\lambda\sum_{n}h_{n},</annotation><annotation encoding="application/x-llamapun" id="S4.E16.m1.1d">italic_V = italic_λ ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(16)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.2">which will have the constructed scars as eigenstates with energy <math alttext="E=\lambda\frac{L_{1}L_{2}}{2}(2i-S)" class="ltx_Math" display="inline" id="S4.p2.2.m1.1"><semantics id="S4.p2.2.m1.1a"><mrow id="S4.p2.2.m1.1.1" xref="S4.p2.2.m1.1.1.cmml"><mi id="S4.p2.2.m1.1.1.3" xref="S4.p2.2.m1.1.1.3.cmml">E</mi><mo id="S4.p2.2.m1.1.1.2" xref="S4.p2.2.m1.1.1.2.cmml">=</mo><mrow id="S4.p2.2.m1.1.1.1" xref="S4.p2.2.m1.1.1.1.cmml"><mi id="S4.p2.2.m1.1.1.1.3" xref="S4.p2.2.m1.1.1.1.3.cmml">λ</mi><mo id="S4.p2.2.m1.1.1.1.2" xref="S4.p2.2.m1.1.1.1.2.cmml">⁢</mo><mfrac id="S4.p2.2.m1.1.1.1.4" xref="S4.p2.2.m1.1.1.1.4.cmml"><mrow id="S4.p2.2.m1.1.1.1.4.2" xref="S4.p2.2.m1.1.1.1.4.2.cmml"><msub id="S4.p2.2.m1.1.1.1.4.2.2" xref="S4.p2.2.m1.1.1.1.4.2.2.cmml"><mi id="S4.p2.2.m1.1.1.1.4.2.2.2" xref="S4.p2.2.m1.1.1.1.4.2.2.2.cmml">L</mi><mn id="S4.p2.2.m1.1.1.1.4.2.2.3" xref="S4.p2.2.m1.1.1.1.4.2.2.3.cmml">1</mn></msub><mo id="S4.p2.2.m1.1.1.1.4.2.1" xref="S4.p2.2.m1.1.1.1.4.2.1.cmml">⁢</mo><msub id="S4.p2.2.m1.1.1.1.4.2.3" xref="S4.p2.2.m1.1.1.1.4.2.3.cmml"><mi id="S4.p2.2.m1.1.1.1.4.2.3.2" xref="S4.p2.2.m1.1.1.1.4.2.3.2.cmml">L</mi><mn id="S4.p2.2.m1.1.1.1.4.2.3.3" xref="S4.p2.2.m1.1.1.1.4.2.3.3.cmml">2</mn></msub></mrow><mn id="S4.p2.2.m1.1.1.1.4.3" xref="S4.p2.2.m1.1.1.1.4.3.cmml">2</mn></mfrac><mo id="S4.p2.2.m1.1.1.1.2a" xref="S4.p2.2.m1.1.1.1.2.cmml">⁢</mo><mrow id="S4.p2.2.m1.1.1.1.1.1" 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id="S4.p2.2.m1.1c">E=\lambda\frac{L_{1}L_{2}}{2}(2i-S)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m1.1d">italic_E = italic_λ divide start_ARG italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ( 2 italic_i - italic_S )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.1">For <math alttext="S=1" class="ltx_Math" display="inline" id="S4.p3.1.m1.1"><semantics id="S4.p3.1.m1.1a"><mrow id="S4.p3.1.m1.1.1" xref="S4.p3.1.m1.1.1.cmml"><mi id="S4.p3.1.m1.1.1.2" xref="S4.p3.1.m1.1.1.2.cmml">S</mi><mo id="S4.p3.1.m1.1.1.1" xref="S4.p3.1.m1.1.1.1.cmml">=</mo><mn id="S4.p3.1.m1.1.1.3" xref="S4.p3.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.1.m1.1b"><apply id="S4.p3.1.m1.1.1.cmml" xref="S4.p3.1.m1.1.1"><eq id="S4.p3.1.m1.1.1.1.cmml" xref="S4.p3.1.m1.1.1.1"></eq><ci id="S4.p3.1.m1.1.1.2.cmml" xref="S4.p3.1.m1.1.1.2">𝑆</ci><cn id="S4.p3.1.m1.1.1.3.cmml" type="integer" xref="S4.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.1.m1.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="S4.p3.1.m1.1d">italic_S = 1</annotation></semantics></math> the two zero-mode tiles are</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx5"> <tbody id="S4.E17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\ket{\psi_{\pm}}=\frac{1}{\sqrt{2}}(\ket{0\ \pm 1}-\ket{\pm 1\ 0})," class="ltx_Math" display="inline" id="S4.E17.m1.4"><semantics id="S4.E17.m1.4a"><mrow id="S4.E17.m1.4.4.1" xref="S4.E17.m1.4.4.1.1.cmml"><mrow id="S4.E17.m1.4.4.1.1" xref="S4.E17.m1.4.4.1.1.cmml"><mrow id="S4.E17.m1.1.1a.3" xref="S4.E17.m1.1.1a.2.cmml"><mo id="S4.E17.m1.1.1a.3.1" stretchy="false" xref="S4.E17.m1.1.1a.2.1.cmml">|</mo><msub id="S4.E17.m1.1.1.1.1" xref="S4.E17.m1.1.1.1.1.cmml"><mi id="S4.E17.m1.1.1.1.1.2" xref="S4.E17.m1.1.1.1.1.2.cmml">ψ</mi><mo id="S4.E17.m1.1.1.1.1.3" xref="S4.E17.m1.1.1.1.1.3.cmml">±</mo></msub><mo id="S4.E17.m1.1.1a.3.2" stretchy="false" xref="S4.E17.m1.1.1a.2.1.cmml">⟩</mo></mrow><mo id="S4.E17.m1.4.4.1.1.2" xref="S4.E17.m1.4.4.1.1.2.cmml">=</mo><mrow id="S4.E17.m1.4.4.1.1.1" xref="S4.E17.m1.4.4.1.1.1.cmml"><mstyle displaystyle="true" id="S4.E17.m1.4.4.1.1.1.3" xref="S4.E17.m1.4.4.1.1.1.3.cmml"><mfrac id="S4.E17.m1.4.4.1.1.1.3a" xref="S4.E17.m1.4.4.1.1.1.3.cmml"><mn id="S4.E17.m1.4.4.1.1.1.3.2" xref="S4.E17.m1.4.4.1.1.1.3.2.cmml">1</mn><msqrt id="S4.E17.m1.4.4.1.1.1.3.3" xref="S4.E17.m1.4.4.1.1.1.3.3.cmml"><mn id="S4.E17.m1.4.4.1.1.1.3.3.2" xref="S4.E17.m1.4.4.1.1.1.3.3.2.cmml">2</mn></msqrt></mfrac></mstyle><mo id="S4.E17.m1.4.4.1.1.1.2" xref="S4.E17.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S4.E17.m1.4.4.1.1.1.1.1" 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xref="S4.E17.m1.2.2a.3"><csymbol cd="latexml" id="S4.E17.m1.2.2a.2.1.cmml" xref="S4.E17.m1.2.2a.3.1">ket</csymbol><apply id="S4.E17.m1.2.2.1.1.cmml" xref="S4.E17.m1.2.2.1.1"><csymbol cd="latexml" id="S4.E17.m1.2.2.1.1.1.cmml" xref="S4.E17.m1.2.2.1.1.1">plus-or-minus</csymbol><cn id="S4.E17.m1.2.2.1.1.2.cmml" type="integer" xref="S4.E17.m1.2.2.1.1.2">0</cn><cn id="S4.E17.m1.2.2.1.1.3.cmml" type="integer" xref="S4.E17.m1.2.2.1.1.3">1</cn></apply></apply><apply id="S4.E17.m1.3.3a.2.cmml" xref="S4.E17.m1.3.3a.3"><csymbol cd="latexml" id="S4.E17.m1.3.3a.2.1.cmml" xref="S4.E17.m1.3.3a.3.1">ket</csymbol><apply id="S4.E17.m1.3.3.1.1.cmml" xref="S4.E17.m1.3.3.1.1"><csymbol cd="latexml" id="S4.E17.m1.3.3.1.1.1.cmml" xref="S4.E17.m1.3.3.1.1">plus-or-minus</csymbol><cn id="S4.E17.m1.3.3.1.1.2.cmml" type="integer" xref="S4.E17.m1.3.3.1.1.2">10</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E17.m1.4c">\displaystyle\ket{\psi_{\pm}}=\frac{1}{\sqrt{2}}(\ket{0\ \pm 1}-\ket{\pm 1\ 0}),</annotation><annotation encoding="application/x-llamapun" id="S4.E17.m1.4d">| start_ARG italic_ψ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT end_ARG ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | start_ARG 0 ± 1 end_ARG ⟩ - | start_ARG ± 1 0 end_ARG ⟩ ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(17)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p3.13">which correspond to choosing <math alttext="i=1" class="ltx_Math" display="inline" id="S4.p3.2.m1.1"><semantics id="S4.p3.2.m1.1a"><mrow id="S4.p3.2.m1.1.1" xref="S4.p3.2.m1.1.1.cmml"><mi id="S4.p3.2.m1.1.1.2" xref="S4.p3.2.m1.1.1.2.cmml">i</mi><mo id="S4.p3.2.m1.1.1.1" xref="S4.p3.2.m1.1.1.1.cmml">=</mo><mn id="S4.p3.2.m1.1.1.3" xref="S4.p3.2.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.2.m1.1b"><apply id="S4.p3.2.m1.1.1.cmml" xref="S4.p3.2.m1.1.1"><eq id="S4.p3.2.m1.1.1.1.cmml" xref="S4.p3.2.m1.1.1.1"></eq><ci id="S4.p3.2.m1.1.1.2.cmml" xref="S4.p3.2.m1.1.1.2">𝑖</ci><cn id="S4.p3.2.m1.1.1.3.cmml" type="integer" xref="S4.p3.2.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.2.m1.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S4.p3.2.m1.1d">italic_i = 1</annotation></semantics></math> and <math alttext="i=0" class="ltx_Math" display="inline" id="S4.p3.3.m2.1"><semantics id="S4.p3.3.m2.1a"><mrow id="S4.p3.3.m2.1.1" xref="S4.p3.3.m2.1.1.cmml"><mi id="S4.p3.3.m2.1.1.2" xref="S4.p3.3.m2.1.1.2.cmml">i</mi><mo id="S4.p3.3.m2.1.1.1" xref="S4.p3.3.m2.1.1.1.cmml">=</mo><mn id="S4.p3.3.m2.1.1.3" xref="S4.p3.3.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.3.m2.1b"><apply id="S4.p3.3.m2.1.1.cmml" xref="S4.p3.3.m2.1.1"><eq id="S4.p3.3.m2.1.1.1.cmml" xref="S4.p3.3.m2.1.1.1"></eq><ci id="S4.p3.3.m2.1.1.2.cmml" xref="S4.p3.3.m2.1.1.2">𝑖</ci><cn id="S4.p3.3.m2.1.1.3.cmml" type="integer" xref="S4.p3.3.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.3.m2.1c">i=0</annotation><annotation encoding="application/x-llamapun" id="S4.p3.3.m2.1d">italic_i = 0</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.E11" title="In III.1 QMBS in TLM With Arbitrary Integer Spin ‣ III Zero-mode scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">11</span></a>). We first study a <math alttext="S=1" class="ltx_Math" display="inline" id="S4.p3.4.m3.1"><semantics id="S4.p3.4.m3.1a"><mrow id="S4.p3.4.m3.1.1" xref="S4.p3.4.m3.1.1.cmml"><mi id="S4.p3.4.m3.1.1.2" xref="S4.p3.4.m3.1.1.2.cmml">S</mi><mo id="S4.p3.4.m3.1.1.1" xref="S4.p3.4.m3.1.1.1.cmml">=</mo><mn id="S4.p3.4.m3.1.1.3" xref="S4.p3.4.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.4.m3.1b"><apply id="S4.p3.4.m3.1.1.cmml" xref="S4.p3.4.m3.1.1"><eq id="S4.p3.4.m3.1.1.1.cmml" xref="S4.p3.4.m3.1.1.1"></eq><ci id="S4.p3.4.m3.1.1.2.cmml" xref="S4.p3.4.m3.1.1.2">𝑆</ci><cn id="S4.p3.4.m3.1.1.3.cmml" type="integer" xref="S4.p3.4.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.4.m3.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="S4.p3.4.m3.1d">italic_S = 1</annotation></semantics></math> ladder with <math alttext="L\times 1" class="ltx_Math" display="inline" id="S4.p3.5.m4.1"><semantics id="S4.p3.5.m4.1a"><mrow id="S4.p3.5.m4.1.1" xref="S4.p3.5.m4.1.1.cmml"><mi id="S4.p3.5.m4.1.1.2" xref="S4.p3.5.m4.1.1.2.cmml">L</mi><mo id="S4.p3.5.m4.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.p3.5.m4.1.1.1.cmml">×</mo><mn id="S4.p3.5.m4.1.1.3" xref="S4.p3.5.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.5.m4.1b"><apply id="S4.p3.5.m4.1.1.cmml" xref="S4.p3.5.m4.1.1"><times id="S4.p3.5.m4.1.1.1.cmml" xref="S4.p3.5.m4.1.1.1"></times><ci id="S4.p3.5.m4.1.1.2.cmml" xref="S4.p3.5.m4.1.1.2">𝐿</ci><cn id="S4.p3.5.m4.1.1.3.cmml" type="integer" xref="S4.p3.5.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.5.m4.1c">L\times 1</annotation><annotation encoding="application/x-llamapun" id="S4.p3.5.m4.1d">italic_L × 1</annotation></semantics></math> plaquettes. This is a chain of plaquettes with periodic boundary conditions in the direction of the chain, but not the direction perpendicular to it. For such a ladder, there are exactly two distinct tilings of <math alttext="2\times 1" class="ltx_Math" display="inline" id="S4.p3.6.m5.1"><semantics id="S4.p3.6.m5.1a"><mrow id="S4.p3.6.m5.1.1" xref="S4.p3.6.m5.1.1.cmml"><mn id="S4.p3.6.m5.1.1.2" xref="S4.p3.6.m5.1.1.2.cmml">2</mn><mo id="S4.p3.6.m5.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.p3.6.m5.1.1.1.cmml">×</mo><mn id="S4.p3.6.m5.1.1.3" xref="S4.p3.6.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.6.m5.1b"><apply id="S4.p3.6.m5.1.1.cmml" xref="S4.p3.6.m5.1.1"><times id="S4.p3.6.m5.1.1.1.cmml" xref="S4.p3.6.m5.1.1.1"></times><cn id="S4.p3.6.m5.1.1.2.cmml" type="integer" xref="S4.p3.6.m5.1.1.2">2</cn><cn id="S4.p3.6.m5.1.1.3.cmml" type="integer" xref="S4.p3.6.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.6.m5.1c">2\times 1</annotation><annotation encoding="application/x-llamapun" id="S4.p3.6.m5.1d">2 × 1</annotation></semantics></math> tiles. They correspond to placing the lower left corner of the tiles in either even or odd sites. These two states, coming from two different tilings with the same <math alttext="i" class="ltx_Math" display="inline" id="S4.p3.7.m6.1"><semantics id="S4.p3.7.m6.1a"><mi id="S4.p3.7.m6.1.1" xref="S4.p3.7.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.p3.7.m6.1b"><ci id="S4.p3.7.m6.1.1.cmml" xref="S4.p3.7.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.7.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.p3.7.m6.1d">italic_i</annotation></semantics></math>, are not orthogonal but generate a two-dimensional scarred Hilbert space. We then expect four scars. Two degenerate low entropy states with energy <math alttext="L\lambda/2" class="ltx_Math" display="inline" id="S4.p3.8.m7.1"><semantics id="S4.p3.8.m7.1a"><mrow id="S4.p3.8.m7.1.1" xref="S4.p3.8.m7.1.1.cmml"><mrow id="S4.p3.8.m7.1.1.2" xref="S4.p3.8.m7.1.1.2.cmml"><mi id="S4.p3.8.m7.1.1.2.2" xref="S4.p3.8.m7.1.1.2.2.cmml">L</mi><mo id="S4.p3.8.m7.1.1.2.1" xref="S4.p3.8.m7.1.1.2.1.cmml">⁢</mo><mi id="S4.p3.8.m7.1.1.2.3" xref="S4.p3.8.m7.1.1.2.3.cmml">λ</mi></mrow><mo id="S4.p3.8.m7.1.1.1" xref="S4.p3.8.m7.1.1.1.cmml">/</mo><mn id="S4.p3.8.m7.1.1.3" xref="S4.p3.8.m7.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.8.m7.1b"><apply id="S4.p3.8.m7.1.1.cmml" xref="S4.p3.8.m7.1.1"><divide id="S4.p3.8.m7.1.1.1.cmml" xref="S4.p3.8.m7.1.1.1"></divide><apply id="S4.p3.8.m7.1.1.2.cmml" xref="S4.p3.8.m7.1.1.2"><times id="S4.p3.8.m7.1.1.2.1.cmml" xref="S4.p3.8.m7.1.1.2.1"></times><ci id="S4.p3.8.m7.1.1.2.2.cmml" xref="S4.p3.8.m7.1.1.2.2">𝐿</ci><ci id="S4.p3.8.m7.1.1.2.3.cmml" xref="S4.p3.8.m7.1.1.2.3">𝜆</ci></apply><cn id="S4.p3.8.m7.1.1.3.cmml" type="integer" xref="S4.p3.8.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.8.m7.1c">L\lambda/2</annotation><annotation encoding="application/x-llamapun" id="S4.p3.8.m7.1d">italic_L italic_λ / 2</annotation></semantics></math> from the tilings of state <math alttext="\ket{\psi_{+}}" class="ltx_Math" display="inline" id="S4.p3.9.m8.1"><semantics id="S4.p3.9.m8.1a"><mrow id="S4.p3.9.m8.1.1.3" xref="S4.p3.9.m8.1.1.2.cmml"><mo id="S4.p3.9.m8.1.1.3.1" stretchy="false" xref="S4.p3.9.m8.1.1.2.1.cmml">|</mo><msub id="S4.p3.9.m8.1.1.1.1" xref="S4.p3.9.m8.1.1.1.1.cmml"><mi id="S4.p3.9.m8.1.1.1.1.2" xref="S4.p3.9.m8.1.1.1.1.2.cmml">ψ</mi><mo id="S4.p3.9.m8.1.1.1.1.3" xref="S4.p3.9.m8.1.1.1.1.3.cmml">+</mo></msub><mo id="S4.p3.9.m8.1.1.3.2" stretchy="false" xref="S4.p3.9.m8.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.9.m8.1b"><apply id="S4.p3.9.m8.1.1.2.cmml" xref="S4.p3.9.m8.1.1.3"><csymbol cd="latexml" id="S4.p3.9.m8.1.1.2.1.cmml" xref="S4.p3.9.m8.1.1.3.1">ket</csymbol><apply id="S4.p3.9.m8.1.1.1.1.cmml" xref="S4.p3.9.m8.1.1.1.1"><csymbol cd="ambiguous" id="S4.p3.9.m8.1.1.1.1.1.cmml" xref="S4.p3.9.m8.1.1.1.1">subscript</csymbol><ci id="S4.p3.9.m8.1.1.1.1.2.cmml" xref="S4.p3.9.m8.1.1.1.1.2">𝜓</ci><plus id="S4.p3.9.m8.1.1.1.1.3.cmml" xref="S4.p3.9.m8.1.1.1.1.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.9.m8.1c">\ket{\psi_{+}}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.9.m8.1d">| start_ARG italic_ψ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> (<math alttext="i=1" class="ltx_Math" display="inline" id="S4.p3.10.m9.1"><semantics id="S4.p3.10.m9.1a"><mrow id="S4.p3.10.m9.1.1" xref="S4.p3.10.m9.1.1.cmml"><mi id="S4.p3.10.m9.1.1.2" xref="S4.p3.10.m9.1.1.2.cmml">i</mi><mo id="S4.p3.10.m9.1.1.1" xref="S4.p3.10.m9.1.1.1.cmml">=</mo><mn id="S4.p3.10.m9.1.1.3" xref="S4.p3.10.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.10.m9.1b"><apply id="S4.p3.10.m9.1.1.cmml" xref="S4.p3.10.m9.1.1"><eq id="S4.p3.10.m9.1.1.1.cmml" xref="S4.p3.10.m9.1.1.1"></eq><ci id="S4.p3.10.m9.1.1.2.cmml" xref="S4.p3.10.m9.1.1.2">𝑖</ci><cn id="S4.p3.10.m9.1.1.3.cmml" type="integer" xref="S4.p3.10.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.10.m9.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S4.p3.10.m9.1d">italic_i = 1</annotation></semantics></math>) and two with energy <math alttext="-L\lambda/2" class="ltx_Math" display="inline" id="S4.p3.11.m10.1"><semantics id="S4.p3.11.m10.1a"><mrow id="S4.p3.11.m10.1.1" xref="S4.p3.11.m10.1.1.cmml"><mo id="S4.p3.11.m10.1.1a" xref="S4.p3.11.m10.1.1.cmml">−</mo><mrow id="S4.p3.11.m10.1.1.2" xref="S4.p3.11.m10.1.1.2.cmml"><mrow id="S4.p3.11.m10.1.1.2.2" xref="S4.p3.11.m10.1.1.2.2.cmml"><mi id="S4.p3.11.m10.1.1.2.2.2" xref="S4.p3.11.m10.1.1.2.2.2.cmml">L</mi><mo id="S4.p3.11.m10.1.1.2.2.1" xref="S4.p3.11.m10.1.1.2.2.1.cmml">⁢</mo><mi id="S4.p3.11.m10.1.1.2.2.3" xref="S4.p3.11.m10.1.1.2.2.3.cmml">λ</mi></mrow><mo id="S4.p3.11.m10.1.1.2.1" xref="S4.p3.11.m10.1.1.2.1.cmml">/</mo><mn id="S4.p3.11.m10.1.1.2.3" xref="S4.p3.11.m10.1.1.2.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.11.m10.1b"><apply id="S4.p3.11.m10.1.1.cmml" xref="S4.p3.11.m10.1.1"><minus id="S4.p3.11.m10.1.1.1.cmml" xref="S4.p3.11.m10.1.1"></minus><apply id="S4.p3.11.m10.1.1.2.cmml" xref="S4.p3.11.m10.1.1.2"><divide id="S4.p3.11.m10.1.1.2.1.cmml" xref="S4.p3.11.m10.1.1.2.1"></divide><apply id="S4.p3.11.m10.1.1.2.2.cmml" xref="S4.p3.11.m10.1.1.2.2"><times id="S4.p3.11.m10.1.1.2.2.1.cmml" xref="S4.p3.11.m10.1.1.2.2.1"></times><ci id="S4.p3.11.m10.1.1.2.2.2.cmml" xref="S4.p3.11.m10.1.1.2.2.2">𝐿</ci><ci id="S4.p3.11.m10.1.1.2.2.3.cmml" xref="S4.p3.11.m10.1.1.2.2.3">𝜆</ci></apply><cn id="S4.p3.11.m10.1.1.2.3.cmml" type="integer" xref="S4.p3.11.m10.1.1.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.11.m10.1c">-L\lambda/2</annotation><annotation encoding="application/x-llamapun" id="S4.p3.11.m10.1d">- italic_L italic_λ / 2</annotation></semantics></math> from tilings of state <math alttext="\ket{\psi_{-}}" class="ltx_Math" display="inline" id="S4.p3.12.m11.1"><semantics id="S4.p3.12.m11.1a"><mrow id="S4.p3.12.m11.1.1.3" xref="S4.p3.12.m11.1.1.2.cmml"><mo id="S4.p3.12.m11.1.1.3.1" stretchy="false" xref="S4.p3.12.m11.1.1.2.1.cmml">|</mo><msub id="S4.p3.12.m11.1.1.1.1" xref="S4.p3.12.m11.1.1.1.1.cmml"><mi id="S4.p3.12.m11.1.1.1.1.2" xref="S4.p3.12.m11.1.1.1.1.2.cmml">ψ</mi><mo id="S4.p3.12.m11.1.1.1.1.3" xref="S4.p3.12.m11.1.1.1.1.3.cmml">−</mo></msub><mo id="S4.p3.12.m11.1.1.3.2" stretchy="false" xref="S4.p3.12.m11.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.12.m11.1b"><apply id="S4.p3.12.m11.1.1.2.cmml" xref="S4.p3.12.m11.1.1.3"><csymbol cd="latexml" id="S4.p3.12.m11.1.1.2.1.cmml" xref="S4.p3.12.m11.1.1.3.1">ket</csymbol><apply id="S4.p3.12.m11.1.1.1.1.cmml" xref="S4.p3.12.m11.1.1.1.1"><csymbol cd="ambiguous" id="S4.p3.12.m11.1.1.1.1.1.cmml" xref="S4.p3.12.m11.1.1.1.1">subscript</csymbol><ci id="S4.p3.12.m11.1.1.1.1.2.cmml" xref="S4.p3.12.m11.1.1.1.1.2">𝜓</ci><minus id="S4.p3.12.m11.1.1.1.1.3.cmml" xref="S4.p3.12.m11.1.1.1.1.3"></minus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.12.m11.1c">\ket{\psi_{-}}</annotation><annotation encoding="application/x-llamapun" id="S4.p3.12.m11.1d">| start_ARG italic_ψ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> (<math alttext="i=0" class="ltx_Math" display="inline" id="S4.p3.13.m12.1"><semantics id="S4.p3.13.m12.1a"><mrow id="S4.p3.13.m12.1.1" xref="S4.p3.13.m12.1.1.cmml"><mi id="S4.p3.13.m12.1.1.2" xref="S4.p3.13.m12.1.1.2.cmml">i</mi><mo id="S4.p3.13.m12.1.1.1" xref="S4.p3.13.m12.1.1.1.cmml">=</mo><mn id="S4.p3.13.m12.1.1.3" xref="S4.p3.13.m12.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.13.m12.1b"><apply id="S4.p3.13.m12.1.1.cmml" xref="S4.p3.13.m12.1.1"><eq id="S4.p3.13.m12.1.1.1.cmml" xref="S4.p3.13.m12.1.1.1"></eq><ci id="S4.p3.13.m12.1.1.2.cmml" xref="S4.p3.13.m12.1.1.2">𝑖</ci><cn id="S4.p3.13.m12.1.1.3.cmml" type="integer" xref="S4.p3.13.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.13.m12.1c">i=0</annotation><annotation encoding="application/x-llamapun" id="S4.p3.13.m12.1d">italic_i = 0</annotation></semantics></math>). This is exactly what is observed when performing exact diagonalization on small systems, the results of which are presented in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S4.F5" title="Figure 5 ‣ IV Numerical results and discussion ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">5</span></a>. The four low-entropy states are clearly identifiable and each of them is composed of the expected states, as shown in the Supplementary Material <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib56" title="">56</a>]</cite> by looking at the amplitude of the scars in the electric field basis.</p> </div> <figure class="ltx_figure" id="S4.F5"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="220" id="S4.F5.g1" src="extracted/5828746/images/entanglementEntropyS=1.jpeg" width="293"/></div> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="220" id="S4.F5.g2" src="extracted/5828746/images/entanglementEntropy6x2.jpeg" width="293"/></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5: </span> Entanglement entropy for two different volumes with periodic boundary conditions in the <math alttext="x" class="ltx_Math" display="inline" id="S4.F5.6.m1.1"><semantics id="S4.F5.6.m1.1b"><mi id="S4.F5.6.m1.1.1" xref="S4.F5.6.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.F5.6.m1.1c"><ci id="S4.F5.6.m1.1.1.cmml" xref="S4.F5.6.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.6.m1.1d">x</annotation><annotation encoding="application/x-llamapun" id="S4.F5.6.m1.1e">italic_x</annotation></semantics></math> direction, <math alttext="S=1" class="ltx_Math" display="inline" id="S4.F5.7.m2.1"><semantics id="S4.F5.7.m2.1b"><mrow id="S4.F5.7.m2.1.1" xref="S4.F5.7.m2.1.1.cmml"><mi id="S4.F5.7.m2.1.1.2" xref="S4.F5.7.m2.1.1.2.cmml">S</mi><mo id="S4.F5.7.m2.1.1.1" xref="S4.F5.7.m2.1.1.1.cmml">=</mo><mn id="S4.F5.7.m2.1.1.3" xref="S4.F5.7.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.7.m2.1c"><apply id="S4.F5.7.m2.1.1.cmml" xref="S4.F5.7.m2.1.1"><eq id="S4.F5.7.m2.1.1.1.cmml" xref="S4.F5.7.m2.1.1.1"></eq><ci id="S4.F5.7.m2.1.1.2.cmml" xref="S4.F5.7.m2.1.1.2">𝑆</ci><cn id="S4.F5.7.m2.1.1.3.cmml" type="integer" xref="S4.F5.7.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.7.m2.1d">S=1</annotation><annotation encoding="application/x-llamapun" id="S4.F5.7.m2.1e">italic_S = 1</annotation></semantics></math>, and height potential (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S4.E16" title="In IV Numerical results and discussion ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">16</span></a>) with <math alttext="\lambda=0.2" class="ltx_Math" display="inline" id="S4.F5.8.m3.1"><semantics id="S4.F5.8.m3.1b"><mrow id="S4.F5.8.m3.1.1" xref="S4.F5.8.m3.1.1.cmml"><mi id="S4.F5.8.m3.1.1.2" xref="S4.F5.8.m3.1.1.2.cmml">λ</mi><mo id="S4.F5.8.m3.1.1.1" xref="S4.F5.8.m3.1.1.1.cmml">=</mo><mn id="S4.F5.8.m3.1.1.3" xref="S4.F5.8.m3.1.1.3.cmml">0.2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.8.m3.1c"><apply id="S4.F5.8.m3.1.1.cmml" xref="S4.F5.8.m3.1.1"><eq id="S4.F5.8.m3.1.1.1.cmml" xref="S4.F5.8.m3.1.1.1"></eq><ci id="S4.F5.8.m3.1.1.2.cmml" xref="S4.F5.8.m3.1.1.2">𝜆</ci><cn id="S4.F5.8.m3.1.1.3.cmml" type="float" xref="S4.F5.8.m3.1.1.3">0.2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.8.m3.1d">\lambda=0.2</annotation><annotation encoding="application/x-llamapun" id="S4.F5.8.m3.1e">italic_λ = 0.2</annotation></semantics></math>. Brighter colors indicate a higher density of states. Left: <math alttext="10\times 1" class="ltx_Math" display="inline" id="S4.F5.9.m4.1"><semantics id="S4.F5.9.m4.1b"><mrow id="S4.F5.9.m4.1.1" xref="S4.F5.9.m4.1.1.cmml"><mn id="S4.F5.9.m4.1.1.2" xref="S4.F5.9.m4.1.1.2.cmml">10</mn><mo id="S4.F5.9.m4.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.F5.9.m4.1.1.1.cmml">×</mo><mn id="S4.F5.9.m4.1.1.3" xref="S4.F5.9.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.9.m4.1c"><apply id="S4.F5.9.m4.1.1.cmml" xref="S4.F5.9.m4.1.1"><times id="S4.F5.9.m4.1.1.1.cmml" xref="S4.F5.9.m4.1.1.1"></times><cn id="S4.F5.9.m4.1.1.2.cmml" type="integer" xref="S4.F5.9.m4.1.1.2">10</cn><cn id="S4.F5.9.m4.1.1.3.cmml" type="integer" xref="S4.F5.9.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.9.m4.1d">10\times 1</annotation><annotation encoding="application/x-llamapun" id="S4.F5.9.m4.1e">10 × 1</annotation></semantics></math> ladder. Four mid-spectrum states with low entropy, shown in red, are visible. They are superpositions of the predicted states. Right: <math alttext="6\times 2" class="ltx_Math" display="inline" id="S4.F5.10.m5.1"><semantics id="S4.F5.10.m5.1b"><mrow id="S4.F5.10.m5.1.1" xref="S4.F5.10.m5.1.1.cmml"><mn id="S4.F5.10.m5.1.1.2" xref="S4.F5.10.m5.1.1.2.cmml">6</mn><mo id="S4.F5.10.m5.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.F5.10.m5.1.1.1.cmml">×</mo><mn id="S4.F5.10.m5.1.1.3" xref="S4.F5.10.m5.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.10.m5.1c"><apply id="S4.F5.10.m5.1.1.cmml" xref="S4.F5.10.m5.1.1"><times id="S4.F5.10.m5.1.1.1.cmml" xref="S4.F5.10.m5.1.1.1"></times><cn id="S4.F5.10.m5.1.1.2.cmml" type="integer" xref="S4.F5.10.m5.1.1.2">6</cn><cn id="S4.F5.10.m5.1.1.3.cmml" type="integer" xref="S4.F5.10.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.10.m5.1d">6\times 2</annotation><annotation encoding="application/x-llamapun" id="S4.F5.10.m5.1e">6 × 2</annotation></semantics></math> volume in the zero momentum sector. Many mid-spectrum states with low entropy are present. The towers marked in red contain the predicted states. </figcaption> </figure> <div class="ltx_para" id="S4.p4"> <p class="ltx_p" id="S4.p4.4">For systems with a larger vertical direction, there are many more ways to tile the plane with <math alttext="2\times 1" class="ltx_Math" display="inline" id="S4.p4.1.m1.1"><semantics id="S4.p4.1.m1.1a"><mrow id="S4.p4.1.m1.1.1" xref="S4.p4.1.m1.1.1.cmml"><mn id="S4.p4.1.m1.1.1.2" xref="S4.p4.1.m1.1.1.2.cmml">2</mn><mo id="S4.p4.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.p4.1.m1.1.1.1.cmml">×</mo><mn id="S4.p4.1.m1.1.1.3" xref="S4.p4.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.1.m1.1b"><apply id="S4.p4.1.m1.1.1.cmml" xref="S4.p4.1.m1.1.1"><times id="S4.p4.1.m1.1.1.1.cmml" xref="S4.p4.1.m1.1.1.1"></times><cn id="S4.p4.1.m1.1.1.2.cmml" type="integer" xref="S4.p4.1.m1.1.1.2">2</cn><cn id="S4.p4.1.m1.1.1.3.cmml" type="integer" xref="S4.p4.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.1.m1.1c">2\times 1</annotation><annotation encoding="application/x-llamapun" id="S4.p4.1.m1.1d">2 × 1</annotation></semantics></math> and <math alttext="1\times 2" class="ltx_Math" display="inline" id="S4.p4.2.m2.1"><semantics id="S4.p4.2.m2.1a"><mrow id="S4.p4.2.m2.1.1" xref="S4.p4.2.m2.1.1.cmml"><mn id="S4.p4.2.m2.1.1.2" xref="S4.p4.2.m2.1.1.2.cmml">1</mn><mo id="S4.p4.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.p4.2.m2.1.1.1.cmml">×</mo><mn id="S4.p4.2.m2.1.1.3" xref="S4.p4.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.2.m2.1b"><apply id="S4.p4.2.m2.1.1.cmml" xref="S4.p4.2.m2.1.1"><times id="S4.p4.2.m2.1.1.1.cmml" xref="S4.p4.2.m2.1.1.1"></times><cn id="S4.p4.2.m2.1.1.2.cmml" type="integer" xref="S4.p4.2.m2.1.1.2">1</cn><cn id="S4.p4.2.m2.1.1.3.cmml" type="integer" xref="S4.p4.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.2.m2.1c">1\times 2</annotation><annotation encoding="application/x-llamapun" id="S4.p4.2.m2.1d">1 × 2</annotation></semantics></math> tiles. Adding a second leg is enough to make the number of possible tilings exponentially large, all of which will mix in the spectrum. Using the sum of the height variables as the potential, we expect towers of low-entropy states at energies <math alttext="\pm L_{1}L_{2}\lambda/2" class="ltx_Math" display="inline" id="S4.p4.3.m3.1"><semantics id="S4.p4.3.m3.1a"><mrow id="S4.p4.3.m3.1.1" xref="S4.p4.3.m3.1.1.cmml"><mo id="S4.p4.3.m3.1.1a" xref="S4.p4.3.m3.1.1.cmml">±</mo><mrow id="S4.p4.3.m3.1.1.2" xref="S4.p4.3.m3.1.1.2.cmml"><mrow id="S4.p4.3.m3.1.1.2.2" xref="S4.p4.3.m3.1.1.2.2.cmml"><msub id="S4.p4.3.m3.1.1.2.2.2" xref="S4.p4.3.m3.1.1.2.2.2.cmml"><mi id="S4.p4.3.m3.1.1.2.2.2.2" xref="S4.p4.3.m3.1.1.2.2.2.2.cmml">L</mi><mn id="S4.p4.3.m3.1.1.2.2.2.3" xref="S4.p4.3.m3.1.1.2.2.2.3.cmml">1</mn></msub><mo id="S4.p4.3.m3.1.1.2.2.1" xref="S4.p4.3.m3.1.1.2.2.1.cmml">⁢</mo><msub id="S4.p4.3.m3.1.1.2.2.3" xref="S4.p4.3.m3.1.1.2.2.3.cmml"><mi id="S4.p4.3.m3.1.1.2.2.3.2" xref="S4.p4.3.m3.1.1.2.2.3.2.cmml">L</mi><mn id="S4.p4.3.m3.1.1.2.2.3.3" xref="S4.p4.3.m3.1.1.2.2.3.3.cmml">2</mn></msub><mo id="S4.p4.3.m3.1.1.2.2.1a" xref="S4.p4.3.m3.1.1.2.2.1.cmml">⁢</mo><mi id="S4.p4.3.m3.1.1.2.2.4" xref="S4.p4.3.m3.1.1.2.2.4.cmml">λ</mi></mrow><mo id="S4.p4.3.m3.1.1.2.1" xref="S4.p4.3.m3.1.1.2.1.cmml">/</mo><mn id="S4.p4.3.m3.1.1.2.3" xref="S4.p4.3.m3.1.1.2.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.3.m3.1b"><apply id="S4.p4.3.m3.1.1.cmml" xref="S4.p4.3.m3.1.1"><csymbol cd="latexml" id="S4.p4.3.m3.1.1.1.cmml" xref="S4.p4.3.m3.1.1">plus-or-minus</csymbol><apply id="S4.p4.3.m3.1.1.2.cmml" xref="S4.p4.3.m3.1.1.2"><divide id="S4.p4.3.m3.1.1.2.1.cmml" xref="S4.p4.3.m3.1.1.2.1"></divide><apply id="S4.p4.3.m3.1.1.2.2.cmml" xref="S4.p4.3.m3.1.1.2.2"><times id="S4.p4.3.m3.1.1.2.2.1.cmml" xref="S4.p4.3.m3.1.1.2.2.1"></times><apply id="S4.p4.3.m3.1.1.2.2.2.cmml" xref="S4.p4.3.m3.1.1.2.2.2"><csymbol cd="ambiguous" id="S4.p4.3.m3.1.1.2.2.2.1.cmml" xref="S4.p4.3.m3.1.1.2.2.2">subscript</csymbol><ci id="S4.p4.3.m3.1.1.2.2.2.2.cmml" xref="S4.p4.3.m3.1.1.2.2.2.2">𝐿</ci><cn id="S4.p4.3.m3.1.1.2.2.2.3.cmml" type="integer" xref="S4.p4.3.m3.1.1.2.2.2.3">1</cn></apply><apply id="S4.p4.3.m3.1.1.2.2.3.cmml" xref="S4.p4.3.m3.1.1.2.2.3"><csymbol cd="ambiguous" id="S4.p4.3.m3.1.1.2.2.3.1.cmml" xref="S4.p4.3.m3.1.1.2.2.3">subscript</csymbol><ci id="S4.p4.3.m3.1.1.2.2.3.2.cmml" xref="S4.p4.3.m3.1.1.2.2.3.2">𝐿</ci><cn id="S4.p4.3.m3.1.1.2.2.3.3.cmml" type="integer" xref="S4.p4.3.m3.1.1.2.2.3.3">2</cn></apply><ci id="S4.p4.3.m3.1.1.2.2.4.cmml" xref="S4.p4.3.m3.1.1.2.2.4">𝜆</ci></apply><cn id="S4.p4.3.m3.1.1.2.3.cmml" type="integer" xref="S4.p4.3.m3.1.1.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.3.m3.1c">\pm L_{1}L_{2}\lambda/2</annotation><annotation encoding="application/x-llamapun" id="S4.p4.3.m3.1d">± italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ / 2</annotation></semantics></math>. These two towers are observed, as depicted in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S4.F5" title="Figure 5 ‣ IV Numerical results and discussion ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">5</span></a>. There are many other low-entropy states at other energies. While we observed these states to be built from zero-mode tiles, we do not expect their construction to be generalizable beyond <math alttext="2\times L" class="ltx_Math" display="inline" id="S4.p4.4.m4.1"><semantics id="S4.p4.4.m4.1a"><mrow id="S4.p4.4.m4.1.1" xref="S4.p4.4.m4.1.1.cmml"><mn id="S4.p4.4.m4.1.1.2" xref="S4.p4.4.m4.1.1.2.cmml">2</mn><mo id="S4.p4.4.m4.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.p4.4.m4.1.1.1.cmml">×</mo><mi id="S4.p4.4.m4.1.1.3" xref="S4.p4.4.m4.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p4.4.m4.1b"><apply id="S4.p4.4.m4.1.1.cmml" xref="S4.p4.4.m4.1.1"><times id="S4.p4.4.m4.1.1.1.cmml" xref="S4.p4.4.m4.1.1.1"></times><cn id="S4.p4.4.m4.1.1.2.cmml" type="integer" xref="S4.p4.4.m4.1.1.2">2</cn><ci id="S4.p4.4.m4.1.1.3.cmml" xref="S4.p4.4.m4.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p4.4.m4.1c">2\times L</annotation><annotation encoding="application/x-llamapun" id="S4.p4.4.m4.1d">2 × italic_L</annotation></semantics></math> systems and will not attempt a detailed characterization here.</p> </div> <div class="ltx_para" id="S4.p5"> <p class="ltx_p" id="S4.p5.3">All scars we have constructed also exist in periodic boundary conditions. Using the sum of the height variables as the potential is not well defined, as previously discussed. Instead, for the scars of the form (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S5.E24" title="In V A 4×4 Scar for the 𝐸² potential ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">24</span></a>), we use the minimum number of kinetic operators that must be applied to the state <math alttext="\ket{\textbf{0}}" class="ltx_Math" display="inline" id="S4.p5.1.m1.1"><semantics id="S4.p5.1.m1.1a"><mrow id="S4.p5.1.m1.1.1.3" xref="S4.p5.1.m1.1.1.2.cmml"><mo id="S4.p5.1.m1.1.1.3.1" stretchy="false" xref="S4.p5.1.m1.1.1.2.1.cmml">|</mo><mtext class="ltx_mathvariant_bold" id="S4.p5.1.m1.1.1.1.1" xref="S4.p5.1.m1.1.1.1.1a.cmml">0</mtext><mo id="S4.p5.1.m1.1.1.3.2" stretchy="false" xref="S4.p5.1.m1.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p5.1.m1.1b"><apply id="S4.p5.1.m1.1.1.2.cmml" xref="S4.p5.1.m1.1.1.3"><csymbol cd="latexml" id="S4.p5.1.m1.1.1.2.1.cmml" xref="S4.p5.1.m1.1.1.3.1">ket</csymbol><ci id="S4.p5.1.m1.1.1.1.1a.cmml" xref="S4.p5.1.m1.1.1.1.1"><mtext class="ltx_mathvariant_bold" id="S4.p5.1.m1.1.1.1.1.cmml" xref="S4.p5.1.m1.1.1.1.1">0</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.1.m1.1c">\ket{\textbf{0}}</annotation><annotation encoding="application/x-llamapun" id="S4.p5.1.m1.1d">| start_ARG 0 end_ARG ⟩</annotation></semantics></math> to arrive at any given electric field basis state <math alttext="\ket{\phi_{i}}" class="ltx_Math" display="inline" id="S4.p5.2.m2.1"><semantics id="S4.p5.2.m2.1a"><mrow id="S4.p5.2.m2.1.1.3" xref="S4.p5.2.m2.1.1.2.cmml"><mo id="S4.p5.2.m2.1.1.3.1" stretchy="false" xref="S4.p5.2.m2.1.1.2.1.cmml">|</mo><msub id="S4.p5.2.m2.1.1.1.1" xref="S4.p5.2.m2.1.1.1.1.cmml"><mi id="S4.p5.2.m2.1.1.1.1.2" xref="S4.p5.2.m2.1.1.1.1.2.cmml">ϕ</mi><mi id="S4.p5.2.m2.1.1.1.1.3" xref="S4.p5.2.m2.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.p5.2.m2.1.1.3.2" stretchy="false" xref="S4.p5.2.m2.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p5.2.m2.1b"><apply id="S4.p5.2.m2.1.1.2.cmml" xref="S4.p5.2.m2.1.1.3"><csymbol cd="latexml" id="S4.p5.2.m2.1.1.2.1.cmml" xref="S4.p5.2.m2.1.1.3.1">ket</csymbol><apply id="S4.p5.2.m2.1.1.1.1.cmml" xref="S4.p5.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.p5.2.m2.1.1.1.1.1.cmml" xref="S4.p5.2.m2.1.1.1.1">subscript</csymbol><ci id="S4.p5.2.m2.1.1.1.1.2.cmml" xref="S4.p5.2.m2.1.1.1.1.2">italic-ϕ</ci><ci id="S4.p5.2.m2.1.1.1.1.3.cmml" xref="S4.p5.2.m2.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.2.m2.1c">\ket{\phi_{i}}</annotation><annotation encoding="application/x-llamapun" id="S4.p5.2.m2.1d">| start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math>. This is similar to the Hamming distance as used in e.g. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib64" title="">64</a>]</cite> generalized for higher spin. We therefore call this the Hamming distance potential. The entropy using this potential is shown in the left panel of Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S4.F6" title="Figure 6 ‣ IV Numerical results and discussion ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">6</span></a>, where the tower of predicted scars is visible. Additional towers are present at non-integer multiples of <math alttext="\lambda" class="ltx_Math" display="inline" id="S4.p5.3.m3.1"><semantics id="S4.p5.3.m3.1a"><mi id="S4.p5.3.m3.1.1" xref="S4.p5.3.m3.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S4.p5.3.m3.1b"><ci id="S4.p5.3.m3.1.1.cmml" xref="S4.p5.3.m3.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p5.3.m3.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S4.p5.3.m3.1d">italic_λ</annotation></semantics></math>, which are not characterized here.</p> </div> <div class="ltx_para" id="S4.p6"> <p class="ltx_p" id="S4.p6.1">For the other types of scars described for <math alttext="S=1" class="ltx_Math" display="inline" id="S4.p6.1.m1.1"><semantics id="S4.p6.1.m1.1a"><mrow id="S4.p6.1.m1.1.1" xref="S4.p6.1.m1.1.1.cmml"><mi id="S4.p6.1.m1.1.1.2" xref="S4.p6.1.m1.1.1.2.cmml">S</mi><mo id="S4.p6.1.m1.1.1.1" xref="S4.p6.1.m1.1.1.1.cmml">=</mo><mn id="S4.p6.1.m1.1.1.3" xref="S4.p6.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p6.1.m1.1b"><apply id="S4.p6.1.m1.1.1.cmml" xref="S4.p6.1.m1.1.1"><eq id="S4.p6.1.m1.1.1.1.cmml" xref="S4.p6.1.m1.1.1.1"></eq><ci id="S4.p6.1.m1.1.1.2.cmml" xref="S4.p6.1.m1.1.1.2">𝑆</ci><cn id="S4.p6.1.m1.1.1.3.cmml" type="integer" xref="S4.p6.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p6.1.m1.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="S4.p6.1.m1.1d">italic_S = 1</annotation></semantics></math>, all contributing states in (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.E12" title="In Diagonal Tiling ‣ III.2 Beyond Zero-Mode Building Blocks ‣ III Zero-mode scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">12</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.E13" title="In Non-Tiling Scar ‣ III.2 Beyond Zero-Mode Building Blocks ‣ III Zero-mode scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">13</span></a>) have the same number of links with electric field zero. In the first case, half of the links are zero while in the second no link is zero. These scars can then be isolated by the standard electric field term</p> <table class="ltx_equation ltx_eqn_table" id="S4.E18"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="V=g\sum_{n}E_{n}^{2}." class="ltx_Math" display="block" id="S4.E18.m1.1"><semantics id="S4.E18.m1.1a"><mrow id="S4.E18.m1.1.1.1" xref="S4.E18.m1.1.1.1.1.cmml"><mrow id="S4.E18.m1.1.1.1.1" xref="S4.E18.m1.1.1.1.1.cmml"><mi id="S4.E18.m1.1.1.1.1.2" xref="S4.E18.m1.1.1.1.1.2.cmml">V</mi><mo id="S4.E18.m1.1.1.1.1.1" xref="S4.E18.m1.1.1.1.1.1.cmml">=</mo><mrow id="S4.E18.m1.1.1.1.1.3" xref="S4.E18.m1.1.1.1.1.3.cmml"><mi id="S4.E18.m1.1.1.1.1.3.2" xref="S4.E18.m1.1.1.1.1.3.2.cmml">g</mi><mo id="S4.E18.m1.1.1.1.1.3.1" xref="S4.E18.m1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S4.E18.m1.1.1.1.1.3.3" xref="S4.E18.m1.1.1.1.1.3.3.cmml"><munder id="S4.E18.m1.1.1.1.1.3.3.1" xref="S4.E18.m1.1.1.1.1.3.3.1.cmml"><mo id="S4.E18.m1.1.1.1.1.3.3.1.2" movablelimits="false" xref="S4.E18.m1.1.1.1.1.3.3.1.2.cmml">∑</mo><mi id="S4.E18.m1.1.1.1.1.3.3.1.3" xref="S4.E18.m1.1.1.1.1.3.3.1.3.cmml">n</mi></munder><msubsup id="S4.E18.m1.1.1.1.1.3.3.2" xref="S4.E18.m1.1.1.1.1.3.3.2.cmml"><mi id="S4.E18.m1.1.1.1.1.3.3.2.2.2" xref="S4.E18.m1.1.1.1.1.3.3.2.2.2.cmml">E</mi><mi id="S4.E18.m1.1.1.1.1.3.3.2.2.3" xref="S4.E18.m1.1.1.1.1.3.3.2.2.3.cmml">n</mi><mn id="S4.E18.m1.1.1.1.1.3.3.2.3" xref="S4.E18.m1.1.1.1.1.3.3.2.3.cmml">2</mn></msubsup></mrow></mrow></mrow><mo id="S4.E18.m1.1.1.1.2" lspace="0em" xref="S4.E18.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E18.m1.1b"><apply id="S4.E18.m1.1.1.1.1.cmml" xref="S4.E18.m1.1.1.1"><eq id="S4.E18.m1.1.1.1.1.1.cmml" xref="S4.E18.m1.1.1.1.1.1"></eq><ci id="S4.E18.m1.1.1.1.1.2.cmml" xref="S4.E18.m1.1.1.1.1.2">𝑉</ci><apply id="S4.E18.m1.1.1.1.1.3.cmml" xref="S4.E18.m1.1.1.1.1.3"><times id="S4.E18.m1.1.1.1.1.3.1.cmml" xref="S4.E18.m1.1.1.1.1.3.1"></times><ci id="S4.E18.m1.1.1.1.1.3.2.cmml" xref="S4.E18.m1.1.1.1.1.3.2">𝑔</ci><apply id="S4.E18.m1.1.1.1.1.3.3.cmml" xref="S4.E18.m1.1.1.1.1.3.3"><apply id="S4.E18.m1.1.1.1.1.3.3.1.cmml" xref="S4.E18.m1.1.1.1.1.3.3.1"><csymbol cd="ambiguous" id="S4.E18.m1.1.1.1.1.3.3.1.1.cmml" xref="S4.E18.m1.1.1.1.1.3.3.1">subscript</csymbol><sum id="S4.E18.m1.1.1.1.1.3.3.1.2.cmml" xref="S4.E18.m1.1.1.1.1.3.3.1.2"></sum><ci id="S4.E18.m1.1.1.1.1.3.3.1.3.cmml" xref="S4.E18.m1.1.1.1.1.3.3.1.3">𝑛</ci></apply><apply id="S4.E18.m1.1.1.1.1.3.3.2.cmml" xref="S4.E18.m1.1.1.1.1.3.3.2"><csymbol cd="ambiguous" id="S4.E18.m1.1.1.1.1.3.3.2.1.cmml" xref="S4.E18.m1.1.1.1.1.3.3.2">superscript</csymbol><apply id="S4.E18.m1.1.1.1.1.3.3.2.2.cmml" xref="S4.E18.m1.1.1.1.1.3.3.2"><csymbol cd="ambiguous" id="S4.E18.m1.1.1.1.1.3.3.2.2.1.cmml" xref="S4.E18.m1.1.1.1.1.3.3.2">subscript</csymbol><ci id="S4.E18.m1.1.1.1.1.3.3.2.2.2.cmml" xref="S4.E18.m1.1.1.1.1.3.3.2.2.2">𝐸</ci><ci id="S4.E18.m1.1.1.1.1.3.3.2.2.3.cmml" xref="S4.E18.m1.1.1.1.1.3.3.2.2.3">𝑛</ci></apply><cn id="S4.E18.m1.1.1.1.1.3.3.2.3.cmml" type="integer" xref="S4.E18.m1.1.1.1.1.3.3.2.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E18.m1.1c">V=g\sum_{n}E_{n}^{2}.</annotation><annotation encoding="application/x-llamapun" id="S4.E18.m1.1d">italic_V = italic_g ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(18)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p6.2">The third scar, described in the Supplemental Material <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib56" title="">56</a>]</cite>, is also an eigenstate of this potential. These are precisely the three scars observed in the right panel of Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S4.F6" title="Figure 6 ‣ IV Numerical results and discussion ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">6</span></a>.</p> </div> <figure class="ltx_figure" id="S4.F6"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="220" id="S4.F6.g1" src="extracted/5828746/images/entropy4x4.jpeg" width="293"/></div> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="220" id="S4.F6.g2" src="extracted/5828746/images/entropyE2.jpeg" width="293"/></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 6: </span> Shannon entropy for a <math alttext="4\times 4" class="ltx_Math" display="inline" id="S4.F6.7.m1.1"><semantics id="S4.F6.7.m1.1b"><mrow id="S4.F6.7.m1.1.1" xref="S4.F6.7.m1.1.1.cmml"><mn id="S4.F6.7.m1.1.1.2" xref="S4.F6.7.m1.1.1.2.cmml">4</mn><mo id="S4.F6.7.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.F6.7.m1.1.1.1.cmml">×</mo><mn id="S4.F6.7.m1.1.1.3" xref="S4.F6.7.m1.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.7.m1.1c"><apply id="S4.F6.7.m1.1.1.cmml" xref="S4.F6.7.m1.1.1"><times id="S4.F6.7.m1.1.1.1.cmml" xref="S4.F6.7.m1.1.1.1"></times><cn id="S4.F6.7.m1.1.1.2.cmml" type="integer" xref="S4.F6.7.m1.1.1.2">4</cn><cn id="S4.F6.7.m1.1.1.3.cmml" type="integer" xref="S4.F6.7.m1.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.7.m1.1d">4\times 4</annotation><annotation encoding="application/x-llamapun" id="S4.F6.7.m1.1e">4 × 4</annotation></semantics></math> system with periodic boundary conditions in both directions, <math alttext="S=1" class="ltx_Math" display="inline" id="S4.F6.8.m2.1"><semantics id="S4.F6.8.m2.1b"><mrow id="S4.F6.8.m2.1.1" xref="S4.F6.8.m2.1.1.cmml"><mi id="S4.F6.8.m2.1.1.2" xref="S4.F6.8.m2.1.1.2.cmml">S</mi><mo id="S4.F6.8.m2.1.1.1" xref="S4.F6.8.m2.1.1.1.cmml">=</mo><mn id="S4.F6.8.m2.1.1.3" xref="S4.F6.8.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.8.m2.1c"><apply id="S4.F6.8.m2.1.1.cmml" xref="S4.F6.8.m2.1.1"><eq id="S4.F6.8.m2.1.1.1.cmml" xref="S4.F6.8.m2.1.1.1"></eq><ci id="S4.F6.8.m2.1.1.2.cmml" xref="S4.F6.8.m2.1.1.2">𝑆</ci><cn id="S4.F6.8.m2.1.1.3.cmml" type="integer" xref="S4.F6.8.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.8.m2.1d">S=1</annotation><annotation encoding="application/x-llamapun" id="S4.F6.8.m2.1e">italic_S = 1</annotation></semantics></math>, and different potentials. Shown states are in the zero momentum and <math alttext="+1" class="ltx_Math" display="inline" id="S4.F6.9.m3.1"><semantics id="S4.F6.9.m3.1b"><mrow id="S4.F6.9.m3.1.1" xref="S4.F6.9.m3.1.1.cmml"><mo id="S4.F6.9.m3.1.1b" xref="S4.F6.9.m3.1.1.cmml">+</mo><mn id="S4.F6.9.m3.1.1.2" xref="S4.F6.9.m3.1.1.2.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.9.m3.1c"><apply id="S4.F6.9.m3.1.1.cmml" xref="S4.F6.9.m3.1.1"><plus id="S4.F6.9.m3.1.1.1.cmml" xref="S4.F6.9.m3.1.1"></plus><cn id="S4.F6.9.m3.1.1.2.cmml" type="integer" xref="S4.F6.9.m3.1.1.2">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.9.m3.1d">+1</annotation><annotation encoding="application/x-llamapun" id="S4.F6.9.m3.1e">+ 1</annotation></semantics></math> charge conjugation sectors. Bright colors indicate a higher density of states. Left: Hamming distance potential with <math alttext="\lambda=0.2" class="ltx_Math" display="inline" id="S4.F6.10.m4.1"><semantics id="S4.F6.10.m4.1b"><mrow id="S4.F6.10.m4.1.1" xref="S4.F6.10.m4.1.1.cmml"><mi id="S4.F6.10.m4.1.1.2" xref="S4.F6.10.m4.1.1.2.cmml">λ</mi><mo id="S4.F6.10.m4.1.1.1" xref="S4.F6.10.m4.1.1.1.cmml">=</mo><mn id="S4.F6.10.m4.1.1.3" xref="S4.F6.10.m4.1.1.3.cmml">0.2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.10.m4.1c"><apply id="S4.F6.10.m4.1.1.cmml" xref="S4.F6.10.m4.1.1"><eq id="S4.F6.10.m4.1.1.1.cmml" xref="S4.F6.10.m4.1.1.1"></eq><ci id="S4.F6.10.m4.1.1.2.cmml" xref="S4.F6.10.m4.1.1.2">𝜆</ci><cn id="S4.F6.10.m4.1.1.3.cmml" type="float" xref="S4.F6.10.m4.1.1.3">0.2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.10.m4.1d">\lambda=0.2</annotation><annotation encoding="application/x-llamapun" id="S4.F6.10.m4.1e">italic_λ = 0.2</annotation></semantics></math>. Many mid-spectrum states with low entropy are visible. The tower colored in red at <math alttext="E=1.6" class="ltx_Math" display="inline" id="S4.F6.11.m5.1"><semantics id="S4.F6.11.m5.1b"><mrow id="S4.F6.11.m5.1.1" xref="S4.F6.11.m5.1.1.cmml"><mi id="S4.F6.11.m5.1.1.2" xref="S4.F6.11.m5.1.1.2.cmml">E</mi><mo id="S4.F6.11.m5.1.1.1" xref="S4.F6.11.m5.1.1.1.cmml">=</mo><mn id="S4.F6.11.m5.1.1.3" xref="S4.F6.11.m5.1.1.3.cmml">1.6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.11.m5.1c"><apply id="S4.F6.11.m5.1.1.cmml" xref="S4.F6.11.m5.1.1"><eq id="S4.F6.11.m5.1.1.1.cmml" xref="S4.F6.11.m5.1.1.1"></eq><ci id="S4.F6.11.m5.1.1.2.cmml" xref="S4.F6.11.m5.1.1.2">𝐸</ci><cn id="S4.F6.11.m5.1.1.3.cmml" type="float" xref="S4.F6.11.m5.1.1.3">1.6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.11.m5.1d">E=1.6</annotation><annotation encoding="application/x-llamapun" id="S4.F6.11.m5.1e">italic_E = 1.6</annotation></semantics></math> contains the predicted scars. Right: Standard electric field term (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S4.E18" title="In IV Numerical results and discussion ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">18</span></a>) with <math alttext="g=0.2" class="ltx_Math" display="inline" id="S4.F6.12.m6.1"><semantics id="S4.F6.12.m6.1b"><mrow id="S4.F6.12.m6.1.1" xref="S4.F6.12.m6.1.1.cmml"><mi id="S4.F6.12.m6.1.1.2" xref="S4.F6.12.m6.1.1.2.cmml">g</mi><mo id="S4.F6.12.m6.1.1.1" xref="S4.F6.12.m6.1.1.1.cmml">=</mo><mn id="S4.F6.12.m6.1.1.3" xref="S4.F6.12.m6.1.1.3.cmml">0.2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F6.12.m6.1c"><apply id="S4.F6.12.m6.1.1.cmml" xref="S4.F6.12.m6.1.1"><eq id="S4.F6.12.m6.1.1.1.cmml" xref="S4.F6.12.m6.1.1.1"></eq><ci id="S4.F6.12.m6.1.1.2.cmml" xref="S4.F6.12.m6.1.1.2">𝑔</ci><cn id="S4.F6.12.m6.1.1.3.cmml" type="float" xref="S4.F6.12.m6.1.1.3">0.2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F6.12.m6.1d">g=0.2</annotation><annotation encoding="application/x-llamapun" id="S4.F6.12.m6.1e">italic_g = 0.2</annotation></semantics></math>. Three mid-spectrum states with low entropy are visible and colored in red.</figcaption> </figure> <div class="ltx_para" id="S4.p7"> <p class="ltx_p" id="S4.p7.9">For <math alttext="S=2" class="ltx_Math" display="inline" id="S4.p7.1.m1.1"><semantics id="S4.p7.1.m1.1a"><mrow id="S4.p7.1.m1.1.1" xref="S4.p7.1.m1.1.1.cmml"><mi id="S4.p7.1.m1.1.1.2" xref="S4.p7.1.m1.1.1.2.cmml">S</mi><mo id="S4.p7.1.m1.1.1.1" xref="S4.p7.1.m1.1.1.1.cmml">=</mo><mn id="S4.p7.1.m1.1.1.3" xref="S4.p7.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p7.1.m1.1b"><apply id="S4.p7.1.m1.1.1.cmml" xref="S4.p7.1.m1.1.1"><eq id="S4.p7.1.m1.1.1.1.cmml" xref="S4.p7.1.m1.1.1.1"></eq><ci id="S4.p7.1.m1.1.1.2.cmml" xref="S4.p7.1.m1.1.1.2">𝑆</ci><cn id="S4.p7.1.m1.1.1.3.cmml" type="integer" xref="S4.p7.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p7.1.m1.1c">S=2</annotation><annotation encoding="application/x-llamapun" id="S4.p7.1.m1.1d">italic_S = 2</annotation></semantics></math> we have studied the <math alttext="6\times 1" class="ltx_Math" display="inline" id="S4.p7.2.m2.1"><semantics id="S4.p7.2.m2.1a"><mrow id="S4.p7.2.m2.1.1" xref="S4.p7.2.m2.1.1.cmml"><mn id="S4.p7.2.m2.1.1.2" xref="S4.p7.2.m2.1.1.2.cmml">6</mn><mo id="S4.p7.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.p7.2.m2.1.1.1.cmml">×</mo><mn id="S4.p7.2.m2.1.1.3" xref="S4.p7.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p7.2.m2.1b"><apply id="S4.p7.2.m2.1.1.cmml" xref="S4.p7.2.m2.1.1"><times id="S4.p7.2.m2.1.1.1.cmml" xref="S4.p7.2.m2.1.1.1"></times><cn id="S4.p7.2.m2.1.1.2.cmml" type="integer" xref="S4.p7.2.m2.1.1.2">6</cn><cn id="S4.p7.2.m2.1.1.3.cmml" type="integer" xref="S4.p7.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p7.2.m2.1c">6\times 1</annotation><annotation encoding="application/x-llamapun" id="S4.p7.2.m2.1d">6 × 1</annotation></semantics></math> ladder in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S4.F7" title="Figure 7 ‣ IV Numerical results and discussion ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">7</span></a> with the height potential (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S4.E16" title="In IV Numerical results and discussion ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">16</span></a>). In this case, we expect scars obtained from choosing <math alttext="i=0" class="ltx_Math" display="inline" id="S4.p7.3.m3.1"><semantics id="S4.p7.3.m3.1a"><mrow id="S4.p7.3.m3.1.1" xref="S4.p7.3.m3.1.1.cmml"><mi id="S4.p7.3.m3.1.1.2" xref="S4.p7.3.m3.1.1.2.cmml">i</mi><mo id="S4.p7.3.m3.1.1.1" xref="S4.p7.3.m3.1.1.1.cmml">=</mo><mn id="S4.p7.3.m3.1.1.3" xref="S4.p7.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p7.3.m3.1b"><apply id="S4.p7.3.m3.1.1.cmml" xref="S4.p7.3.m3.1.1"><eq id="S4.p7.3.m3.1.1.1.cmml" xref="S4.p7.3.m3.1.1.1"></eq><ci id="S4.p7.3.m3.1.1.2.cmml" xref="S4.p7.3.m3.1.1.2">𝑖</ci><cn id="S4.p7.3.m3.1.1.3.cmml" type="integer" xref="S4.p7.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p7.3.m3.1c">i=0</annotation><annotation encoding="application/x-llamapun" id="S4.p7.3.m3.1d">italic_i = 0</annotation></semantics></math>, <math alttext="i=1" class="ltx_Math" display="inline" id="S4.p7.4.m4.1"><semantics id="S4.p7.4.m4.1a"><mrow id="S4.p7.4.m4.1.1" xref="S4.p7.4.m4.1.1.cmml"><mi id="S4.p7.4.m4.1.1.2" xref="S4.p7.4.m4.1.1.2.cmml">i</mi><mo id="S4.p7.4.m4.1.1.1" xref="S4.p7.4.m4.1.1.1.cmml">=</mo><mn id="S4.p7.4.m4.1.1.3" xref="S4.p7.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p7.4.m4.1b"><apply id="S4.p7.4.m4.1.1.cmml" xref="S4.p7.4.m4.1.1"><eq id="S4.p7.4.m4.1.1.1.cmml" xref="S4.p7.4.m4.1.1.1"></eq><ci id="S4.p7.4.m4.1.1.2.cmml" xref="S4.p7.4.m4.1.1.2">𝑖</ci><cn id="S4.p7.4.m4.1.1.3.cmml" type="integer" xref="S4.p7.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p7.4.m4.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S4.p7.4.m4.1d">italic_i = 1</annotation></semantics></math>, and <math alttext="i=2" class="ltx_Math" display="inline" id="S4.p7.5.m5.1"><semantics id="S4.p7.5.m5.1a"><mrow id="S4.p7.5.m5.1.1" xref="S4.p7.5.m5.1.1.cmml"><mi id="S4.p7.5.m5.1.1.2" xref="S4.p7.5.m5.1.1.2.cmml">i</mi><mo id="S4.p7.5.m5.1.1.1" xref="S4.p7.5.m5.1.1.1.cmml">=</mo><mn id="S4.p7.5.m5.1.1.3" xref="S4.p7.5.m5.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p7.5.m5.1b"><apply id="S4.p7.5.m5.1.1.cmml" xref="S4.p7.5.m5.1.1"><eq id="S4.p7.5.m5.1.1.1.cmml" xref="S4.p7.5.m5.1.1.1"></eq><ci id="S4.p7.5.m5.1.1.2.cmml" xref="S4.p7.5.m5.1.1.2">𝑖</ci><cn id="S4.p7.5.m5.1.1.3.cmml" type="integer" xref="S4.p7.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p7.5.m5.1c">i=2</annotation><annotation encoding="application/x-llamapun" id="S4.p7.5.m5.1d">italic_i = 2</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.E11" title="In III.1 QMBS in TLM With Arbitrary Integer Spin ‣ III Zero-mode scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">11</span></a>). The <math alttext="i=1" class="ltx_Math" display="inline" id="S4.p7.6.m6.1"><semantics id="S4.p7.6.m6.1a"><mrow id="S4.p7.6.m6.1.1" xref="S4.p7.6.m6.1.1.cmml"><mi id="S4.p7.6.m6.1.1.2" xref="S4.p7.6.m6.1.1.2.cmml">i</mi><mo id="S4.p7.6.m6.1.1.1" xref="S4.p7.6.m6.1.1.1.cmml">=</mo><mn id="S4.p7.6.m6.1.1.3" xref="S4.p7.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p7.6.m6.1b"><apply id="S4.p7.6.m6.1.1.cmml" xref="S4.p7.6.m6.1.1"><eq id="S4.p7.6.m6.1.1.1.cmml" xref="S4.p7.6.m6.1.1.1"></eq><ci id="S4.p7.6.m6.1.1.2.cmml" xref="S4.p7.6.m6.1.1.2">𝑖</ci><cn id="S4.p7.6.m6.1.1.3.cmml" type="integer" xref="S4.p7.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p7.6.m6.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S4.p7.6.m6.1d">italic_i = 1</annotation></semantics></math> scar has zero energy for any value of <math alttext="\lambda" class="ltx_Math" display="inline" id="S4.p7.7.m7.1"><semantics id="S4.p7.7.m7.1a"><mi id="S4.p7.7.m7.1.1" xref="S4.p7.7.m7.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S4.p7.7.m7.1b"><ci id="S4.p7.7.m7.1.1.cmml" xref="S4.p7.7.m7.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p7.7.m7.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S4.p7.7.m7.1d">italic_λ</annotation></semantics></math>, while the other two will have <math alttext="E_{2,1}=\pm 6\lambda" class="ltx_Math" display="inline" id="S4.p7.8.m8.2"><semantics id="S4.p7.8.m8.2a"><mrow id="S4.p7.8.m8.2.3" xref="S4.p7.8.m8.2.3.cmml"><msub id="S4.p7.8.m8.2.3.2" xref="S4.p7.8.m8.2.3.2.cmml"><mi id="S4.p7.8.m8.2.3.2.2" xref="S4.p7.8.m8.2.3.2.2.cmml">E</mi><mrow id="S4.p7.8.m8.2.2.2.4" xref="S4.p7.8.m8.2.2.2.3.cmml"><mn id="S4.p7.8.m8.1.1.1.1" xref="S4.p7.8.m8.1.1.1.1.cmml">2</mn><mo id="S4.p7.8.m8.2.2.2.4.1" xref="S4.p7.8.m8.2.2.2.3.cmml">,</mo><mn id="S4.p7.8.m8.2.2.2.2" xref="S4.p7.8.m8.2.2.2.2.cmml">1</mn></mrow></msub><mo id="S4.p7.8.m8.2.3.1" xref="S4.p7.8.m8.2.3.1.cmml">=</mo><mrow id="S4.p7.8.m8.2.3.3" xref="S4.p7.8.m8.2.3.3.cmml"><mo id="S4.p7.8.m8.2.3.3a" xref="S4.p7.8.m8.2.3.3.cmml">±</mo><mrow id="S4.p7.8.m8.2.3.3.2" xref="S4.p7.8.m8.2.3.3.2.cmml"><mn id="S4.p7.8.m8.2.3.3.2.2" xref="S4.p7.8.m8.2.3.3.2.2.cmml">6</mn><mo id="S4.p7.8.m8.2.3.3.2.1" xref="S4.p7.8.m8.2.3.3.2.1.cmml">⁢</mo><mi id="S4.p7.8.m8.2.3.3.2.3" xref="S4.p7.8.m8.2.3.3.2.3.cmml">λ</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p7.8.m8.2b"><apply id="S4.p7.8.m8.2.3.cmml" xref="S4.p7.8.m8.2.3"><eq id="S4.p7.8.m8.2.3.1.cmml" xref="S4.p7.8.m8.2.3.1"></eq><apply id="S4.p7.8.m8.2.3.2.cmml" xref="S4.p7.8.m8.2.3.2"><csymbol cd="ambiguous" id="S4.p7.8.m8.2.3.2.1.cmml" xref="S4.p7.8.m8.2.3.2">subscript</csymbol><ci id="S4.p7.8.m8.2.3.2.2.cmml" xref="S4.p7.8.m8.2.3.2.2">𝐸</ci><list id="S4.p7.8.m8.2.2.2.3.cmml" xref="S4.p7.8.m8.2.2.2.4"><cn id="S4.p7.8.m8.1.1.1.1.cmml" type="integer" xref="S4.p7.8.m8.1.1.1.1">2</cn><cn id="S4.p7.8.m8.2.2.2.2.cmml" type="integer" xref="S4.p7.8.m8.2.2.2.2">1</cn></list></apply><apply id="S4.p7.8.m8.2.3.3.cmml" xref="S4.p7.8.m8.2.3.3"><csymbol cd="latexml" id="S4.p7.8.m8.2.3.3.1.cmml" xref="S4.p7.8.m8.2.3.3">plus-or-minus</csymbol><apply id="S4.p7.8.m8.2.3.3.2.cmml" xref="S4.p7.8.m8.2.3.3.2"><times id="S4.p7.8.m8.2.3.3.2.1.cmml" xref="S4.p7.8.m8.2.3.3.2.1"></times><cn id="S4.p7.8.m8.2.3.3.2.2.cmml" type="integer" xref="S4.p7.8.m8.2.3.3.2.2">6</cn><ci id="S4.p7.8.m8.2.3.3.2.3.cmml" xref="S4.p7.8.m8.2.3.3.2.3">𝜆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p7.8.m8.2c">E_{2,1}=\pm 6\lambda</annotation><annotation encoding="application/x-llamapun" id="S4.p7.8.m8.2d">italic_E start_POSTSUBSCRIPT 2 , 1 end_POSTSUBSCRIPT = ± 6 italic_λ</annotation></semantics></math>. The latter two are evident in the figure. The <math alttext="i=1" class="ltx_Math" display="inline" id="S4.p7.9.m9.1"><semantics id="S4.p7.9.m9.1a"><mrow id="S4.p7.9.m9.1.1" xref="S4.p7.9.m9.1.1.cmml"><mi id="S4.p7.9.m9.1.1.2" xref="S4.p7.9.m9.1.1.2.cmml">i</mi><mo id="S4.p7.9.m9.1.1.1" xref="S4.p7.9.m9.1.1.1.cmml">=</mo><mn id="S4.p7.9.m9.1.1.3" xref="S4.p7.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p7.9.m9.1b"><apply id="S4.p7.9.m9.1.1.cmml" xref="S4.p7.9.m9.1.1"><eq id="S4.p7.9.m9.1.1.1.cmml" xref="S4.p7.9.m9.1.1.1"></eq><ci id="S4.p7.9.m9.1.1.2.cmml" xref="S4.p7.9.m9.1.1.2">𝑖</ci><cn id="S4.p7.9.m9.1.1.3.cmml" type="integer" xref="S4.p7.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p7.9.m9.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S4.p7.9.m9.1d">italic_i = 1</annotation></semantics></math> scar is not visible, since it is in superposition with a tower of zero-modes.</p> </div> <div class="ltx_para" id="S4.p8"> <p class="ltx_p" id="S4.p8.1">This numerically demonstrates the existence of low entanglement entropy zero-modes in systems beyond <math alttext="S=1" class="ltx_Math" display="inline" id="S4.p8.1.m1.1"><semantics id="S4.p8.1.m1.1a"><mrow id="S4.p8.1.m1.1.1" xref="S4.p8.1.m1.1.1.cmml"><mi id="S4.p8.1.m1.1.1.2" xref="S4.p8.1.m1.1.1.2.cmml">S</mi><mo id="S4.p8.1.m1.1.1.1" xref="S4.p8.1.m1.1.1.1.cmml">=</mo><mn id="S4.p8.1.m1.1.1.3" xref="S4.p8.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p8.1.m1.1b"><apply id="S4.p8.1.m1.1.1.cmml" xref="S4.p8.1.m1.1.1"><eq id="S4.p8.1.m1.1.1.1.cmml" xref="S4.p8.1.m1.1.1.1"></eq><ci id="S4.p8.1.m1.1.1.2.cmml" xref="S4.p8.1.m1.1.1.2">𝑆</ci><cn id="S4.p8.1.m1.1.1.3.cmml" type="integer" xref="S4.p8.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p8.1.m1.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="S4.p8.1.m1.1d">italic_S = 1</annotation></semantics></math>. For larger volume or higher spin systems, we have established the existence of such states analytically in the earlier sections of this article.</p> </div> <figure class="ltx_figure" id="S4.F7"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="220" id="S4.F7.g1" src="extracted/5828746/images/entanglementEntropyS=2.jpeg" width="293"/></div> <div class="ltx_flex_cell ltx_flex_size_2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" height="220" id="S4.F7.g2" src="extracted/5828746/images/entropyS=2.jpeg" width="293"/></div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 7: </span> Entropies for a <math alttext="S=2" class="ltx_Math" display="inline" id="S4.F7.4.m1.1"><semantics id="S4.F7.4.m1.1b"><mrow id="S4.F7.4.m1.1.1" xref="S4.F7.4.m1.1.1.cmml"><mi id="S4.F7.4.m1.1.1.2" xref="S4.F7.4.m1.1.1.2.cmml">S</mi><mo id="S4.F7.4.m1.1.1.1" xref="S4.F7.4.m1.1.1.1.cmml">=</mo><mn id="S4.F7.4.m1.1.1.3" xref="S4.F7.4.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F7.4.m1.1c"><apply id="S4.F7.4.m1.1.1.cmml" xref="S4.F7.4.m1.1.1"><eq id="S4.F7.4.m1.1.1.1.cmml" xref="S4.F7.4.m1.1.1.1"></eq><ci id="S4.F7.4.m1.1.1.2.cmml" xref="S4.F7.4.m1.1.1.2">𝑆</ci><cn id="S4.F7.4.m1.1.1.3.cmml" type="integer" xref="S4.F7.4.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F7.4.m1.1d">S=2</annotation><annotation encoding="application/x-llamapun" id="S4.F7.4.m1.1e">italic_S = 2</annotation></semantics></math> ladder with 6 plaquettes and <math alttext="\lambda=0.2" class="ltx_Math" display="inline" id="S4.F7.5.m2.1"><semantics id="S4.F7.5.m2.1b"><mrow id="S4.F7.5.m2.1.1" xref="S4.F7.5.m2.1.1.cmml"><mi id="S4.F7.5.m2.1.1.2" xref="S4.F7.5.m2.1.1.2.cmml">λ</mi><mo id="S4.F7.5.m2.1.1.1" xref="S4.F7.5.m2.1.1.1.cmml">=</mo><mn id="S4.F7.5.m2.1.1.3" xref="S4.F7.5.m2.1.1.3.cmml">0.2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F7.5.m2.1c"><apply id="S4.F7.5.m2.1.1.cmml" xref="S4.F7.5.m2.1.1"><eq id="S4.F7.5.m2.1.1.1.cmml" xref="S4.F7.5.m2.1.1.1"></eq><ci id="S4.F7.5.m2.1.1.2.cmml" xref="S4.F7.5.m2.1.1.2">𝜆</ci><cn id="S4.F7.5.m2.1.1.3.cmml" type="float" xref="S4.F7.5.m2.1.1.3">0.2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F7.5.m2.1d">\lambda=0.2</annotation><annotation encoding="application/x-llamapun" id="S4.F7.5.m2.1e">italic_λ = 0.2</annotation></semantics></math>. Brighter colors indicate a higher density of states. Left: Entanglement entropy. Right: Shannon entropy. Four mid-spectrum states with low entropy are visible, colored in red. The remaining two predicted states are mixed with a tower of zero-modes at <math alttext="E=0" class="ltx_Math" display="inline" id="S4.F7.6.m3.1"><semantics id="S4.F7.6.m3.1b"><mrow id="S4.F7.6.m3.1.1" xref="S4.F7.6.m3.1.1.cmml"><mi id="S4.F7.6.m3.1.1.2" xref="S4.F7.6.m3.1.1.2.cmml">E</mi><mo id="S4.F7.6.m3.1.1.1" xref="S4.F7.6.m3.1.1.1.cmml">=</mo><mn id="S4.F7.6.m3.1.1.3" xref="S4.F7.6.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F7.6.m3.1c"><apply id="S4.F7.6.m3.1.1.cmml" xref="S4.F7.6.m3.1.1"><eq id="S4.F7.6.m3.1.1.1.cmml" xref="S4.F7.6.m3.1.1.1"></eq><ci id="S4.F7.6.m3.1.1.2.cmml" xref="S4.F7.6.m3.1.1.2">𝐸</ci><cn id="S4.F7.6.m3.1.1.3.cmml" type="integer" xref="S4.F7.6.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F7.6.m3.1d">E=0</annotation><annotation encoding="application/x-llamapun" id="S4.F7.6.m3.1e">italic_E = 0</annotation></semantics></math>. </figcaption> </figure> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">V </span>Conclusions and outlook</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">In this work, we demonstrate that Quantum Many-Body Scars (QMBS) exist across numerous 2D Abelian pure gauge theories. In the limit of zero coupling, a spectral symmetry emerges, leading to an exponentially large number of zero-modes. We show that it is possible to construct many states with area-law entanglement entropy within this subspace. Since these are mid-spectrum states of a non-integrable model, they constitute QMBS. The number of scars we could identify grows exponentially with the volume, but comprise an exponentially small fraction of the Hilbert space. This indicates weak breaking of the Eigenstate Thermalization Hypothesis.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1">Through the analytical construction of the scars, we can strategically add a potential to the Hamiltonian that isolates them from other eigenstates. This allows for the numerical verification of their presence for moderate volumes and values of the spin. Different choices of the potential can isolate different scars. If the new potential does not have scar states as eigenstates, they will disappear from the spectrum. An interesting question for future work is whether this type of analytical construction can be generalized in the presence of bosonic or fermionic matter.</p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.3">For <math alttext="S=1" class="ltx_Math" display="inline" id="S5.p3.1.m1.1"><semantics id="S5.p3.1.m1.1a"><mrow id="S5.p3.1.m1.1.1" xref="S5.p3.1.m1.1.1.cmml"><mi id="S5.p3.1.m1.1.1.2" xref="S5.p3.1.m1.1.1.2.cmml">S</mi><mo id="S5.p3.1.m1.1.1.1" xref="S5.p3.1.m1.1.1.1.cmml">=</mo><mn id="S5.p3.1.m1.1.1.3" xref="S5.p3.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.1.m1.1b"><apply id="S5.p3.1.m1.1.1.cmml" xref="S5.p3.1.m1.1.1"><eq id="S5.p3.1.m1.1.1.1.cmml" xref="S5.p3.1.m1.1.1.1"></eq><ci id="S5.p3.1.m1.1.1.2.cmml" xref="S5.p3.1.m1.1.1.2">𝑆</ci><cn id="S5.p3.1.m1.1.1.3.cmml" type="integer" xref="S5.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.1.m1.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="S5.p3.1.m1.1d">italic_S = 1</annotation></semantics></math>, some scars survive the presence of the standard <math alttext="g\sum_{n}E^{2}_{n}" class="ltx_Math" display="inline" id="S5.p3.2.m2.1"><semantics id="S5.p3.2.m2.1a"><mrow id="S5.p3.2.m2.1.1" xref="S5.p3.2.m2.1.1.cmml"><mi id="S5.p3.2.m2.1.1.2" xref="S5.p3.2.m2.1.1.2.cmml">g</mi><mo id="S5.p3.2.m2.1.1.1" xref="S5.p3.2.m2.1.1.1.cmml">⁢</mo><mrow id="S5.p3.2.m2.1.1.3" xref="S5.p3.2.m2.1.1.3.cmml"><msub id="S5.p3.2.m2.1.1.3.1" xref="S5.p3.2.m2.1.1.3.1.cmml"><mo id="S5.p3.2.m2.1.1.3.1.2" xref="S5.p3.2.m2.1.1.3.1.2.cmml">∑</mo><mi id="S5.p3.2.m2.1.1.3.1.3" xref="S5.p3.2.m2.1.1.3.1.3.cmml">n</mi></msub><msubsup id="S5.p3.2.m2.1.1.3.2" xref="S5.p3.2.m2.1.1.3.2.cmml"><mi id="S5.p3.2.m2.1.1.3.2.2.2" xref="S5.p3.2.m2.1.1.3.2.2.2.cmml">E</mi><mi id="S5.p3.2.m2.1.1.3.2.3" xref="S5.p3.2.m2.1.1.3.2.3.cmml">n</mi><mn id="S5.p3.2.m2.1.1.3.2.2.3" xref="S5.p3.2.m2.1.1.3.2.2.3.cmml">2</mn></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.2.m2.1b"><apply id="S5.p3.2.m2.1.1.cmml" xref="S5.p3.2.m2.1.1"><times id="S5.p3.2.m2.1.1.1.cmml" xref="S5.p3.2.m2.1.1.1"></times><ci id="S5.p3.2.m2.1.1.2.cmml" xref="S5.p3.2.m2.1.1.2">𝑔</ci><apply id="S5.p3.2.m2.1.1.3.cmml" xref="S5.p3.2.m2.1.1.3"><apply id="S5.p3.2.m2.1.1.3.1.cmml" xref="S5.p3.2.m2.1.1.3.1"><csymbol cd="ambiguous" id="S5.p3.2.m2.1.1.3.1.1.cmml" xref="S5.p3.2.m2.1.1.3.1">subscript</csymbol><sum id="S5.p3.2.m2.1.1.3.1.2.cmml" xref="S5.p3.2.m2.1.1.3.1.2"></sum><ci id="S5.p3.2.m2.1.1.3.1.3.cmml" xref="S5.p3.2.m2.1.1.3.1.3">𝑛</ci></apply><apply id="S5.p3.2.m2.1.1.3.2.cmml" xref="S5.p3.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.p3.2.m2.1.1.3.2.1.cmml" xref="S5.p3.2.m2.1.1.3.2">subscript</csymbol><apply id="S5.p3.2.m2.1.1.3.2.2.cmml" xref="S5.p3.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.p3.2.m2.1.1.3.2.2.1.cmml" xref="S5.p3.2.m2.1.1.3.2">superscript</csymbol><ci id="S5.p3.2.m2.1.1.3.2.2.2.cmml" xref="S5.p3.2.m2.1.1.3.2.2.2">𝐸</ci><cn id="S5.p3.2.m2.1.1.3.2.2.3.cmml" type="integer" xref="S5.p3.2.m2.1.1.3.2.2.3">2</cn></apply><ci id="S5.p3.2.m2.1.1.3.2.3.cmml" xref="S5.p3.2.m2.1.1.3.2.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.2.m2.1c">g\sum_{n}E^{2}_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.2.m2.1d">italic_g ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_E start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> potential, if the lattice size is even in both directions and under periodic boundary conditions. It would be interesting to understand if a similar dependence on the geometry could occur in other gauge theories, such as <math alttext="SU\left(2\right)" class="ltx_Math" display="inline" id="S5.p3.3.m3.1"><semantics id="S5.p3.3.m3.1a"><mrow id="S5.p3.3.m3.1.2" xref="S5.p3.3.m3.1.2.cmml"><mi id="S5.p3.3.m3.1.2.2" xref="S5.p3.3.m3.1.2.2.cmml">S</mi><mo id="S5.p3.3.m3.1.2.1" xref="S5.p3.3.m3.1.2.1.cmml">⁢</mo><mi id="S5.p3.3.m3.1.2.3" xref="S5.p3.3.m3.1.2.3.cmml">U</mi><mo id="S5.p3.3.m3.1.2.1a" xref="S5.p3.3.m3.1.2.1.cmml">⁢</mo><mrow id="S5.p3.3.m3.1.2.4.2" xref="S5.p3.3.m3.1.2.cmml"><mo id="S5.p3.3.m3.1.2.4.2.1" xref="S5.p3.3.m3.1.2.cmml">(</mo><mn id="S5.p3.3.m3.1.1" xref="S5.p3.3.m3.1.1.cmml">2</mn><mo id="S5.p3.3.m3.1.2.4.2.2" xref="S5.p3.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.3.m3.1b"><apply id="S5.p3.3.m3.1.2.cmml" xref="S5.p3.3.m3.1.2"><times id="S5.p3.3.m3.1.2.1.cmml" xref="S5.p3.3.m3.1.2.1"></times><ci id="S5.p3.3.m3.1.2.2.cmml" xref="S5.p3.3.m3.1.2.2">𝑆</ci><ci id="S5.p3.3.m3.1.2.3.cmml" xref="S5.p3.3.m3.1.2.3">𝑈</ci><cn id="S5.p3.3.m3.1.1.cmml" type="integer" xref="S5.p3.3.m3.1.1">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.3.m3.1c">SU\left(2\right)</annotation><annotation encoding="application/x-llamapun" id="S5.p3.3.m3.1d">italic_S italic_U ( 2 )</annotation></semantics></math>, where scars were not found for larger truncations in single-leg ladders <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib65" title="">65</a>]</cite>.</p> </div> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.1">Our analytical construction of QMBS extends to arbitrary truncated spins and arbitrarily large volumes. This result establishes the presence of scars well beyond the reach of existing numerical methods. It can be used to guide future quantum simulations of gauge theories, in a regime beyond classical simulations, towards interesting non-equilibrium phenomena. From the theoretical point of view, it constitutes a step towards understanding the role of scars when taking the continuum limit, where the Hilbert space per link is not bounded.</p> </div> <div class="ltx_para" id="S5.p5"> <p class="ltx_p" id="S5.p5.1"><span class="ltx_text ltx_font_italic" id="S5.p5.1.1">Acknowledgements.</span>—We are grateful to Debasish Banerjee for insightful discussions. T.B. thanks the Galileo Galilei Institute for Theoretical Physics for the hospitality during the completion of part of this work. This research was supported by the Munich Institute for Astro-, Particle and BioPhysics (MIAPbP), which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany´s Excellence Strategy – EXC-2094 – 390783311. We acknowledge access to Piz Daint at the Swiss National Supercomputing Centre, Switzerland under the ETHZ’s share with the project ID eth8. Support from the Google Research Scholar Award in Quantum Computing and the Quantum Center at ETH Zurich is gratefully acknowledged.</p> </div> <div class="ltx_para" id="S5.p6"> <p class="ltx_p" id="S5.p6.1"><span class="ltx_text ltx_font_italic" id="S5.p6.1.1">Note.</span>—During the final stages of our manuscript, we became aware of another work <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib66" title="">66</a>]</cite> on quantum many-body scarring in a 2+1D U(1) gauge theory with dynamical matter. <br class="ltx_break"/></p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Deutsch [1991]</span> <span class="ltx_bibblock">J. M. 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We will always have</p> </div> <div class="ltx_para" id="S1a.p2"> <ul class="ltx_itemize" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i1.p1"> <p class="ltx_p" id="S1.I1.i1.p1.2"><span class="ltx_text ltx_font_bold" id="S1.I1.i1.p1.2.1">Translation Symmetry</span> along the horizontal axis <math alttext="E_{ni}\rightarrow E_{n+\hat{1}i}" class="ltx_Math" display="inline" id="S1.I1.i1.p1.1.m1.1"><semantics id="S1.I1.i1.p1.1.m1.1a"><mrow id="S1.I1.i1.p1.1.m1.1.1" xref="S1.I1.i1.p1.1.m1.1.1.cmml"><msub id="S1.I1.i1.p1.1.m1.1.1.2" xref="S1.I1.i1.p1.1.m1.1.1.2.cmml"><mi id="S1.I1.i1.p1.1.m1.1.1.2.2" xref="S1.I1.i1.p1.1.m1.1.1.2.2.cmml">E</mi><mrow id="S1.I1.i1.p1.1.m1.1.1.2.3" xref="S1.I1.i1.p1.1.m1.1.1.2.3.cmml"><mi id="S1.I1.i1.p1.1.m1.1.1.2.3.2" xref="S1.I1.i1.p1.1.m1.1.1.2.3.2.cmml">n</mi><mo id="S1.I1.i1.p1.1.m1.1.1.2.3.1" xref="S1.I1.i1.p1.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S1.I1.i1.p1.1.m1.1.1.2.3.3" xref="S1.I1.i1.p1.1.m1.1.1.2.3.3.cmml">i</mi></mrow></msub><mo id="S1.I1.i1.p1.1.m1.1.1.1" stretchy="false" xref="S1.I1.i1.p1.1.m1.1.1.1.cmml">→</mo><msub id="S1.I1.i1.p1.1.m1.1.1.3" xref="S1.I1.i1.p1.1.m1.1.1.3.cmml"><mi id="S1.I1.i1.p1.1.m1.1.1.3.2" xref="S1.I1.i1.p1.1.m1.1.1.3.2.cmml">E</mi><mrow id="S1.I1.i1.p1.1.m1.1.1.3.3" xref="S1.I1.i1.p1.1.m1.1.1.3.3.cmml"><mi id="S1.I1.i1.p1.1.m1.1.1.3.3.2" xref="S1.I1.i1.p1.1.m1.1.1.3.3.2.cmml">n</mi><mo id="S1.I1.i1.p1.1.m1.1.1.3.3.1" xref="S1.I1.i1.p1.1.m1.1.1.3.3.1.cmml">+</mo><mrow id="S1.I1.i1.p1.1.m1.1.1.3.3.3" xref="S1.I1.i1.p1.1.m1.1.1.3.3.3.cmml"><mover accent="true" id="S1.I1.i1.p1.1.m1.1.1.3.3.3.2" xref="S1.I1.i1.p1.1.m1.1.1.3.3.3.2.cmml"><mn id="S1.I1.i1.p1.1.m1.1.1.3.3.3.2.2" xref="S1.I1.i1.p1.1.m1.1.1.3.3.3.2.2.cmml">1</mn><mo id="S1.I1.i1.p1.1.m1.1.1.3.3.3.2.1" xref="S1.I1.i1.p1.1.m1.1.1.3.3.3.2.1.cmml">^</mo></mover><mo id="S1.I1.i1.p1.1.m1.1.1.3.3.3.1" xref="S1.I1.i1.p1.1.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="S1.I1.i1.p1.1.m1.1.1.3.3.3.3" xref="S1.I1.i1.p1.1.m1.1.1.3.3.3.3.cmml">i</mi></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.1.m1.1b"><apply id="S1.I1.i1.p1.1.m1.1.1.cmml" xref="S1.I1.i1.p1.1.m1.1.1"><ci id="S1.I1.i1.p1.1.m1.1.1.1.cmml" xref="S1.I1.i1.p1.1.m1.1.1.1">→</ci><apply id="S1.I1.i1.p1.1.m1.1.1.2.cmml" xref="S1.I1.i1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S1.I1.i1.p1.1.m1.1.1.2.1.cmml" xref="S1.I1.i1.p1.1.m1.1.1.2">subscript</csymbol><ci id="S1.I1.i1.p1.1.m1.1.1.2.2.cmml" xref="S1.I1.i1.p1.1.m1.1.1.2.2">𝐸</ci><apply id="S1.I1.i1.p1.1.m1.1.1.2.3.cmml" xref="S1.I1.i1.p1.1.m1.1.1.2.3"><times id="S1.I1.i1.p1.1.m1.1.1.2.3.1.cmml" xref="S1.I1.i1.p1.1.m1.1.1.2.3.1"></times><ci id="S1.I1.i1.p1.1.m1.1.1.2.3.2.cmml" xref="S1.I1.i1.p1.1.m1.1.1.2.3.2">𝑛</ci><ci id="S1.I1.i1.p1.1.m1.1.1.2.3.3.cmml" xref="S1.I1.i1.p1.1.m1.1.1.2.3.3">𝑖</ci></apply></apply><apply id="S1.I1.i1.p1.1.m1.1.1.3.cmml" xref="S1.I1.i1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.I1.i1.p1.1.m1.1.1.3.1.cmml" xref="S1.I1.i1.p1.1.m1.1.1.3">subscript</csymbol><ci id="S1.I1.i1.p1.1.m1.1.1.3.2.cmml" xref="S1.I1.i1.p1.1.m1.1.1.3.2">𝐸</ci><apply id="S1.I1.i1.p1.1.m1.1.1.3.3.cmml" xref="S1.I1.i1.p1.1.m1.1.1.3.3"><plus id="S1.I1.i1.p1.1.m1.1.1.3.3.1.cmml" xref="S1.I1.i1.p1.1.m1.1.1.3.3.1"></plus><ci id="S1.I1.i1.p1.1.m1.1.1.3.3.2.cmml" xref="S1.I1.i1.p1.1.m1.1.1.3.3.2">𝑛</ci><apply id="S1.I1.i1.p1.1.m1.1.1.3.3.3.cmml" xref="S1.I1.i1.p1.1.m1.1.1.3.3.3"><times id="S1.I1.i1.p1.1.m1.1.1.3.3.3.1.cmml" xref="S1.I1.i1.p1.1.m1.1.1.3.3.3.1"></times><apply id="S1.I1.i1.p1.1.m1.1.1.3.3.3.2.cmml" xref="S1.I1.i1.p1.1.m1.1.1.3.3.3.2"><ci id="S1.I1.i1.p1.1.m1.1.1.3.3.3.2.1.cmml" xref="S1.I1.i1.p1.1.m1.1.1.3.3.3.2.1">^</ci><cn id="S1.I1.i1.p1.1.m1.1.1.3.3.3.2.2.cmml" type="integer" xref="S1.I1.i1.p1.1.m1.1.1.3.3.3.2.2">1</cn></apply><ci id="S1.I1.i1.p1.1.m1.1.1.3.3.3.3.cmml" xref="S1.I1.i1.p1.1.m1.1.1.3.3.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.1.m1.1c">E_{ni}\rightarrow E_{n+\hat{1}i}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.1.m1.1d">italic_E start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT → italic_E start_POSTSUBSCRIPT italic_n + over^ start_ARG 1 end_ARG italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="U_{n+\hat{1}i}\rightarrow U_{n+\hat{1}i}" class="ltx_Math" display="inline" id="S1.I1.i1.p1.2.m2.1"><semantics id="S1.I1.i1.p1.2.m2.1a"><mrow id="S1.I1.i1.p1.2.m2.1.1" xref="S1.I1.i1.p1.2.m2.1.1.cmml"><msub id="S1.I1.i1.p1.2.m2.1.1.2" xref="S1.I1.i1.p1.2.m2.1.1.2.cmml"><mi id="S1.I1.i1.p1.2.m2.1.1.2.2" xref="S1.I1.i1.p1.2.m2.1.1.2.2.cmml">U</mi><mrow id="S1.I1.i1.p1.2.m2.1.1.2.3" xref="S1.I1.i1.p1.2.m2.1.1.2.3.cmml"><mi id="S1.I1.i1.p1.2.m2.1.1.2.3.2" xref="S1.I1.i1.p1.2.m2.1.1.2.3.2.cmml">n</mi><mo id="S1.I1.i1.p1.2.m2.1.1.2.3.1" xref="S1.I1.i1.p1.2.m2.1.1.2.3.1.cmml">+</mo><mrow id="S1.I1.i1.p1.2.m2.1.1.2.3.3" xref="S1.I1.i1.p1.2.m2.1.1.2.3.3.cmml"><mover accent="true" id="S1.I1.i1.p1.2.m2.1.1.2.3.3.2" xref="S1.I1.i1.p1.2.m2.1.1.2.3.3.2.cmml"><mn id="S1.I1.i1.p1.2.m2.1.1.2.3.3.2.2" xref="S1.I1.i1.p1.2.m2.1.1.2.3.3.2.2.cmml">1</mn><mo id="S1.I1.i1.p1.2.m2.1.1.2.3.3.2.1" xref="S1.I1.i1.p1.2.m2.1.1.2.3.3.2.1.cmml">^</mo></mover><mo id="S1.I1.i1.p1.2.m2.1.1.2.3.3.1" xref="S1.I1.i1.p1.2.m2.1.1.2.3.3.1.cmml">⁢</mo><mi id="S1.I1.i1.p1.2.m2.1.1.2.3.3.3" xref="S1.I1.i1.p1.2.m2.1.1.2.3.3.3.cmml">i</mi></mrow></mrow></msub><mo id="S1.I1.i1.p1.2.m2.1.1.1" stretchy="false" xref="S1.I1.i1.p1.2.m2.1.1.1.cmml">→</mo><msub id="S1.I1.i1.p1.2.m2.1.1.3" xref="S1.I1.i1.p1.2.m2.1.1.3.cmml"><mi id="S1.I1.i1.p1.2.m2.1.1.3.2" xref="S1.I1.i1.p1.2.m2.1.1.3.2.cmml">U</mi><mrow id="S1.I1.i1.p1.2.m2.1.1.3.3" xref="S1.I1.i1.p1.2.m2.1.1.3.3.cmml"><mi id="S1.I1.i1.p1.2.m2.1.1.3.3.2" xref="S1.I1.i1.p1.2.m2.1.1.3.3.2.cmml">n</mi><mo id="S1.I1.i1.p1.2.m2.1.1.3.3.1" xref="S1.I1.i1.p1.2.m2.1.1.3.3.1.cmml">+</mo><mrow id="S1.I1.i1.p1.2.m2.1.1.3.3.3" xref="S1.I1.i1.p1.2.m2.1.1.3.3.3.cmml"><mover accent="true" id="S1.I1.i1.p1.2.m2.1.1.3.3.3.2" xref="S1.I1.i1.p1.2.m2.1.1.3.3.3.2.cmml"><mn id="S1.I1.i1.p1.2.m2.1.1.3.3.3.2.2" xref="S1.I1.i1.p1.2.m2.1.1.3.3.3.2.2.cmml">1</mn><mo id="S1.I1.i1.p1.2.m2.1.1.3.3.3.2.1" xref="S1.I1.i1.p1.2.m2.1.1.3.3.3.2.1.cmml">^</mo></mover><mo id="S1.I1.i1.p1.2.m2.1.1.3.3.3.1" xref="S1.I1.i1.p1.2.m2.1.1.3.3.3.1.cmml">⁢</mo><mi id="S1.I1.i1.p1.2.m2.1.1.3.3.3.3" xref="S1.I1.i1.p1.2.m2.1.1.3.3.3.3.cmml">i</mi></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.2.m2.1b"><apply id="S1.I1.i1.p1.2.m2.1.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1"><ci id="S1.I1.i1.p1.2.m2.1.1.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1.1">→</ci><apply id="S1.I1.i1.p1.2.m2.1.1.2.cmml" xref="S1.I1.i1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.I1.i1.p1.2.m2.1.1.2.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1.2">subscript</csymbol><ci id="S1.I1.i1.p1.2.m2.1.1.2.2.cmml" xref="S1.I1.i1.p1.2.m2.1.1.2.2">𝑈</ci><apply id="S1.I1.i1.p1.2.m2.1.1.2.3.cmml" xref="S1.I1.i1.p1.2.m2.1.1.2.3"><plus id="S1.I1.i1.p1.2.m2.1.1.2.3.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1.2.3.1"></plus><ci id="S1.I1.i1.p1.2.m2.1.1.2.3.2.cmml" xref="S1.I1.i1.p1.2.m2.1.1.2.3.2">𝑛</ci><apply id="S1.I1.i1.p1.2.m2.1.1.2.3.3.cmml" xref="S1.I1.i1.p1.2.m2.1.1.2.3.3"><times id="S1.I1.i1.p1.2.m2.1.1.2.3.3.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1.2.3.3.1"></times><apply id="S1.I1.i1.p1.2.m2.1.1.2.3.3.2.cmml" xref="S1.I1.i1.p1.2.m2.1.1.2.3.3.2"><ci id="S1.I1.i1.p1.2.m2.1.1.2.3.3.2.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1.2.3.3.2.1">^</ci><cn id="S1.I1.i1.p1.2.m2.1.1.2.3.3.2.2.cmml" type="integer" xref="S1.I1.i1.p1.2.m2.1.1.2.3.3.2.2">1</cn></apply><ci id="S1.I1.i1.p1.2.m2.1.1.2.3.3.3.cmml" xref="S1.I1.i1.p1.2.m2.1.1.2.3.3.3">𝑖</ci></apply></apply></apply><apply id="S1.I1.i1.p1.2.m2.1.1.3.cmml" xref="S1.I1.i1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.I1.i1.p1.2.m2.1.1.3.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1.3">subscript</csymbol><ci id="S1.I1.i1.p1.2.m2.1.1.3.2.cmml" xref="S1.I1.i1.p1.2.m2.1.1.3.2">𝑈</ci><apply id="S1.I1.i1.p1.2.m2.1.1.3.3.cmml" xref="S1.I1.i1.p1.2.m2.1.1.3.3"><plus id="S1.I1.i1.p1.2.m2.1.1.3.3.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1.3.3.1"></plus><ci id="S1.I1.i1.p1.2.m2.1.1.3.3.2.cmml" xref="S1.I1.i1.p1.2.m2.1.1.3.3.2">𝑛</ci><apply id="S1.I1.i1.p1.2.m2.1.1.3.3.3.cmml" xref="S1.I1.i1.p1.2.m2.1.1.3.3.3"><times id="S1.I1.i1.p1.2.m2.1.1.3.3.3.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1.3.3.3.1"></times><apply id="S1.I1.i1.p1.2.m2.1.1.3.3.3.2.cmml" xref="S1.I1.i1.p1.2.m2.1.1.3.3.3.2"><ci id="S1.I1.i1.p1.2.m2.1.1.3.3.3.2.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1.3.3.3.2.1">^</ci><cn id="S1.I1.i1.p1.2.m2.1.1.3.3.3.2.2.cmml" type="integer" xref="S1.I1.i1.p1.2.m2.1.1.3.3.3.2.2">1</cn></apply><ci id="S1.I1.i1.p1.2.m2.1.1.3.3.3.3.cmml" xref="S1.I1.i1.p1.2.m2.1.1.3.3.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.2.m2.1c">U_{n+\hat{1}i}\rightarrow U_{n+\hat{1}i}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.2.m2.1d">italic_U start_POSTSUBSCRIPT italic_n + over^ start_ARG 1 end_ARG italic_i end_POSTSUBSCRIPT → italic_U start_POSTSUBSCRIPT italic_n + over^ start_ARG 1 end_ARG italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Translation along the vertical axis is not always present due to boundary conditions;</p> </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i2.p1"> <p class="ltx_p" id="S1.I1.i2.p1.8"><span class="ltx_text ltx_font_bold" id="S1.I1.i2.p1.8.1">Reflection Symmetries</span> with respect to the two axis <math alttext="{\cal I}_{x}" class="ltx_Math" display="inline" id="S1.I1.i2.p1.1.m1.1"><semantics id="S1.I1.i2.p1.1.m1.1a"><msub id="S1.I1.i2.p1.1.m1.1.1" xref="S1.I1.i2.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.I1.i2.p1.1.m1.1.1.2" xref="S1.I1.i2.p1.1.m1.1.1.2.cmml">ℐ</mi><mi id="S1.I1.i2.p1.1.m1.1.1.3" xref="S1.I1.i2.p1.1.m1.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.1.m1.1b"><apply id="S1.I1.i2.p1.1.m1.1.1.cmml" xref="S1.I1.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p1.1.m1.1.1.1.cmml" xref="S1.I1.i2.p1.1.m1.1.1">subscript</csymbol><ci id="S1.I1.i2.p1.1.m1.1.1.2.cmml" xref="S1.I1.i2.p1.1.m1.1.1.2">ℐ</ci><ci id="S1.I1.i2.p1.1.m1.1.1.3.cmml" xref="S1.I1.i2.p1.1.m1.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.1.m1.1c">{\cal I}_{x}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.1.m1.1d">caligraphic_I start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="{\cal I}_{y}" class="ltx_Math" display="inline" id="S1.I1.i2.p1.2.m2.1"><semantics id="S1.I1.i2.p1.2.m2.1a"><msub id="S1.I1.i2.p1.2.m2.1.1" xref="S1.I1.i2.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.I1.i2.p1.2.m2.1.1.2" xref="S1.I1.i2.p1.2.m2.1.1.2.cmml">ℐ</mi><mi id="S1.I1.i2.p1.2.m2.1.1.3" xref="S1.I1.i2.p1.2.m2.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.2.m2.1b"><apply id="S1.I1.i2.p1.2.m2.1.1.cmml" xref="S1.I1.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p1.2.m2.1.1.1.cmml" xref="S1.I1.i2.p1.2.m2.1.1">subscript</csymbol><ci id="S1.I1.i2.p1.2.m2.1.1.2.cmml" xref="S1.I1.i2.p1.2.m2.1.1.2">ℐ</ci><ci id="S1.I1.i2.p1.2.m2.1.1.3.cmml" xref="S1.I1.i2.p1.2.m2.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.2.m2.1c">{\cal I}_{y}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.2.m2.1d">caligraphic_I start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math>. Explicitly, <math alttext="{\cal I}_{x}" class="ltx_Math" display="inline" id="S1.I1.i2.p1.3.m3.1"><semantics id="S1.I1.i2.p1.3.m3.1a"><msub id="S1.I1.i2.p1.3.m3.1.1" xref="S1.I1.i2.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.I1.i2.p1.3.m3.1.1.2" xref="S1.I1.i2.p1.3.m3.1.1.2.cmml">ℐ</mi><mi id="S1.I1.i2.p1.3.m3.1.1.3" xref="S1.I1.i2.p1.3.m3.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.3.m3.1b"><apply id="S1.I1.i2.p1.3.m3.1.1.cmml" xref="S1.I1.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p1.3.m3.1.1.1.cmml" xref="S1.I1.i2.p1.3.m3.1.1">subscript</csymbol><ci id="S1.I1.i2.p1.3.m3.1.1.2.cmml" xref="S1.I1.i2.p1.3.m3.1.1.2">ℐ</ci><ci id="S1.I1.i2.p1.3.m3.1.1.3.cmml" xref="S1.I1.i2.p1.3.m3.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.3.m3.1c">{\cal I}_{x}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.3.m3.1d">caligraphic_I start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> is given by <math alttext="E_{\left(n_{1},n_{2}\right)1}\rightarrow E_{\left(n_{1},-n_{2}\right)1}" class="ltx_Math" display="inline" id="S1.I1.i2.p1.4.m4.4"><semantics id="S1.I1.i2.p1.4.m4.4a"><mrow id="S1.I1.i2.p1.4.m4.4.5" xref="S1.I1.i2.p1.4.m4.4.5.cmml"><msub id="S1.I1.i2.p1.4.m4.4.5.2" xref="S1.I1.i2.p1.4.m4.4.5.2.cmml"><mi id="S1.I1.i2.p1.4.m4.4.5.2.2" xref="S1.I1.i2.p1.4.m4.4.5.2.2.cmml">E</mi><mrow id="S1.I1.i2.p1.4.m4.2.2.2" xref="S1.I1.i2.p1.4.m4.2.2.2.cmml"><mrow id="S1.I1.i2.p1.4.m4.2.2.2.2.2" xref="S1.I1.i2.p1.4.m4.2.2.2.2.3.cmml"><mo id="S1.I1.i2.p1.4.m4.2.2.2.2.2.3" xref="S1.I1.i2.p1.4.m4.2.2.2.2.3.cmml">(</mo><msub id="S1.I1.i2.p1.4.m4.1.1.1.1.1.1" xref="S1.I1.i2.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S1.I1.i2.p1.4.m4.1.1.1.1.1.1.2" xref="S1.I1.i2.p1.4.m4.1.1.1.1.1.1.2.cmml">n</mi><mn id="S1.I1.i2.p1.4.m4.1.1.1.1.1.1.3" xref="S1.I1.i2.p1.4.m4.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S1.I1.i2.p1.4.m4.2.2.2.2.2.4" xref="S1.I1.i2.p1.4.m4.2.2.2.2.3.cmml">,</mo><msub id="S1.I1.i2.p1.4.m4.2.2.2.2.2.2" xref="S1.I1.i2.p1.4.m4.2.2.2.2.2.2.cmml"><mi id="S1.I1.i2.p1.4.m4.2.2.2.2.2.2.2" xref="S1.I1.i2.p1.4.m4.2.2.2.2.2.2.2.cmml">n</mi><mn id="S1.I1.i2.p1.4.m4.2.2.2.2.2.2.3" xref="S1.I1.i2.p1.4.m4.2.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S1.I1.i2.p1.4.m4.2.2.2.2.2.5" xref="S1.I1.i2.p1.4.m4.2.2.2.2.3.cmml">)</mo></mrow><mo id="S1.I1.i2.p1.4.m4.2.2.2.3" xref="S1.I1.i2.p1.4.m4.2.2.2.3.cmml">⁢</mo><mn id="S1.I1.i2.p1.4.m4.2.2.2.4" xref="S1.I1.i2.p1.4.m4.2.2.2.4.cmml">1</mn></mrow></msub><mo id="S1.I1.i2.p1.4.m4.4.5.1" stretchy="false" xref="S1.I1.i2.p1.4.m4.4.5.1.cmml">→</mo><msub id="S1.I1.i2.p1.4.m4.4.5.3" xref="S1.I1.i2.p1.4.m4.4.5.3.cmml"><mi id="S1.I1.i2.p1.4.m4.4.5.3.2" xref="S1.I1.i2.p1.4.m4.4.5.3.2.cmml">E</mi><mrow id="S1.I1.i2.p1.4.m4.4.4.2" xref="S1.I1.i2.p1.4.m4.4.4.2.cmml"><mrow id="S1.I1.i2.p1.4.m4.4.4.2.2.2" xref="S1.I1.i2.p1.4.m4.4.4.2.2.3.cmml"><mo id="S1.I1.i2.p1.4.m4.4.4.2.2.2.3" xref="S1.I1.i2.p1.4.m4.4.4.2.2.3.cmml">(</mo><msub id="S1.I1.i2.p1.4.m4.3.3.1.1.1.1" xref="S1.I1.i2.p1.4.m4.3.3.1.1.1.1.cmml"><mi id="S1.I1.i2.p1.4.m4.3.3.1.1.1.1.2" xref="S1.I1.i2.p1.4.m4.3.3.1.1.1.1.2.cmml">n</mi><mn id="S1.I1.i2.p1.4.m4.3.3.1.1.1.1.3" xref="S1.I1.i2.p1.4.m4.3.3.1.1.1.1.3.cmml">1</mn></msub><mo id="S1.I1.i2.p1.4.m4.4.4.2.2.2.4" xref="S1.I1.i2.p1.4.m4.4.4.2.2.3.cmml">,</mo><mrow id="S1.I1.i2.p1.4.m4.4.4.2.2.2.2" xref="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.cmml"><mo id="S1.I1.i2.p1.4.m4.4.4.2.2.2.2a" xref="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.cmml">−</mo><msub id="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.2" xref="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.2.cmml"><mi id="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.2.2" xref="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.2.2.cmml">n</mi><mn id="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.2.3" xref="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S1.I1.i2.p1.4.m4.4.4.2.2.2.5" xref="S1.I1.i2.p1.4.m4.4.4.2.2.3.cmml">)</mo></mrow><mo id="S1.I1.i2.p1.4.m4.4.4.2.3" xref="S1.I1.i2.p1.4.m4.4.4.2.3.cmml">⁢</mo><mn id="S1.I1.i2.p1.4.m4.4.4.2.4" xref="S1.I1.i2.p1.4.m4.4.4.2.4.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.4.m4.4b"><apply id="S1.I1.i2.p1.4.m4.4.5.cmml" xref="S1.I1.i2.p1.4.m4.4.5"><ci id="S1.I1.i2.p1.4.m4.4.5.1.cmml" xref="S1.I1.i2.p1.4.m4.4.5.1">→</ci><apply id="S1.I1.i2.p1.4.m4.4.5.2.cmml" xref="S1.I1.i2.p1.4.m4.4.5.2"><csymbol cd="ambiguous" id="S1.I1.i2.p1.4.m4.4.5.2.1.cmml" xref="S1.I1.i2.p1.4.m4.4.5.2">subscript</csymbol><ci id="S1.I1.i2.p1.4.m4.4.5.2.2.cmml" xref="S1.I1.i2.p1.4.m4.4.5.2.2">𝐸</ci><apply id="S1.I1.i2.p1.4.m4.2.2.2.cmml" xref="S1.I1.i2.p1.4.m4.2.2.2"><times id="S1.I1.i2.p1.4.m4.2.2.2.3.cmml" xref="S1.I1.i2.p1.4.m4.2.2.2.3"></times><interval closure="open" id="S1.I1.i2.p1.4.m4.2.2.2.2.3.cmml" xref="S1.I1.i2.p1.4.m4.2.2.2.2.2"><apply id="S1.I1.i2.p1.4.m4.1.1.1.1.1.1.cmml" xref="S1.I1.i2.p1.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S1.I1.i2.p1.4.m4.1.1.1.1.1.1">subscript</csymbol><ci id="S1.I1.i2.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S1.I1.i2.p1.4.m4.1.1.1.1.1.1.2">𝑛</ci><cn id="S1.I1.i2.p1.4.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="S1.I1.i2.p1.4.m4.1.1.1.1.1.1.3">1</cn></apply><apply id="S1.I1.i2.p1.4.m4.2.2.2.2.2.2.cmml" xref="S1.I1.i2.p1.4.m4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S1.I1.i2.p1.4.m4.2.2.2.2.2.2.1.cmml" xref="S1.I1.i2.p1.4.m4.2.2.2.2.2.2">subscript</csymbol><ci id="S1.I1.i2.p1.4.m4.2.2.2.2.2.2.2.cmml" xref="S1.I1.i2.p1.4.m4.2.2.2.2.2.2.2">𝑛</ci><cn id="S1.I1.i2.p1.4.m4.2.2.2.2.2.2.3.cmml" type="integer" xref="S1.I1.i2.p1.4.m4.2.2.2.2.2.2.3">2</cn></apply></interval><cn id="S1.I1.i2.p1.4.m4.2.2.2.4.cmml" type="integer" xref="S1.I1.i2.p1.4.m4.2.2.2.4">1</cn></apply></apply><apply id="S1.I1.i2.p1.4.m4.4.5.3.cmml" xref="S1.I1.i2.p1.4.m4.4.5.3"><csymbol cd="ambiguous" id="S1.I1.i2.p1.4.m4.4.5.3.1.cmml" xref="S1.I1.i2.p1.4.m4.4.5.3">subscript</csymbol><ci id="S1.I1.i2.p1.4.m4.4.5.3.2.cmml" xref="S1.I1.i2.p1.4.m4.4.5.3.2">𝐸</ci><apply id="S1.I1.i2.p1.4.m4.4.4.2.cmml" xref="S1.I1.i2.p1.4.m4.4.4.2"><times id="S1.I1.i2.p1.4.m4.4.4.2.3.cmml" xref="S1.I1.i2.p1.4.m4.4.4.2.3"></times><interval closure="open" id="S1.I1.i2.p1.4.m4.4.4.2.2.3.cmml" xref="S1.I1.i2.p1.4.m4.4.4.2.2.2"><apply id="S1.I1.i2.p1.4.m4.3.3.1.1.1.1.cmml" xref="S1.I1.i2.p1.4.m4.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p1.4.m4.3.3.1.1.1.1.1.cmml" xref="S1.I1.i2.p1.4.m4.3.3.1.1.1.1">subscript</csymbol><ci id="S1.I1.i2.p1.4.m4.3.3.1.1.1.1.2.cmml" xref="S1.I1.i2.p1.4.m4.3.3.1.1.1.1.2">𝑛</ci><cn id="S1.I1.i2.p1.4.m4.3.3.1.1.1.1.3.cmml" type="integer" xref="S1.I1.i2.p1.4.m4.3.3.1.1.1.1.3">1</cn></apply><apply id="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.cmml" xref="S1.I1.i2.p1.4.m4.4.4.2.2.2.2"><minus id="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.1.cmml" xref="S1.I1.i2.p1.4.m4.4.4.2.2.2.2"></minus><apply id="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.2.cmml" xref="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.2"><csymbol cd="ambiguous" id="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.2.1.cmml" xref="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.2">subscript</csymbol><ci id="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.2.2.cmml" xref="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.2.2">𝑛</ci><cn id="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.2.3.cmml" type="integer" xref="S1.I1.i2.p1.4.m4.4.4.2.2.2.2.2.3">2</cn></apply></apply></interval><cn id="S1.I1.i2.p1.4.m4.4.4.2.4.cmml" type="integer" xref="S1.I1.i2.p1.4.m4.4.4.2.4">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.4.m4.4c">E_{\left(n_{1},n_{2}\right)1}\rightarrow E_{\left(n_{1},-n_{2}\right)1}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.4.m4.4d">italic_E start_POSTSUBSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) 1 end_POSTSUBSCRIPT → italic_E start_POSTSUBSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , - italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="E_{\left(n_{1},n_{2}\right)2}\rightarrow-E_{\left(n_{1},-n_{2}\right)2}" class="ltx_Math" display="inline" id="S1.I1.i2.p1.5.m5.4"><semantics id="S1.I1.i2.p1.5.m5.4a"><mrow id="S1.I1.i2.p1.5.m5.4.5" xref="S1.I1.i2.p1.5.m5.4.5.cmml"><msub id="S1.I1.i2.p1.5.m5.4.5.2" xref="S1.I1.i2.p1.5.m5.4.5.2.cmml"><mi id="S1.I1.i2.p1.5.m5.4.5.2.2" xref="S1.I1.i2.p1.5.m5.4.5.2.2.cmml">E</mi><mrow id="S1.I1.i2.p1.5.m5.2.2.2" xref="S1.I1.i2.p1.5.m5.2.2.2.cmml"><mrow id="S1.I1.i2.p1.5.m5.2.2.2.2.2" xref="S1.I1.i2.p1.5.m5.2.2.2.2.3.cmml"><mo id="S1.I1.i2.p1.5.m5.2.2.2.2.2.3" xref="S1.I1.i2.p1.5.m5.2.2.2.2.3.cmml">(</mo><msub id="S1.I1.i2.p1.5.m5.1.1.1.1.1.1" xref="S1.I1.i2.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="S1.I1.i2.p1.5.m5.1.1.1.1.1.1.2" xref="S1.I1.i2.p1.5.m5.1.1.1.1.1.1.2.cmml">n</mi><mn id="S1.I1.i2.p1.5.m5.1.1.1.1.1.1.3" xref="S1.I1.i2.p1.5.m5.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S1.I1.i2.p1.5.m5.2.2.2.2.2.4" 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id="S1.I1.i2.p1.5.m5.4.4.2.2.2.5" xref="S1.I1.i2.p1.5.m5.4.4.2.2.3.cmml">)</mo></mrow><mo id="S1.I1.i2.p1.5.m5.4.4.2.3" xref="S1.I1.i2.p1.5.m5.4.4.2.3.cmml">⁢</mo><mn id="S1.I1.i2.p1.5.m5.4.4.2.4" xref="S1.I1.i2.p1.5.m5.4.4.2.4.cmml">2</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.5.m5.4b"><apply id="S1.I1.i2.p1.5.m5.4.5.cmml" xref="S1.I1.i2.p1.5.m5.4.5"><ci id="S1.I1.i2.p1.5.m5.4.5.1.cmml" xref="S1.I1.i2.p1.5.m5.4.5.1">→</ci><apply id="S1.I1.i2.p1.5.m5.4.5.2.cmml" xref="S1.I1.i2.p1.5.m5.4.5.2"><csymbol cd="ambiguous" id="S1.I1.i2.p1.5.m5.4.5.2.1.cmml" xref="S1.I1.i2.p1.5.m5.4.5.2">subscript</csymbol><ci id="S1.I1.i2.p1.5.m5.4.5.2.2.cmml" xref="S1.I1.i2.p1.5.m5.4.5.2.2">𝐸</ci><apply id="S1.I1.i2.p1.5.m5.2.2.2.cmml" xref="S1.I1.i2.p1.5.m5.2.2.2"><times id="S1.I1.i2.p1.5.m5.2.2.2.3.cmml" xref="S1.I1.i2.p1.5.m5.2.2.2.3"></times><interval closure="open" id="S1.I1.i2.p1.5.m5.2.2.2.2.3.cmml" xref="S1.I1.i2.p1.5.m5.2.2.2.2.2"><apply 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id="S1.I1.i2.p1.5.m5.4.5.3.1.cmml" xref="S1.I1.i2.p1.5.m5.4.5.3"></minus><apply id="S1.I1.i2.p1.5.m5.4.5.3.2.cmml" xref="S1.I1.i2.p1.5.m5.4.5.3.2"><csymbol cd="ambiguous" id="S1.I1.i2.p1.5.m5.4.5.3.2.1.cmml" xref="S1.I1.i2.p1.5.m5.4.5.3.2">subscript</csymbol><ci id="S1.I1.i2.p1.5.m5.4.5.3.2.2.cmml" xref="S1.I1.i2.p1.5.m5.4.5.3.2.2">𝐸</ci><apply id="S1.I1.i2.p1.5.m5.4.4.2.cmml" xref="S1.I1.i2.p1.5.m5.4.4.2"><times id="S1.I1.i2.p1.5.m5.4.4.2.3.cmml" xref="S1.I1.i2.p1.5.m5.4.4.2.3"></times><interval closure="open" id="S1.I1.i2.p1.5.m5.4.4.2.2.3.cmml" xref="S1.I1.i2.p1.5.m5.4.4.2.2.2"><apply id="S1.I1.i2.p1.5.m5.3.3.1.1.1.1.cmml" xref="S1.I1.i2.p1.5.m5.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p1.5.m5.3.3.1.1.1.1.1.cmml" xref="S1.I1.i2.p1.5.m5.3.3.1.1.1.1">subscript</csymbol><ci id="S1.I1.i2.p1.5.m5.3.3.1.1.1.1.2.cmml" xref="S1.I1.i2.p1.5.m5.3.3.1.1.1.1.2">𝑛</ci><cn id="S1.I1.i2.p1.5.m5.3.3.1.1.1.1.3.cmml" type="integer" xref="S1.I1.i2.p1.5.m5.3.3.1.1.1.1.3">1</cn></apply><apply id="S1.I1.i2.p1.5.m5.4.4.2.2.2.2.cmml" xref="S1.I1.i2.p1.5.m5.4.4.2.2.2.2"><minus id="S1.I1.i2.p1.5.m5.4.4.2.2.2.2.1.cmml" xref="S1.I1.i2.p1.5.m5.4.4.2.2.2.2"></minus><apply id="S1.I1.i2.p1.5.m5.4.4.2.2.2.2.2.cmml" xref="S1.I1.i2.p1.5.m5.4.4.2.2.2.2.2"><csymbol cd="ambiguous" id="S1.I1.i2.p1.5.m5.4.4.2.2.2.2.2.1.cmml" xref="S1.I1.i2.p1.5.m5.4.4.2.2.2.2.2">subscript</csymbol><ci id="S1.I1.i2.p1.5.m5.4.4.2.2.2.2.2.2.cmml" xref="S1.I1.i2.p1.5.m5.4.4.2.2.2.2.2.2">𝑛</ci><cn id="S1.I1.i2.p1.5.m5.4.4.2.2.2.2.2.3.cmml" type="integer" xref="S1.I1.i2.p1.5.m5.4.4.2.2.2.2.2.3">2</cn></apply></apply></interval><cn id="S1.I1.i2.p1.5.m5.4.4.2.4.cmml" type="integer" xref="S1.I1.i2.p1.5.m5.4.4.2.4">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.5.m5.4c">E_{\left(n_{1},n_{2}\right)2}\rightarrow-E_{\left(n_{1},-n_{2}\right)2}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.5.m5.4d">italic_E start_POSTSUBSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) 2 end_POSTSUBSCRIPT → - italic_E start_POSTSUBSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , - italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) 2 end_POSTSUBSCRIPT</annotation></semantics></math> <math alttext="U_{\left(n_{1},n_{2}\right)1}\rightarrow U^{\dagger}_{\left(n_{1},-n_{2}\right% )1}" class="ltx_Math" display="inline" id="S1.I1.i2.p1.6.m6.4"><semantics id="S1.I1.i2.p1.6.m6.4a"><mrow id="S1.I1.i2.p1.6.m6.4.5" xref="S1.I1.i2.p1.6.m6.4.5.cmml"><msub id="S1.I1.i2.p1.6.m6.4.5.2" xref="S1.I1.i2.p1.6.m6.4.5.2.cmml"><mi id="S1.I1.i2.p1.6.m6.4.5.2.2" xref="S1.I1.i2.p1.6.m6.4.5.2.2.cmml">U</mi><mrow id="S1.I1.i2.p1.6.m6.2.2.2" xref="S1.I1.i2.p1.6.m6.2.2.2.cmml"><mrow id="S1.I1.i2.p1.6.m6.2.2.2.2.2" xref="S1.I1.i2.p1.6.m6.2.2.2.2.3.cmml"><mo id="S1.I1.i2.p1.6.m6.2.2.2.2.2.3" xref="S1.I1.i2.p1.6.m6.2.2.2.2.3.cmml">(</mo><msub id="S1.I1.i2.p1.6.m6.1.1.1.1.1.1" 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xref="S1.I1.i2.p1.6.m6.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p1.6.m6.3.3.1.1.1.1.1.cmml" xref="S1.I1.i2.p1.6.m6.3.3.1.1.1.1">subscript</csymbol><ci id="S1.I1.i2.p1.6.m6.3.3.1.1.1.1.2.cmml" xref="S1.I1.i2.p1.6.m6.3.3.1.1.1.1.2">𝑛</ci><cn id="S1.I1.i2.p1.6.m6.3.3.1.1.1.1.3.cmml" type="integer" xref="S1.I1.i2.p1.6.m6.3.3.1.1.1.1.3">1</cn></apply><apply id="S1.I1.i2.p1.6.m6.4.4.2.2.2.2.cmml" xref="S1.I1.i2.p1.6.m6.4.4.2.2.2.2"><minus id="S1.I1.i2.p1.6.m6.4.4.2.2.2.2.1.cmml" xref="S1.I1.i2.p1.6.m6.4.4.2.2.2.2"></minus><apply id="S1.I1.i2.p1.6.m6.4.4.2.2.2.2.2.cmml" xref="S1.I1.i2.p1.6.m6.4.4.2.2.2.2.2"><csymbol cd="ambiguous" id="S1.I1.i2.p1.6.m6.4.4.2.2.2.2.2.1.cmml" xref="S1.I1.i2.p1.6.m6.4.4.2.2.2.2.2">subscript</csymbol><ci id="S1.I1.i2.p1.6.m6.4.4.2.2.2.2.2.2.cmml" xref="S1.I1.i2.p1.6.m6.4.4.2.2.2.2.2.2">𝑛</ci><cn id="S1.I1.i2.p1.6.m6.4.4.2.2.2.2.2.3.cmml" type="integer" xref="S1.I1.i2.p1.6.m6.4.4.2.2.2.2.2.3">2</cn></apply></apply></interval><cn 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end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , - italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) 2 end_POSTSUBSCRIPT</annotation></semantics></math> and analogous for <math alttext="{\cal I}_{y}" class="ltx_Math" display="inline" id="S1.I1.i2.p1.8.m8.1"><semantics id="S1.I1.i2.p1.8.m8.1a"><msub id="S1.I1.i2.p1.8.m8.1.1" xref="S1.I1.i2.p1.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.I1.i2.p1.8.m8.1.1.2" xref="S1.I1.i2.p1.8.m8.1.1.2.cmml">ℐ</mi><mi id="S1.I1.i2.p1.8.m8.1.1.3" xref="S1.I1.i2.p1.8.m8.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.8.m8.1b"><apply id="S1.I1.i2.p1.8.m8.1.1.cmml" xref="S1.I1.i2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p1.8.m8.1.1.1.cmml" xref="S1.I1.i2.p1.8.m8.1.1">subscript</csymbol><ci id="S1.I1.i2.p1.8.m8.1.1.2.cmml" xref="S1.I1.i2.p1.8.m8.1.1.2">ℐ</ci><ci id="S1.I1.i2.p1.8.m8.1.1.3.cmml" xref="S1.I1.i2.p1.8.m8.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.8.m8.1c">{\cal I}_{y}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.8.m8.1d">caligraphic_I start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math>. These symmetries are important to prove the existence of an exponential number of zero-modes;</p> </div> </li> <li class="ltx_item" id="S1.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i3.p1"> <p class="ltx_p" id="S1.I1.i3.p1.2"><span class="ltx_text ltx_font_bold" id="S1.I1.i3.p1.2.1">Charge Conjugation</span>, which is characterized by the field transformations <math alttext="E_{ni}\rightarrow-E_{ni}" class="ltx_Math" display="inline" id="S1.I1.i3.p1.1.m1.1"><semantics id="S1.I1.i3.p1.1.m1.1a"><mrow id="S1.I1.i3.p1.1.m1.1.1" xref="S1.I1.i3.p1.1.m1.1.1.cmml"><msub id="S1.I1.i3.p1.1.m1.1.1.2" xref="S1.I1.i3.p1.1.m1.1.1.2.cmml"><mi id="S1.I1.i3.p1.1.m1.1.1.2.2" xref="S1.I1.i3.p1.1.m1.1.1.2.2.cmml">E</mi><mrow id="S1.I1.i3.p1.1.m1.1.1.2.3" xref="S1.I1.i3.p1.1.m1.1.1.2.3.cmml"><mi id="S1.I1.i3.p1.1.m1.1.1.2.3.2" xref="S1.I1.i3.p1.1.m1.1.1.2.3.2.cmml">n</mi><mo id="S1.I1.i3.p1.1.m1.1.1.2.3.1" xref="S1.I1.i3.p1.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S1.I1.i3.p1.1.m1.1.1.2.3.3" xref="S1.I1.i3.p1.1.m1.1.1.2.3.3.cmml">i</mi></mrow></msub><mo id="S1.I1.i3.p1.1.m1.1.1.1" stretchy="false" xref="S1.I1.i3.p1.1.m1.1.1.1.cmml">→</mo><mrow id="S1.I1.i3.p1.1.m1.1.1.3" xref="S1.I1.i3.p1.1.m1.1.1.3.cmml"><mo id="S1.I1.i3.p1.1.m1.1.1.3a" xref="S1.I1.i3.p1.1.m1.1.1.3.cmml">−</mo><msub id="S1.I1.i3.p1.1.m1.1.1.3.2" xref="S1.I1.i3.p1.1.m1.1.1.3.2.cmml"><mi id="S1.I1.i3.p1.1.m1.1.1.3.2.2" xref="S1.I1.i3.p1.1.m1.1.1.3.2.2.cmml">E</mi><mrow id="S1.I1.i3.p1.1.m1.1.1.3.2.3" xref="S1.I1.i3.p1.1.m1.1.1.3.2.3.cmml"><mi id="S1.I1.i3.p1.1.m1.1.1.3.2.3.2" xref="S1.I1.i3.p1.1.m1.1.1.3.2.3.2.cmml">n</mi><mo id="S1.I1.i3.p1.1.m1.1.1.3.2.3.1" xref="S1.I1.i3.p1.1.m1.1.1.3.2.3.1.cmml">⁢</mo><mi id="S1.I1.i3.p1.1.m1.1.1.3.2.3.3" xref="S1.I1.i3.p1.1.m1.1.1.3.2.3.3.cmml">i</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i3.p1.1.m1.1b"><apply id="S1.I1.i3.p1.1.m1.1.1.cmml" xref="S1.I1.i3.p1.1.m1.1.1"><ci id="S1.I1.i3.p1.1.m1.1.1.1.cmml" xref="S1.I1.i3.p1.1.m1.1.1.1">→</ci><apply id="S1.I1.i3.p1.1.m1.1.1.2.cmml" xref="S1.I1.i3.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S1.I1.i3.p1.1.m1.1.1.2.1.cmml" xref="S1.I1.i3.p1.1.m1.1.1.2">subscript</csymbol><ci id="S1.I1.i3.p1.1.m1.1.1.2.2.cmml" xref="S1.I1.i3.p1.1.m1.1.1.2.2">𝐸</ci><apply id="S1.I1.i3.p1.1.m1.1.1.2.3.cmml" xref="S1.I1.i3.p1.1.m1.1.1.2.3"><times id="S1.I1.i3.p1.1.m1.1.1.2.3.1.cmml" xref="S1.I1.i3.p1.1.m1.1.1.2.3.1"></times><ci id="S1.I1.i3.p1.1.m1.1.1.2.3.2.cmml" xref="S1.I1.i3.p1.1.m1.1.1.2.3.2">𝑛</ci><ci id="S1.I1.i3.p1.1.m1.1.1.2.3.3.cmml" xref="S1.I1.i3.p1.1.m1.1.1.2.3.3">𝑖</ci></apply></apply><apply id="S1.I1.i3.p1.1.m1.1.1.3.cmml" xref="S1.I1.i3.p1.1.m1.1.1.3"><minus id="S1.I1.i3.p1.1.m1.1.1.3.1.cmml" xref="S1.I1.i3.p1.1.m1.1.1.3"></minus><apply id="S1.I1.i3.p1.1.m1.1.1.3.2.cmml" xref="S1.I1.i3.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S1.I1.i3.p1.1.m1.1.1.3.2.1.cmml" xref="S1.I1.i3.p1.1.m1.1.1.3.2">subscript</csymbol><ci id="S1.I1.i3.p1.1.m1.1.1.3.2.2.cmml" xref="S1.I1.i3.p1.1.m1.1.1.3.2.2">𝐸</ci><apply id="S1.I1.i3.p1.1.m1.1.1.3.2.3.cmml" xref="S1.I1.i3.p1.1.m1.1.1.3.2.3"><times id="S1.I1.i3.p1.1.m1.1.1.3.2.3.1.cmml" xref="S1.I1.i3.p1.1.m1.1.1.3.2.3.1"></times><ci id="S1.I1.i3.p1.1.m1.1.1.3.2.3.2.cmml" xref="S1.I1.i3.p1.1.m1.1.1.3.2.3.2">𝑛</ci><ci id="S1.I1.i3.p1.1.m1.1.1.3.2.3.3.cmml" xref="S1.I1.i3.p1.1.m1.1.1.3.2.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i3.p1.1.m1.1c">E_{ni}\rightarrow-E_{ni}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i3.p1.1.m1.1d">italic_E start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT → - italic_E start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="U_{ni}\rightarrow U^{\dagger}_{ni}" class="ltx_Math" display="inline" id="S1.I1.i3.p1.2.m2.1"><semantics id="S1.I1.i3.p1.2.m2.1a"><mrow id="S1.I1.i3.p1.2.m2.1.1" xref="S1.I1.i3.p1.2.m2.1.1.cmml"><msub id="S1.I1.i3.p1.2.m2.1.1.2" xref="S1.I1.i3.p1.2.m2.1.1.2.cmml"><mi id="S1.I1.i3.p1.2.m2.1.1.2.2" xref="S1.I1.i3.p1.2.m2.1.1.2.2.cmml">U</mi><mrow id="S1.I1.i3.p1.2.m2.1.1.2.3" xref="S1.I1.i3.p1.2.m2.1.1.2.3.cmml"><mi id="S1.I1.i3.p1.2.m2.1.1.2.3.2" xref="S1.I1.i3.p1.2.m2.1.1.2.3.2.cmml">n</mi><mo id="S1.I1.i3.p1.2.m2.1.1.2.3.1" xref="S1.I1.i3.p1.2.m2.1.1.2.3.1.cmml">⁢</mo><mi id="S1.I1.i3.p1.2.m2.1.1.2.3.3" xref="S1.I1.i3.p1.2.m2.1.1.2.3.3.cmml">i</mi></mrow></msub><mo id="S1.I1.i3.p1.2.m2.1.1.1" stretchy="false" xref="S1.I1.i3.p1.2.m2.1.1.1.cmml">→</mo><msubsup id="S1.I1.i3.p1.2.m2.1.1.3" xref="S1.I1.i3.p1.2.m2.1.1.3.cmml"><mi id="S1.I1.i3.p1.2.m2.1.1.3.2.2" xref="S1.I1.i3.p1.2.m2.1.1.3.2.2.cmml">U</mi><mrow id="S1.I1.i3.p1.2.m2.1.1.3.3" xref="S1.I1.i3.p1.2.m2.1.1.3.3.cmml"><mi id="S1.I1.i3.p1.2.m2.1.1.3.3.2" xref="S1.I1.i3.p1.2.m2.1.1.3.3.2.cmml">n</mi><mo id="S1.I1.i3.p1.2.m2.1.1.3.3.1" xref="S1.I1.i3.p1.2.m2.1.1.3.3.1.cmml">⁢</mo><mi id="S1.I1.i3.p1.2.m2.1.1.3.3.3" xref="S1.I1.i3.p1.2.m2.1.1.3.3.3.cmml">i</mi></mrow><mo id="S1.I1.i3.p1.2.m2.1.1.3.2.3" xref="S1.I1.i3.p1.2.m2.1.1.3.2.3.cmml">†</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i3.p1.2.m2.1b"><apply id="S1.I1.i3.p1.2.m2.1.1.cmml" xref="S1.I1.i3.p1.2.m2.1.1"><ci id="S1.I1.i3.p1.2.m2.1.1.1.cmml" xref="S1.I1.i3.p1.2.m2.1.1.1">→</ci><apply id="S1.I1.i3.p1.2.m2.1.1.2.cmml" xref="S1.I1.i3.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.I1.i3.p1.2.m2.1.1.2.1.cmml" xref="S1.I1.i3.p1.2.m2.1.1.2">subscript</csymbol><ci id="S1.I1.i3.p1.2.m2.1.1.2.2.cmml" xref="S1.I1.i3.p1.2.m2.1.1.2.2">𝑈</ci><apply id="S1.I1.i3.p1.2.m2.1.1.2.3.cmml" xref="S1.I1.i3.p1.2.m2.1.1.2.3"><times id="S1.I1.i3.p1.2.m2.1.1.2.3.1.cmml" xref="S1.I1.i3.p1.2.m2.1.1.2.3.1"></times><ci id="S1.I1.i3.p1.2.m2.1.1.2.3.2.cmml" xref="S1.I1.i3.p1.2.m2.1.1.2.3.2">𝑛</ci><ci id="S1.I1.i3.p1.2.m2.1.1.2.3.3.cmml" xref="S1.I1.i3.p1.2.m2.1.1.2.3.3">𝑖</ci></apply></apply><apply id="S1.I1.i3.p1.2.m2.1.1.3.cmml" xref="S1.I1.i3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.I1.i3.p1.2.m2.1.1.3.1.cmml" xref="S1.I1.i3.p1.2.m2.1.1.3">subscript</csymbol><apply id="S1.I1.i3.p1.2.m2.1.1.3.2.cmml" xref="S1.I1.i3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.I1.i3.p1.2.m2.1.1.3.2.1.cmml" xref="S1.I1.i3.p1.2.m2.1.1.3">superscript</csymbol><ci id="S1.I1.i3.p1.2.m2.1.1.3.2.2.cmml" xref="S1.I1.i3.p1.2.m2.1.1.3.2.2">𝑈</ci><ci id="S1.I1.i3.p1.2.m2.1.1.3.2.3.cmml" xref="S1.I1.i3.p1.2.m2.1.1.3.2.3">†</ci></apply><apply id="S1.I1.i3.p1.2.m2.1.1.3.3.cmml" xref="S1.I1.i3.p1.2.m2.1.1.3.3"><times id="S1.I1.i3.p1.2.m2.1.1.3.3.1.cmml" xref="S1.I1.i3.p1.2.m2.1.1.3.3.1"></times><ci id="S1.I1.i3.p1.2.m2.1.1.3.3.2.cmml" xref="S1.I1.i3.p1.2.m2.1.1.3.3.2">𝑛</ci><ci id="S1.I1.i3.p1.2.m2.1.1.3.3.3.cmml" xref="S1.I1.i3.p1.2.m2.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i3.p1.2.m2.1c">U_{ni}\rightarrow U^{\dagger}_{ni}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i3.p1.2.m2.1d">italic_U start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT → italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> </ul> </div> <div class="ltx_para" id="S1a.p3"> <p class="ltx_p" id="S1a.p3.1">When explicitly referenced, we restricted our exact diagonalization to the zero momentum sector and to the +1 charge conjugation sector. Otherwise, the plots are in the electric field basis and no symmetries beyond applying Gauss’ law and being restricted to the zero winding sector are used.</p> </div> </section> <section class="ltx_section" id="S2a"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span>Index Theorem and an Exponential Number of zero-modes</h2> <div class="ltx_para" id="S2a.p1"> <p class="ltx_p" id="S2a.p1.1">In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib62" title="">62</a>]</cite> it was shown that non-integrable models can still exhibit an exponential number of eigenstates with zero energy. We review this result and adapt it to our models of interest. For this section, we will refer to plaquette flipping terms as the kinetic part of the Hamiltonian</p> </div> <div class="ltx_para" id="S2a.p2"> <table class="ltx_equation ltx_eqn_table" id="S2.E19"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="K=\sum_{n}\left(U_{n1}^{\dagger}U_{n+\hat{1}2}^{\dagger}U_{n2}U_{n+\hat{2}1}+% \mathrm{h.c.}\right)." class="ltx_math_unparsed" display="block" id="S2.E19.m1.1"><semantics id="S2.E19.m1.1a"><mrow id="S2.E19.m1.1b"><mi id="S2.E19.m1.1.1">K</mi><mo id="S2.E19.m1.1.2" rspace="0.111em">=</mo><munder id="S2.E19.m1.1.3"><mo id="S2.E19.m1.1.3.2" movablelimits="false" rspace="0em">∑</mo><mi id="S2.E19.m1.1.3.3">n</mi></munder><mrow id="S2.E19.m1.1.4"><mo id="S2.E19.m1.1.4.1">(</mo><msubsup id="S2.E19.m1.1.4.2"><mi id="S2.E19.m1.1.4.2.2.2">U</mi><mrow id="S2.E19.m1.1.4.2.2.3"><mi id="S2.E19.m1.1.4.2.2.3.2">n</mi><mo id="S2.E19.m1.1.4.2.2.3.1">⁢</mo><mn id="S2.E19.m1.1.4.2.2.3.3">1</mn></mrow><mo id="S2.E19.m1.1.4.2.3">†</mo></msubsup><msubsup id="S2.E19.m1.1.4.3"><mi id="S2.E19.m1.1.4.3.2.2">U</mi><mrow id="S2.E19.m1.1.4.3.2.3"><mi id="S2.E19.m1.1.4.3.2.3.2">n</mi><mo id="S2.E19.m1.1.4.3.2.3.1">+</mo><mrow id="S2.E19.m1.1.4.3.2.3.3"><mover accent="true" id="S2.E19.m1.1.4.3.2.3.3.2"><mn id="S2.E19.m1.1.4.3.2.3.3.2.2">1</mn><mo id="S2.E19.m1.1.4.3.2.3.3.2.1">^</mo></mover><mo id="S2.E19.m1.1.4.3.2.3.3.1">⁢</mo><mn id="S2.E19.m1.1.4.3.2.3.3.3">2</mn></mrow></mrow><mo id="S2.E19.m1.1.4.3.3">†</mo></msubsup><msub id="S2.E19.m1.1.4.4"><mi id="S2.E19.m1.1.4.4.2">U</mi><mrow id="S2.E19.m1.1.4.4.3"><mi id="S2.E19.m1.1.4.4.3.2">n</mi><mo id="S2.E19.m1.1.4.4.3.1">⁢</mo><mn id="S2.E19.m1.1.4.4.3.3">2</mn></mrow></msub><msub id="S2.E19.m1.1.4.5"><mi id="S2.E19.m1.1.4.5.2">U</mi><mrow id="S2.E19.m1.1.4.5.3"><mi id="S2.E19.m1.1.4.5.3.2">n</mi><mo id="S2.E19.m1.1.4.5.3.1">+</mo><mrow id="S2.E19.m1.1.4.5.3.3"><mover accent="true" id="S2.E19.m1.1.4.5.3.3.2"><mn id="S2.E19.m1.1.4.5.3.3.2.2">2</mn><mo id="S2.E19.m1.1.4.5.3.3.2.1">^</mo></mover><mo id="S2.E19.m1.1.4.5.3.3.1">⁢</mo><mn id="S2.E19.m1.1.4.5.3.3.3">1</mn></mrow></mrow></msub><mo id="S2.E19.m1.1.4.6">+</mo><mi id="S2.E19.m1.1.4.7" mathvariant="normal">h</mi><mo id="S2.E19.m1.1.4.8" lspace="0em" rspace="0.167em">.</mo><mi id="S2.E19.m1.1.4.9" mathvariant="normal">c</mi><mo id="S2.E19.m1.1.4.10" lspace="0em" rspace="0.167em">.</mo><mo id="S2.E19.m1.1.4.11">)</mo></mrow><mo id="S2.E19.m1.1.5" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="S2.E19.m1.1c">K=\sum_{n}\left(U_{n1}^{\dagger}U_{n+\hat{1}2}^{\dagger}U_{n2}U_{n+\hat{2}1}+% \mathrm{h.c.}\right).</annotation><annotation encoding="application/x-llamapun" id="S2.E19.m1.1d">italic_K = ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_n 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_U start_POSTSUBSCRIPT italic_n + over^ start_ARG 1 end_ARG 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_U start_POSTSUBSCRIPT italic_n 2 end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_n + over^ start_ARG 2 end_ARG 1 end_POSTSUBSCRIPT + roman_h . roman_c . ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(19)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2a.p2.1">We only need to assume that <math alttext="U_{ni}/U_{ni}^{\dagger}" class="ltx_Math" display="inline" id="S2a.p2.1.m1.1"><semantics id="S2a.p2.1.m1.1a"><mrow id="S2a.p2.1.m1.1.1" xref="S2a.p2.1.m1.1.1.cmml"><msub id="S2a.p2.1.m1.1.1.2" xref="S2a.p2.1.m1.1.1.2.cmml"><mi id="S2a.p2.1.m1.1.1.2.2" xref="S2a.p2.1.m1.1.1.2.2.cmml">U</mi><mrow id="S2a.p2.1.m1.1.1.2.3" xref="S2a.p2.1.m1.1.1.2.3.cmml"><mi id="S2a.p2.1.m1.1.1.2.3.2" xref="S2a.p2.1.m1.1.1.2.3.2.cmml">n</mi><mo id="S2a.p2.1.m1.1.1.2.3.1" xref="S2a.p2.1.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S2a.p2.1.m1.1.1.2.3.3" xref="S2a.p2.1.m1.1.1.2.3.3.cmml">i</mi></mrow></msub><mo id="S2a.p2.1.m1.1.1.1" xref="S2a.p2.1.m1.1.1.1.cmml">/</mo><msubsup id="S2a.p2.1.m1.1.1.3" xref="S2a.p2.1.m1.1.1.3.cmml"><mi id="S2a.p2.1.m1.1.1.3.2.2" xref="S2a.p2.1.m1.1.1.3.2.2.cmml">U</mi><mrow id="S2a.p2.1.m1.1.1.3.2.3" xref="S2a.p2.1.m1.1.1.3.2.3.cmml"><mi id="S2a.p2.1.m1.1.1.3.2.3.2" xref="S2a.p2.1.m1.1.1.3.2.3.2.cmml">n</mi><mo id="S2a.p2.1.m1.1.1.3.2.3.1" xref="S2a.p2.1.m1.1.1.3.2.3.1.cmml">⁢</mo><mi id="S2a.p2.1.m1.1.1.3.2.3.3" xref="S2a.p2.1.m1.1.1.3.2.3.3.cmml">i</mi></mrow><mo id="S2a.p2.1.m1.1.1.3.3" xref="S2a.p2.1.m1.1.1.3.3.cmml">†</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2a.p2.1.m1.1b"><apply id="S2a.p2.1.m1.1.1.cmml" xref="S2a.p2.1.m1.1.1"><divide id="S2a.p2.1.m1.1.1.1.cmml" xref="S2a.p2.1.m1.1.1.1"></divide><apply id="S2a.p2.1.m1.1.1.2.cmml" xref="S2a.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2a.p2.1.m1.1.1.2.1.cmml" xref="S2a.p2.1.m1.1.1.2">subscript</csymbol><ci id="S2a.p2.1.m1.1.1.2.2.cmml" xref="S2a.p2.1.m1.1.1.2.2">𝑈</ci><apply id="S2a.p2.1.m1.1.1.2.3.cmml" xref="S2a.p2.1.m1.1.1.2.3"><times id="S2a.p2.1.m1.1.1.2.3.1.cmml" xref="S2a.p2.1.m1.1.1.2.3.1"></times><ci id="S2a.p2.1.m1.1.1.2.3.2.cmml" xref="S2a.p2.1.m1.1.1.2.3.2">𝑛</ci><ci id="S2a.p2.1.m1.1.1.2.3.3.cmml" xref="S2a.p2.1.m1.1.1.2.3.3">𝑖</ci></apply></apply><apply id="S2a.p2.1.m1.1.1.3.cmml" xref="S2a.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2a.p2.1.m1.1.1.3.1.cmml" xref="S2a.p2.1.m1.1.1.3">superscript</csymbol><apply id="S2a.p2.1.m1.1.1.3.2.cmml" xref="S2a.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2a.p2.1.m1.1.1.3.2.1.cmml" xref="S2a.p2.1.m1.1.1.3">subscript</csymbol><ci id="S2a.p2.1.m1.1.1.3.2.2.cmml" xref="S2a.p2.1.m1.1.1.3.2.2">𝑈</ci><apply id="S2a.p2.1.m1.1.1.3.2.3.cmml" xref="S2a.p2.1.m1.1.1.3.2.3"><times id="S2a.p2.1.m1.1.1.3.2.3.1.cmml" xref="S2a.p2.1.m1.1.1.3.2.3.1"></times><ci id="S2a.p2.1.m1.1.1.3.2.3.2.cmml" xref="S2a.p2.1.m1.1.1.3.2.3.2">𝑛</ci><ci id="S2a.p2.1.m1.1.1.3.2.3.3.cmml" xref="S2a.p2.1.m1.1.1.3.2.3.3">𝑖</ci></apply></apply><ci id="S2a.p2.1.m1.1.1.3.3.cmml" xref="S2a.p2.1.m1.1.1.3.3">†</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2a.p2.1.m1.1c">U_{ni}/U_{ni}^{\dagger}</annotation><annotation encoding="application/x-llamapun" id="S2a.p2.1.m1.1d">italic_U start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT / italic_U start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT</annotation></semantics></math> are operators that raise/lower the value of the electric field on the link by 1. This makes the result very robust with respect to a wide range of formulations.</p> </div> <div class="ltx_para" id="S2a.p3"> <p class="ltx_p" id="S2a.p3.1">We define a set of link operators <math alttext="\zeta_{ni}" class="ltx_Math" display="inline" id="S2a.p3.1.m1.1"><semantics id="S2a.p3.1.m1.1a"><msub id="S2a.p3.1.m1.1.1" xref="S2a.p3.1.m1.1.1.cmml"><mi id="S2a.p3.1.m1.1.1.2" xref="S2a.p3.1.m1.1.1.2.cmml">ζ</mi><mrow id="S2a.p3.1.m1.1.1.3" xref="S2a.p3.1.m1.1.1.3.cmml"><mi id="S2a.p3.1.m1.1.1.3.2" xref="S2a.p3.1.m1.1.1.3.2.cmml">n</mi><mo id="S2a.p3.1.m1.1.1.3.1" xref="S2a.p3.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S2a.p3.1.m1.1.1.3.3" xref="S2a.p3.1.m1.1.1.3.3.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2a.p3.1.m1.1b"><apply id="S2a.p3.1.m1.1.1.cmml" xref="S2a.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S2a.p3.1.m1.1.1.1.cmml" xref="S2a.p3.1.m1.1.1">subscript</csymbol><ci id="S2a.p3.1.m1.1.1.2.cmml" xref="S2a.p3.1.m1.1.1.2">𝜁</ci><apply id="S2a.p3.1.m1.1.1.3.cmml" xref="S2a.p3.1.m1.1.1.3"><times id="S2a.p3.1.m1.1.1.3.1.cmml" xref="S2a.p3.1.m1.1.1.3.1"></times><ci id="S2a.p3.1.m1.1.1.3.2.cmml" xref="S2a.p3.1.m1.1.1.3.2">𝑛</ci><ci id="S2a.p3.1.m1.1.1.3.3.cmml" xref="S2a.p3.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2a.p3.1.m1.1c">\zeta_{ni}</annotation><annotation encoding="application/x-llamapun" id="S2a.p3.1.m1.1d">italic_ζ start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, which are diagonal in the electric field basis. For each link we have</p> </div> <div class="ltx_para" id="S2a.p4"> <table class="ltx_equation ltx_eqn_table" id="S2.E20"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\zeta\ket{\varepsilon}=\left(-1\right)^{\varepsilon}\ket{\varepsilon}." class="ltx_Math" display="block" id="S2.E20.m1.3"><semantics id="S2.E20.m1.3a"><mrow id="S2.E20.m1.3.3.1" xref="S2.E20.m1.3.3.1.1.cmml"><mrow id="S2.E20.m1.3.3.1.1" xref="S2.E20.m1.3.3.1.1.cmml"><mrow id="S2.E20.m1.3.3.1.1.3" xref="S2.E20.m1.3.3.1.1.3.cmml"><mi id="S2.E20.m1.3.3.1.1.3.2" xref="S2.E20.m1.3.3.1.1.3.2.cmml">ζ</mi><mo id="S2.E20.m1.3.3.1.1.3.1" xref="S2.E20.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S2.E20.m1.1.1.3" xref="S2.E20.m1.1.1.2.cmml"><mo id="S2.E20.m1.1.1.3.1" stretchy="false" xref="S2.E20.m1.1.1.2.1.cmml">|</mo><mi id="S2.E20.m1.1.1.1.1" xref="S2.E20.m1.1.1.1.1.cmml">ε</mi><mo id="S2.E20.m1.1.1.3.2" stretchy="false" xref="S2.E20.m1.1.1.2.1.cmml">⟩</mo></mrow></mrow><mo id="S2.E20.m1.3.3.1.1.2" xref="S2.E20.m1.3.3.1.1.2.cmml">=</mo><mrow id="S2.E20.m1.3.3.1.1.1" xref="S2.E20.m1.3.3.1.1.1.cmml"><msup id="S2.E20.m1.3.3.1.1.1.1" xref="S2.E20.m1.3.3.1.1.1.1.cmml"><mrow id="S2.E20.m1.3.3.1.1.1.1.1.1" xref="S2.E20.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="S2.E20.m1.3.3.1.1.1.1.1.1.2" xref="S2.E20.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E20.m1.3.3.1.1.1.1.1.1.1" xref="S2.E20.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="S2.E20.m1.3.3.1.1.1.1.1.1.1a" xref="S2.E20.m1.3.3.1.1.1.1.1.1.1.cmml">−</mo><mn id="S2.E20.m1.3.3.1.1.1.1.1.1.1.2" xref="S2.E20.m1.3.3.1.1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S2.E20.m1.3.3.1.1.1.1.1.1.3" xref="S2.E20.m1.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S2.E20.m1.3.3.1.1.1.1.3" xref="S2.E20.m1.3.3.1.1.1.1.3.cmml">ε</mi></msup><mo id="S2.E20.m1.3.3.1.1.1.2" xref="S2.E20.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S2.E20.m1.2.2.3" xref="S2.E20.m1.2.2.2.cmml"><mo id="S2.E20.m1.2.2.3.1" stretchy="false" xref="S2.E20.m1.2.2.2.1.cmml">|</mo><mi id="S2.E20.m1.2.2.1.1" xref="S2.E20.m1.2.2.1.1.cmml">ε</mi><mo id="S2.E20.m1.2.2.3.2" stretchy="false" xref="S2.E20.m1.2.2.2.1.cmml">⟩</mo></mrow></mrow></mrow><mo id="S2.E20.m1.3.3.1.2" lspace="0em" xref="S2.E20.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E20.m1.3b"><apply id="S2.E20.m1.3.3.1.1.cmml" xref="S2.E20.m1.3.3.1"><eq id="S2.E20.m1.3.3.1.1.2.cmml" xref="S2.E20.m1.3.3.1.1.2"></eq><apply id="S2.E20.m1.3.3.1.1.3.cmml" xref="S2.E20.m1.3.3.1.1.3"><times id="S2.E20.m1.3.3.1.1.3.1.cmml" xref="S2.E20.m1.3.3.1.1.3.1"></times><ci id="S2.E20.m1.3.3.1.1.3.2.cmml" xref="S2.E20.m1.3.3.1.1.3.2">𝜁</ci><apply id="S2.E20.m1.1.1.2.cmml" xref="S2.E20.m1.1.1.3"><csymbol cd="latexml" id="S2.E20.m1.1.1.2.1.cmml" xref="S2.E20.m1.1.1.3.1">ket</csymbol><ci id="S2.E20.m1.1.1.1.1.cmml" xref="S2.E20.m1.1.1.1.1">𝜀</ci></apply></apply><apply id="S2.E20.m1.3.3.1.1.1.cmml" xref="S2.E20.m1.3.3.1.1.1"><times id="S2.E20.m1.3.3.1.1.1.2.cmml" xref="S2.E20.m1.3.3.1.1.1.2"></times><apply id="S2.E20.m1.3.3.1.1.1.1.cmml" xref="S2.E20.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S2.E20.m1.3.3.1.1.1.1.2.cmml" xref="S2.E20.m1.3.3.1.1.1.1">superscript</csymbol><apply id="S2.E20.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S2.E20.m1.3.3.1.1.1.1.1.1"><minus id="S2.E20.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S2.E20.m1.3.3.1.1.1.1.1.1"></minus><cn id="S2.E20.m1.3.3.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S2.E20.m1.3.3.1.1.1.1.1.1.1.2">1</cn></apply><ci id="S2.E20.m1.3.3.1.1.1.1.3.cmml" xref="S2.E20.m1.3.3.1.1.1.1.3">𝜀</ci></apply><apply id="S2.E20.m1.2.2.2.cmml" xref="S2.E20.m1.2.2.3"><csymbol cd="latexml" id="S2.E20.m1.2.2.2.1.cmml" xref="S2.E20.m1.2.2.3.1">ket</csymbol><ci id="S2.E20.m1.2.2.1.1.cmml" xref="S2.E20.m1.2.2.1.1">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E20.m1.3c">\zeta\ket{\varepsilon}=\left(-1\right)^{\varepsilon}\ket{\varepsilon}.</annotation><annotation encoding="application/x-llamapun" id="S2.E20.m1.3d">italic_ζ | start_ARG italic_ε end_ARG ⟩ = ( - 1 ) start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT | start_ARG italic_ε end_ARG ⟩ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(20)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2a.p4.1">We further construct</p> </div> <div class="ltx_para" id="S2a.p5"> <table class="ltx_equation ltx_eqn_table" id="S2.E21"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="{\cal C}=\prod_{n=0}^{L_{1}-1}\prod_{a=0}^{L_{2}/2-1}\zeta_{\left(n,2a\right)1}," class="ltx_Math" display="block" id="S2.E21.m1.3"><semantics id="S2.E21.m1.3a"><mrow id="S2.E21.m1.3.3.1" xref="S2.E21.m1.3.3.1.1.cmml"><mrow id="S2.E21.m1.3.3.1.1" xref="S2.E21.m1.3.3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E21.m1.3.3.1.1.2" xref="S2.E21.m1.3.3.1.1.2.cmml">𝒞</mi><mo id="S2.E21.m1.3.3.1.1.1" rspace="0.111em" xref="S2.E21.m1.3.3.1.1.1.cmml">=</mo><mrow id="S2.E21.m1.3.3.1.1.3" xref="S2.E21.m1.3.3.1.1.3.cmml"><munderover id="S2.E21.m1.3.3.1.1.3.1" xref="S2.E21.m1.3.3.1.1.3.1.cmml"><mo id="S2.E21.m1.3.3.1.1.3.1.2.2" movablelimits="false" rspace="0em" xref="S2.E21.m1.3.3.1.1.3.1.2.2.cmml">∏</mo><mrow id="S2.E21.m1.3.3.1.1.3.1.2.3" xref="S2.E21.m1.3.3.1.1.3.1.2.3.cmml"><mi id="S2.E21.m1.3.3.1.1.3.1.2.3.2" xref="S2.E21.m1.3.3.1.1.3.1.2.3.2.cmml">n</mi><mo id="S2.E21.m1.3.3.1.1.3.1.2.3.1" xref="S2.E21.m1.3.3.1.1.3.1.2.3.1.cmml">=</mo><mn id="S2.E21.m1.3.3.1.1.3.1.2.3.3" xref="S2.E21.m1.3.3.1.1.3.1.2.3.3.cmml">0</mn></mrow><mrow id="S2.E21.m1.3.3.1.1.3.1.3" 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end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT ∏ start_POSTSUBSCRIPT italic_a = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT / 2 - 1 end_POSTSUPERSCRIPT italic_ζ start_POSTSUBSCRIPT ( italic_n , 2 italic_a ) 1 end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(21)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2a.p5.9">where we have assumed <math alttext="L_{1}" class="ltx_Math" display="inline" id="S2a.p5.1.m1.1"><semantics id="S2a.p5.1.m1.1a"><msub id="S2a.p5.1.m1.1.1" xref="S2a.p5.1.m1.1.1.cmml"><mi id="S2a.p5.1.m1.1.1.2" xref="S2a.p5.1.m1.1.1.2.cmml">L</mi><mn id="S2a.p5.1.m1.1.1.3" xref="S2a.p5.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2a.p5.1.m1.1b"><apply id="S2a.p5.1.m1.1.1.cmml" xref="S2a.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S2a.p5.1.m1.1.1.1.cmml" xref="S2a.p5.1.m1.1.1">subscript</csymbol><ci id="S2a.p5.1.m1.1.1.2.cmml" xref="S2a.p5.1.m1.1.1.2">𝐿</ci><cn id="S2a.p5.1.m1.1.1.3.cmml" type="integer" xref="S2a.p5.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2a.p5.1.m1.1c">L_{1}</annotation><annotation encoding="application/x-llamapun" id="S2a.p5.1.m1.1d">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> to be even. We can see that <math alttext="{\cal C}" class="ltx_Math" display="inline" id="S2a.p5.2.m2.1"><semantics id="S2a.p5.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2a.p5.2.m2.1.1" xref="S2a.p5.2.m2.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2a.p5.2.m2.1b"><ci id="S2a.p5.2.m2.1.1.cmml" xref="S2a.p5.2.m2.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2a.p5.2.m2.1c">{\cal C}</annotation><annotation encoding="application/x-llamapun" id="S2a.p5.2.m2.1d">caligraphic_C</annotation></semantics></math> anti-commutes with the kinetic Hamiltonian <math alttext="\left\{{\cal C},K\right\}=0" class="ltx_Math" display="inline" id="S2a.p5.3.m3.2"><semantics id="S2a.p5.3.m3.2a"><mrow id="S2a.p5.3.m3.2.3" xref="S2a.p5.3.m3.2.3.cmml"><mrow id="S2a.p5.3.m3.2.3.2.2" xref="S2a.p5.3.m3.2.3.2.1.cmml"><mo id="S2a.p5.3.m3.2.3.2.2.1" xref="S2a.p5.3.m3.2.3.2.1.cmml">{</mo><mi class="ltx_font_mathcaligraphic" id="S2a.p5.3.m3.1.1" xref="S2a.p5.3.m3.1.1.cmml">𝒞</mi><mo id="S2a.p5.3.m3.2.3.2.2.2" xref="S2a.p5.3.m3.2.3.2.1.cmml">,</mo><mi id="S2a.p5.3.m3.2.2" xref="S2a.p5.3.m3.2.2.cmml">K</mi><mo id="S2a.p5.3.m3.2.3.2.2.3" xref="S2a.p5.3.m3.2.3.2.1.cmml">}</mo></mrow><mo id="S2a.p5.3.m3.2.3.1" xref="S2a.p5.3.m3.2.3.1.cmml">=</mo><mn id="S2a.p5.3.m3.2.3.3" xref="S2a.p5.3.m3.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2a.p5.3.m3.2b"><apply id="S2a.p5.3.m3.2.3.cmml" xref="S2a.p5.3.m3.2.3"><eq id="S2a.p5.3.m3.2.3.1.cmml" xref="S2a.p5.3.m3.2.3.1"></eq><set id="S2a.p5.3.m3.2.3.2.1.cmml" xref="S2a.p5.3.m3.2.3.2.2"><ci id="S2a.p5.3.m3.1.1.cmml" xref="S2a.p5.3.m3.1.1">𝒞</ci><ci id="S2a.p5.3.m3.2.2.cmml" xref="S2a.p5.3.m3.2.2">𝐾</ci></set><cn id="S2a.p5.3.m3.2.3.3.cmml" type="integer" xref="S2a.p5.3.m3.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2a.p5.3.m3.2c">\left\{{\cal C},K\right\}=0</annotation><annotation encoding="application/x-llamapun" id="S2a.p5.3.m3.2d">{ caligraphic_C , italic_K } = 0</annotation></semantics></math>. This implies that if <math alttext="\ket{E}" class="ltx_Math" display="inline" id="S2a.p5.4.m4.1"><semantics id="S2a.p5.4.m4.1a"><mrow id="S2a.p5.4.m4.1.1.3" xref="S2a.p5.4.m4.1.1.2.cmml"><mo id="S2a.p5.4.m4.1.1.3.1" stretchy="false" xref="S2a.p5.4.m4.1.1.2.1.cmml">|</mo><mi id="S2a.p5.4.m4.1.1.1.1" xref="S2a.p5.4.m4.1.1.1.1.cmml">E</mi><mo id="S2a.p5.4.m4.1.1.3.2" stretchy="false" xref="S2a.p5.4.m4.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S2a.p5.4.m4.1b"><apply id="S2a.p5.4.m4.1.1.2.cmml" xref="S2a.p5.4.m4.1.1.3"><csymbol cd="latexml" id="S2a.p5.4.m4.1.1.2.1.cmml" xref="S2a.p5.4.m4.1.1.3.1">ket</csymbol><ci id="S2a.p5.4.m4.1.1.1.1.cmml" xref="S2a.p5.4.m4.1.1.1.1">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2a.p5.4.m4.1c">\ket{E}</annotation><annotation encoding="application/x-llamapun" id="S2a.p5.4.m4.1d">| start_ARG italic_E end_ARG ⟩</annotation></semantics></math> is an eigenstate with energy <math alttext="E" class="ltx_Math" display="inline" id="S2a.p5.5.m5.1"><semantics id="S2a.p5.5.m5.1a"><mi id="S2a.p5.5.m5.1.1" xref="S2a.p5.5.m5.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S2a.p5.5.m5.1b"><ci id="S2a.p5.5.m5.1.1.cmml" xref="S2a.p5.5.m5.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S2a.p5.5.m5.1c">E</annotation><annotation encoding="application/x-llamapun" id="S2a.p5.5.m5.1d">italic_E</annotation></semantics></math>, then <math alttext="{\cal C}\ket{E}" class="ltx_Math" display="inline" id="S2a.p5.6.m6.1"><semantics id="S2a.p5.6.m6.1a"><mrow id="S2a.p5.6.m6.1.2" xref="S2a.p5.6.m6.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2a.p5.6.m6.1.2.2" xref="S2a.p5.6.m6.1.2.2.cmml">𝒞</mi><mo id="S2a.p5.6.m6.1.2.1" xref="S2a.p5.6.m6.1.2.1.cmml">⁢</mo><mrow id="S2a.p5.6.m6.1.1.3" xref="S2a.p5.6.m6.1.1.2.cmml"><mo id="S2a.p5.6.m6.1.1.3.1" stretchy="false" xref="S2a.p5.6.m6.1.1.2.1.cmml">|</mo><mi id="S2a.p5.6.m6.1.1.1.1" xref="S2a.p5.6.m6.1.1.1.1.cmml">E</mi><mo id="S2a.p5.6.m6.1.1.3.2" stretchy="false" xref="S2a.p5.6.m6.1.1.2.1.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2a.p5.6.m6.1b"><apply id="S2a.p5.6.m6.1.2.cmml" xref="S2a.p5.6.m6.1.2"><times id="S2a.p5.6.m6.1.2.1.cmml" xref="S2a.p5.6.m6.1.2.1"></times><ci id="S2a.p5.6.m6.1.2.2.cmml" xref="S2a.p5.6.m6.1.2.2">𝒞</ci><apply id="S2a.p5.6.m6.1.1.2.cmml" xref="S2a.p5.6.m6.1.1.3"><csymbol cd="latexml" id="S2a.p5.6.m6.1.1.2.1.cmml" xref="S2a.p5.6.m6.1.1.3.1">ket</csymbol><ci id="S2a.p5.6.m6.1.1.1.1.cmml" xref="S2a.p5.6.m6.1.1.1.1">𝐸</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2a.p5.6.m6.1c">{\cal C}\ket{E}</annotation><annotation encoding="application/x-llamapun" id="S2a.p5.6.m6.1d">caligraphic_C | start_ARG italic_E end_ARG ⟩</annotation></semantics></math> has energy <math alttext="-E" class="ltx_Math" display="inline" id="S2a.p5.7.m7.1"><semantics id="S2a.p5.7.m7.1a"><mrow id="S2a.p5.7.m7.1.1" xref="S2a.p5.7.m7.1.1.cmml"><mo id="S2a.p5.7.m7.1.1a" xref="S2a.p5.7.m7.1.1.cmml">−</mo><mi id="S2a.p5.7.m7.1.1.2" xref="S2a.p5.7.m7.1.1.2.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S2a.p5.7.m7.1b"><apply id="S2a.p5.7.m7.1.1.cmml" xref="S2a.p5.7.m7.1.1"><minus id="S2a.p5.7.m7.1.1.1.cmml" xref="S2a.p5.7.m7.1.1"></minus><ci id="S2a.p5.7.m7.1.1.2.cmml" xref="S2a.p5.7.m7.1.1.2">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2a.p5.7.m7.1c">-E</annotation><annotation encoding="application/x-llamapun" id="S2a.p5.7.m7.1d">- italic_E</annotation></semantics></math>. In the subspace generated by eigenstates with zero energy, we can diagonalize <math alttext="{\cal C}" class="ltx_Math" display="inline" id="S2a.p5.8.m8.1"><semantics id="S2a.p5.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S2a.p5.8.m8.1.1" xref="S2a.p5.8.m8.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2a.p5.8.m8.1b"><ci id="S2a.p5.8.m8.1.1.cmml" xref="S2a.p5.8.m8.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2a.p5.8.m8.1c">{\cal C}</annotation><annotation encoding="application/x-llamapun" id="S2a.p5.8.m8.1d">caligraphic_C</annotation></semantics></math> and <math alttext="K" class="ltx_Math" display="inline" id="S2a.p5.9.m9.1"><semantics id="S2a.p5.9.m9.1a"><mi id="S2a.p5.9.m9.1.1" xref="S2a.p5.9.m9.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S2a.p5.9.m9.1b"><ci id="S2a.p5.9.m9.1.1.cmml" xref="S2a.p5.9.m9.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2a.p5.9.m9.1c">K</annotation><annotation encoding="application/x-llamapun" id="S2a.p5.9.m9.1d">italic_K</annotation></semantics></math> together.</p> </div> <div class="ltx_para" id="S2a.p6"> <p class="ltx_p" id="S2a.p6.4">Let now <math alttext="{\cal I}_{y}" class="ltx_Math" display="inline" id="S2a.p6.1.m1.1"><semantics id="S2a.p6.1.m1.1a"><msub id="S2a.p6.1.m1.1.1" xref="S2a.p6.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2a.p6.1.m1.1.1.2" xref="S2a.p6.1.m1.1.1.2.cmml">ℐ</mi><mi id="S2a.p6.1.m1.1.1.3" xref="S2a.p6.1.m1.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S2a.p6.1.m1.1b"><apply id="S2a.p6.1.m1.1.1.cmml" xref="S2a.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S2a.p6.1.m1.1.1.1.cmml" xref="S2a.p6.1.m1.1.1">subscript</csymbol><ci id="S2a.p6.1.m1.1.1.2.cmml" xref="S2a.p6.1.m1.1.1.2">ℐ</ci><ci id="S2a.p6.1.m1.1.1.3.cmml" xref="S2a.p6.1.m1.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2a.p6.1.m1.1c">{\cal I}_{y}</annotation><annotation encoding="application/x-llamapun" id="S2a.p6.1.m1.1d">caligraphic_I start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> be the unitary transformation that reflects along the vertical axes, as defined above. This is a symmetry of the Hamiltonian and also commutes with <math alttext="{\cal C}" class="ltx_Math" display="inline" id="S2a.p6.2.m2.1"><semantics id="S2a.p6.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2a.p6.2.m2.1.1" xref="S2a.p6.2.m2.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2a.p6.2.m2.1b"><ci id="S2a.p6.2.m2.1.1.cmml" xref="S2a.p6.2.m2.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2a.p6.2.m2.1c">{\cal C}</annotation><annotation encoding="application/x-llamapun" id="S2a.p6.2.m2.1d">caligraphic_C</annotation></semantics></math>. We can now use the existence of <math alttext="{\cal I}_{y}" class="ltx_Math" display="inline" id="S2a.p6.3.m3.1"><semantics id="S2a.p6.3.m3.1a"><msub id="S2a.p6.3.m3.1.1" xref="S2a.p6.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2a.p6.3.m3.1.1.2" xref="S2a.p6.3.m3.1.1.2.cmml">ℐ</mi><mi id="S2a.p6.3.m3.1.1.3" xref="S2a.p6.3.m3.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S2a.p6.3.m3.1b"><apply id="S2a.p6.3.m3.1.1.cmml" xref="S2a.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S2a.p6.3.m3.1.1.1.cmml" xref="S2a.p6.3.m3.1.1">subscript</csymbol><ci id="S2a.p6.3.m3.1.1.2.cmml" xref="S2a.p6.3.m3.1.1.2">ℐ</ci><ci id="S2a.p6.3.m3.1.1.3.cmml" xref="S2a.p6.3.m3.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2a.p6.3.m3.1c">{\cal I}_{y}</annotation><annotation encoding="application/x-llamapun" id="S2a.p6.3.m3.1d">caligraphic_I start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="{\cal C}" class="ltx_Math" display="inline" id="S2a.p6.4.m4.1"><semantics id="S2a.p6.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S2a.p6.4.m4.1.1" xref="S2a.p6.4.m4.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2a.p6.4.m4.1b"><ci id="S2a.p6.4.m4.1.1.cmml" xref="S2a.p6.4.m4.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2a.p6.4.m4.1c">{\cal C}</annotation><annotation encoding="application/x-llamapun" id="S2a.p6.4.m4.1d">caligraphic_C</annotation></semantics></math> to show the existence of an exponential number of zero-modes.</p> </div> <div class="ltx_para" id="S2a.p7"> <p class="ltx_p" id="S2a.p7.7">First, note that the number of zero-modes <math alttext="N_{0}" class="ltx_Math" display="inline" id="S2a.p7.1.m1.1"><semantics id="S2a.p7.1.m1.1a"><msub id="S2a.p7.1.m1.1.1" xref="S2a.p7.1.m1.1.1.cmml"><mi id="S2a.p7.1.m1.1.1.2" xref="S2a.p7.1.m1.1.1.2.cmml">N</mi><mn id="S2a.p7.1.m1.1.1.3" xref="S2a.p7.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2a.p7.1.m1.1b"><apply id="S2a.p7.1.m1.1.1.cmml" xref="S2a.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S2a.p7.1.m1.1.1.1.cmml" xref="S2a.p7.1.m1.1.1">subscript</csymbol><ci id="S2a.p7.1.m1.1.1.2.cmml" xref="S2a.p7.1.m1.1.1.2">𝑁</ci><cn id="S2a.p7.1.m1.1.1.3.cmml" type="integer" xref="S2a.p7.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2a.p7.1.m1.1c">N_{0}</annotation><annotation encoding="application/x-llamapun" id="S2a.p7.1.m1.1d">italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, is bounded from below by <math alttext="N_{0}\geq\left|\mathrm{tr}\left({\cal CI}_{y}\right)\right|" class="ltx_Math" display="inline" id="S2a.p7.2.m2.1"><semantics id="S2a.p7.2.m2.1a"><mrow id="S2a.p7.2.m2.1.1" xref="S2a.p7.2.m2.1.1.cmml"><msub id="S2a.p7.2.m2.1.1.3" xref="S2a.p7.2.m2.1.1.3.cmml"><mi id="S2a.p7.2.m2.1.1.3.2" xref="S2a.p7.2.m2.1.1.3.2.cmml">N</mi><mn id="S2a.p7.2.m2.1.1.3.3" xref="S2a.p7.2.m2.1.1.3.3.cmml">0</mn></msub><mo id="S2a.p7.2.m2.1.1.2" xref="S2a.p7.2.m2.1.1.2.cmml">≥</mo><mrow id="S2a.p7.2.m2.1.1.1.1" xref="S2a.p7.2.m2.1.1.1.2.cmml"><mo id="S2a.p7.2.m2.1.1.1.1.2" xref="S2a.p7.2.m2.1.1.1.2.1.cmml">|</mo><mrow id="S2a.p7.2.m2.1.1.1.1.1" xref="S2a.p7.2.m2.1.1.1.1.1.cmml"><mi id="S2a.p7.2.m2.1.1.1.1.1.3" xref="S2a.p7.2.m2.1.1.1.1.1.3.cmml">tr</mi><mo id="S2a.p7.2.m2.1.1.1.1.1.2" xref="S2a.p7.2.m2.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2a.p7.2.m2.1.1.1.1.1.1.1" xref="S2a.p7.2.m2.1.1.1.1.1.1.1.1.cmml"><mo id="S2a.p7.2.m2.1.1.1.1.1.1.1.2" xref="S2a.p7.2.m2.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2a.p7.2.m2.1.1.1.1.1.1.1.1" xref="S2a.p7.2.m2.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2a.p7.2.m2.1.1.1.1.1.1.1.1.2" xref="S2a.p7.2.m2.1.1.1.1.1.1.1.1.2.cmml">𝒞</mi><mo id="S2a.p7.2.m2.1.1.1.1.1.1.1.1.1" xref="S2a.p7.2.m2.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="S2a.p7.2.m2.1.1.1.1.1.1.1.1.3" xref="S2a.p7.2.m2.1.1.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2a.p7.2.m2.1.1.1.1.1.1.1.1.3.2" xref="S2a.p7.2.m2.1.1.1.1.1.1.1.1.3.2.cmml">ℐ</mi><mi id="S2a.p7.2.m2.1.1.1.1.1.1.1.1.3.3" xref="S2a.p7.2.m2.1.1.1.1.1.1.1.1.3.3.cmml">y</mi></msub></mrow><mo id="S2a.p7.2.m2.1.1.1.1.1.1.1.3" xref="S2a.p7.2.m2.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2a.p7.2.m2.1.1.1.1.3" xref="S2a.p7.2.m2.1.1.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2a.p7.2.m2.1b"><apply id="S2a.p7.2.m2.1.1.cmml" xref="S2a.p7.2.m2.1.1"><geq id="S2a.p7.2.m2.1.1.2.cmml" xref="S2a.p7.2.m2.1.1.2"></geq><apply id="S2a.p7.2.m2.1.1.3.cmml" xref="S2a.p7.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2a.p7.2.m2.1.1.3.1.cmml" xref="S2a.p7.2.m2.1.1.3">subscript</csymbol><ci id="S2a.p7.2.m2.1.1.3.2.cmml" xref="S2a.p7.2.m2.1.1.3.2">𝑁</ci><cn id="S2a.p7.2.m2.1.1.3.3.cmml" type="integer" xref="S2a.p7.2.m2.1.1.3.3">0</cn></apply><apply id="S2a.p7.2.m2.1.1.1.2.cmml" xref="S2a.p7.2.m2.1.1.1.1"><abs id="S2a.p7.2.m2.1.1.1.2.1.cmml" xref="S2a.p7.2.m2.1.1.1.1.2"></abs><apply id="S2a.p7.2.m2.1.1.1.1.1.cmml" xref="S2a.p7.2.m2.1.1.1.1.1"><times id="S2a.p7.2.m2.1.1.1.1.1.2.cmml" xref="S2a.p7.2.m2.1.1.1.1.1.2"></times><ci id="S2a.p7.2.m2.1.1.1.1.1.3.cmml" xref="S2a.p7.2.m2.1.1.1.1.1.3">tr</ci><apply id="S2a.p7.2.m2.1.1.1.1.1.1.1.1.cmml" xref="S2a.p7.2.m2.1.1.1.1.1.1.1"><times id="S2a.p7.2.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S2a.p7.2.m2.1.1.1.1.1.1.1.1.1"></times><ci id="S2a.p7.2.m2.1.1.1.1.1.1.1.1.2.cmml" xref="S2a.p7.2.m2.1.1.1.1.1.1.1.1.2">𝒞</ci><apply id="S2a.p7.2.m2.1.1.1.1.1.1.1.1.3.cmml" xref="S2a.p7.2.m2.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2a.p7.2.m2.1.1.1.1.1.1.1.1.3.1.cmml" xref="S2a.p7.2.m2.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2a.p7.2.m2.1.1.1.1.1.1.1.1.3.2.cmml" xref="S2a.p7.2.m2.1.1.1.1.1.1.1.1.3.2">ℐ</ci><ci id="S2a.p7.2.m2.1.1.1.1.1.1.1.1.3.3.cmml" xref="S2a.p7.2.m2.1.1.1.1.1.1.1.1.3.3">𝑦</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2a.p7.2.m2.1c">N_{0}\geq\left|\mathrm{tr}\left({\cal CI}_{y}\right)\right|</annotation><annotation encoding="application/x-llamapun" id="S2a.p7.2.m2.1d">italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≥ | roman_tr ( caligraphic_C caligraphic_I start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ) |</annotation></semantics></math>. We can see this by adopting a basis that diagonalizes both <math alttext="K" class="ltx_Math" display="inline" id="S2a.p7.3.m3.1"><semantics id="S2a.p7.3.m3.1a"><mi id="S2a.p7.3.m3.1.1" xref="S2a.p7.3.m3.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S2a.p7.3.m3.1b"><ci id="S2a.p7.3.m3.1.1.cmml" xref="S2a.p7.3.m3.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S2a.p7.3.m3.1c">K</annotation><annotation encoding="application/x-llamapun" id="S2a.p7.3.m3.1d">italic_K</annotation></semantics></math> and <math alttext="{\cal I}_{y}" class="ltx_Math" display="inline" id="S2a.p7.4.m4.1"><semantics id="S2a.p7.4.m4.1a"><msub id="S2a.p7.4.m4.1.1" xref="S2a.p7.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2a.p7.4.m4.1.1.2" xref="S2a.p7.4.m4.1.1.2.cmml">ℐ</mi><mi id="S2a.p7.4.m4.1.1.3" xref="S2a.p7.4.m4.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S2a.p7.4.m4.1b"><apply id="S2a.p7.4.m4.1.1.cmml" xref="S2a.p7.4.m4.1.1"><csymbol cd="ambiguous" id="S2a.p7.4.m4.1.1.1.cmml" xref="S2a.p7.4.m4.1.1">subscript</csymbol><ci id="S2a.p7.4.m4.1.1.2.cmml" xref="S2a.p7.4.m4.1.1.2">ℐ</ci><ci id="S2a.p7.4.m4.1.1.3.cmml" xref="S2a.p7.4.m4.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2a.p7.4.m4.1c">{\cal I}_{y}</annotation><annotation encoding="application/x-llamapun" id="S2a.p7.4.m4.1d">caligraphic_I start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math>. Such eigenstates are always mapped to a different energy eigenstate by <math alttext="{\cal C}" class="ltx_Math" display="inline" id="S2a.p7.5.m5.1"><semantics id="S2a.p7.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S2a.p7.5.m5.1.1" xref="S2a.p7.5.m5.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2a.p7.5.m5.1b"><ci id="S2a.p7.5.m5.1.1.cmml" xref="S2a.p7.5.m5.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2a.p7.5.m5.1c">{\cal C}</annotation><annotation encoding="application/x-llamapun" id="S2a.p7.5.m5.1d">caligraphic_C</annotation></semantics></math> unless the energy is zero. As a consequence, only zero-modes contribute to the trace. Each eigenstate, on that sector, will contribute with a value in the interval <math alttext="\left[-1,1\right]" class="ltx_Math" display="inline" id="S2a.p7.6.m6.2"><semantics id="S2a.p7.6.m6.2a"><mrow id="S2a.p7.6.m6.2.2.1" xref="S2a.p7.6.m6.2.2.2.cmml"><mo id="S2a.p7.6.m6.2.2.1.2" xref="S2a.p7.6.m6.2.2.2.cmml">[</mo><mrow id="S2a.p7.6.m6.2.2.1.1" xref="S2a.p7.6.m6.2.2.1.1.cmml"><mo id="S2a.p7.6.m6.2.2.1.1a" xref="S2a.p7.6.m6.2.2.1.1.cmml">−</mo><mn id="S2a.p7.6.m6.2.2.1.1.2" xref="S2a.p7.6.m6.2.2.1.1.2.cmml">1</mn></mrow><mo id="S2a.p7.6.m6.2.2.1.3" xref="S2a.p7.6.m6.2.2.2.cmml">,</mo><mn id="S2a.p7.6.m6.1.1" xref="S2a.p7.6.m6.1.1.cmml">1</mn><mo id="S2a.p7.6.m6.2.2.1.4" xref="S2a.p7.6.m6.2.2.2.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S2a.p7.6.m6.2b"><interval closure="closed" id="S2a.p7.6.m6.2.2.2.cmml" xref="S2a.p7.6.m6.2.2.1"><apply id="S2a.p7.6.m6.2.2.1.1.cmml" xref="S2a.p7.6.m6.2.2.1.1"><minus id="S2a.p7.6.m6.2.2.1.1.1.cmml" xref="S2a.p7.6.m6.2.2.1.1"></minus><cn id="S2a.p7.6.m6.2.2.1.1.2.cmml" type="integer" xref="S2a.p7.6.m6.2.2.1.1.2">1</cn></apply><cn id="S2a.p7.6.m6.1.1.cmml" type="integer" xref="S2a.p7.6.m6.1.1">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="S2a.p7.6.m6.2c">\left[-1,1\right]</annotation><annotation encoding="application/x-llamapun" id="S2a.p7.6.m6.2d">[ - 1 , 1 ]</annotation></semantics></math>. This shows that the number of zero-modes can never be smaller than <math alttext="\left|\mathrm{tr}\left({\cal CI}_{y}\right)\right|" class="ltx_Math" display="inline" id="S2a.p7.7.m7.1"><semantics id="S2a.p7.7.m7.1a"><mrow id="S2a.p7.7.m7.1.1.1" xref="S2a.p7.7.m7.1.1.2.cmml"><mo id="S2a.p7.7.m7.1.1.1.2" xref="S2a.p7.7.m7.1.1.2.1.cmml">|</mo><mrow id="S2a.p7.7.m7.1.1.1.1" xref="S2a.p7.7.m7.1.1.1.1.cmml"><mi id="S2a.p7.7.m7.1.1.1.1.3" xref="S2a.p7.7.m7.1.1.1.1.3.cmml">tr</mi><mo id="S2a.p7.7.m7.1.1.1.1.2" xref="S2a.p7.7.m7.1.1.1.1.2.cmml">⁢</mo><mrow id="S2a.p7.7.m7.1.1.1.1.1.1" xref="S2a.p7.7.m7.1.1.1.1.1.1.1.cmml"><mo id="S2a.p7.7.m7.1.1.1.1.1.1.2" xref="S2a.p7.7.m7.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2a.p7.7.m7.1.1.1.1.1.1.1" xref="S2a.p7.7.m7.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2a.p7.7.m7.1.1.1.1.1.1.1.2" xref="S2a.p7.7.m7.1.1.1.1.1.1.1.2.cmml">𝒞</mi><mo id="S2a.p7.7.m7.1.1.1.1.1.1.1.1" xref="S2a.p7.7.m7.1.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="S2a.p7.7.m7.1.1.1.1.1.1.1.3" xref="S2a.p7.7.m7.1.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2a.p7.7.m7.1.1.1.1.1.1.1.3.2" xref="S2a.p7.7.m7.1.1.1.1.1.1.1.3.2.cmml">ℐ</mi><mi id="S2a.p7.7.m7.1.1.1.1.1.1.1.3.3" xref="S2a.p7.7.m7.1.1.1.1.1.1.1.3.3.cmml">y</mi></msub></mrow><mo id="S2a.p7.7.m7.1.1.1.1.1.1.3" xref="S2a.p7.7.m7.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2a.p7.7.m7.1.1.1.3" xref="S2a.p7.7.m7.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2a.p7.7.m7.1b"><apply id="S2a.p7.7.m7.1.1.2.cmml" xref="S2a.p7.7.m7.1.1.1"><abs id="S2a.p7.7.m7.1.1.2.1.cmml" xref="S2a.p7.7.m7.1.1.1.2"></abs><apply id="S2a.p7.7.m7.1.1.1.1.cmml" xref="S2a.p7.7.m7.1.1.1.1"><times id="S2a.p7.7.m7.1.1.1.1.2.cmml" xref="S2a.p7.7.m7.1.1.1.1.2"></times><ci id="S2a.p7.7.m7.1.1.1.1.3.cmml" xref="S2a.p7.7.m7.1.1.1.1.3">tr</ci><apply id="S2a.p7.7.m7.1.1.1.1.1.1.1.cmml" xref="S2a.p7.7.m7.1.1.1.1.1.1"><times id="S2a.p7.7.m7.1.1.1.1.1.1.1.1.cmml" xref="S2a.p7.7.m7.1.1.1.1.1.1.1.1"></times><ci id="S2a.p7.7.m7.1.1.1.1.1.1.1.2.cmml" xref="S2a.p7.7.m7.1.1.1.1.1.1.1.2">𝒞</ci><apply id="S2a.p7.7.m7.1.1.1.1.1.1.1.3.cmml" xref="S2a.p7.7.m7.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2a.p7.7.m7.1.1.1.1.1.1.1.3.1.cmml" xref="S2a.p7.7.m7.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2a.p7.7.m7.1.1.1.1.1.1.1.3.2.cmml" xref="S2a.p7.7.m7.1.1.1.1.1.1.1.3.2">ℐ</ci><ci id="S2a.p7.7.m7.1.1.1.1.1.1.1.3.3.cmml" xref="S2a.p7.7.m7.1.1.1.1.1.1.1.3.3">𝑦</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2a.p7.7.m7.1c">\left|\mathrm{tr}\left({\cal CI}_{y}\right)\right|</annotation><annotation encoding="application/x-llamapun" id="S2a.p7.7.m7.1d">| roman_tr ( caligraphic_C caligraphic_I start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ) |</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2a.p8"> <p class="ltx_p" id="S2a.p8.1">We can now show that the trace is actually exponential on the volume by considering the electric field basis. These are all eigenstates of <math alttext="{\cal C}" class="ltx_Math" display="inline" id="S2a.p8.1.m1.1"><semantics id="S2a.p8.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2a.p8.1.m1.1.1" xref="S2a.p8.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S2a.p8.1.m1.1b"><ci id="S2a.p8.1.m1.1.1.cmml" xref="S2a.p8.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S2a.p8.1.m1.1c">{\cal C}</annotation><annotation encoding="application/x-llamapun" id="S2a.p8.1.m1.1d">caligraphic_C</annotation></semantics></math>, but the only ones surviving the trace are the ones that respect a reflection symmetry along the vertical axis. Such states are fixed by determining the state in half of the volume. The number of these states corresponds to the square root of the total number of states in the whole Hilbert space. By restricting ourselves to specific sectors of the Hilbert space, the two halves are not independent. Nonetheless, we still expect the number of zero-modes to remain exponentially large on the volume.</p> </div> </section> <section class="ltx_section" id="S3a"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span>Low Entropy Zero-Modes in Truncated Link Models</h2> <div class="ltx_para" id="S3a.p1"> <p class="ltx_p" id="S3a.p1.3">Here we describe a generic mechanism to obtain low-entropy mid-spectrum states in the TLM, from the vast space of zero-modes. We start by searching for zero-modes <math alttext="\ket{\psi_{z}}" class="ltx_Math" display="inline" id="S3a.p1.1.m1.1"><semantics id="S3a.p1.1.m1.1a"><mrow id="S3a.p1.1.m1.1.1.3" xref="S3a.p1.1.m1.1.1.2.cmml"><mo id="S3a.p1.1.m1.1.1.3.1" stretchy="false" xref="S3a.p1.1.m1.1.1.2.1.cmml">|</mo><msub id="S3a.p1.1.m1.1.1.1.1" xref="S3a.p1.1.m1.1.1.1.1.cmml"><mi id="S3a.p1.1.m1.1.1.1.1.2" xref="S3a.p1.1.m1.1.1.1.1.2.cmml">ψ</mi><mi id="S3a.p1.1.m1.1.1.1.1.3" xref="S3a.p1.1.m1.1.1.1.1.3.cmml">z</mi></msub><mo id="S3a.p1.1.m1.1.1.3.2" stretchy="false" xref="S3a.p1.1.m1.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S3a.p1.1.m1.1b"><apply id="S3a.p1.1.m1.1.1.2.cmml" xref="S3a.p1.1.m1.1.1.3"><csymbol cd="latexml" id="S3a.p1.1.m1.1.1.2.1.cmml" xref="S3a.p1.1.m1.1.1.3.1">ket</csymbol><apply id="S3a.p1.1.m1.1.1.1.1.cmml" xref="S3a.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p1.1.m1.1.1.1.1.1.cmml" xref="S3a.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S3a.p1.1.m1.1.1.1.1.2.cmml" xref="S3a.p1.1.m1.1.1.1.1.2">𝜓</ci><ci id="S3a.p1.1.m1.1.1.1.1.3.cmml" xref="S3a.p1.1.m1.1.1.1.1.3">𝑧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p1.1.m1.1c">\ket{\psi_{z}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p1.1.m1.1d">| start_ARG italic_ψ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math>, with a generic decomposition following <math alttext="\ket{\psi_{z}}=\sum_{n}c_{i}\ket{\phi_{i}}" class="ltx_Math" display="inline" id="S3a.p1.2.m2.2"><semantics id="S3a.p1.2.m2.2a"><mrow id="S3a.p1.2.m2.2.3" xref="S3a.p1.2.m2.2.3.cmml"><mrow id="S3a.p1.2.m2.1.1.3" xref="S3a.p1.2.m2.1.1.2.cmml"><mo id="S3a.p1.2.m2.1.1.3.1" stretchy="false" xref="S3a.p1.2.m2.1.1.2.1.cmml">|</mo><msub id="S3a.p1.2.m2.1.1.1.1" xref="S3a.p1.2.m2.1.1.1.1.cmml"><mi id="S3a.p1.2.m2.1.1.1.1.2" xref="S3a.p1.2.m2.1.1.1.1.2.cmml">ψ</mi><mi id="S3a.p1.2.m2.1.1.1.1.3" xref="S3a.p1.2.m2.1.1.1.1.3.cmml">z</mi></msub><mo id="S3a.p1.2.m2.1.1.3.2" stretchy="false" xref="S3a.p1.2.m2.1.1.2.1.cmml">⟩</mo></mrow><mo id="S3a.p1.2.m2.2.3.1" rspace="0.111em" xref="S3a.p1.2.m2.2.3.1.cmml">=</mo><mrow id="S3a.p1.2.m2.2.3.2" xref="S3a.p1.2.m2.2.3.2.cmml"><msub id="S3a.p1.2.m2.2.3.2.1" xref="S3a.p1.2.m2.2.3.2.1.cmml"><mo id="S3a.p1.2.m2.2.3.2.1.2" xref="S3a.p1.2.m2.2.3.2.1.2.cmml">∑</mo><mi id="S3a.p1.2.m2.2.3.2.1.3" xref="S3a.p1.2.m2.2.3.2.1.3.cmml">n</mi></msub><mrow id="S3a.p1.2.m2.2.3.2.2" xref="S3a.p1.2.m2.2.3.2.2.cmml"><msub id="S3a.p1.2.m2.2.3.2.2.2" xref="S3a.p1.2.m2.2.3.2.2.2.cmml"><mi id="S3a.p1.2.m2.2.3.2.2.2.2" xref="S3a.p1.2.m2.2.3.2.2.2.2.cmml">c</mi><mi id="S3a.p1.2.m2.2.3.2.2.2.3" xref="S3a.p1.2.m2.2.3.2.2.2.3.cmml">i</mi></msub><mo id="S3a.p1.2.m2.2.3.2.2.1" xref="S3a.p1.2.m2.2.3.2.2.1.cmml">⁢</mo><mrow id="S3a.p1.2.m2.2.2.3" xref="S3a.p1.2.m2.2.2.2.cmml"><mo id="S3a.p1.2.m2.2.2.3.1" stretchy="false" xref="S3a.p1.2.m2.2.2.2.1.cmml">|</mo><msub 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id="S3a.p1.2.m2.2.2.1.1.cmml" xref="S3a.p1.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S3a.p1.2.m2.2.2.1.1.1.cmml" xref="S3a.p1.2.m2.2.2.1.1">subscript</csymbol><ci id="S3a.p1.2.m2.2.2.1.1.2.cmml" xref="S3a.p1.2.m2.2.2.1.1.2">italic-ϕ</ci><ci id="S3a.p1.2.m2.2.2.1.1.3.cmml" xref="S3a.p1.2.m2.2.2.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p1.2.m2.2c">\ket{\psi_{z}}=\sum_{n}c_{i}\ket{\phi_{i}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p1.2.m2.2d">| start_ARG italic_ψ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_ARG ⟩ = ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math>, where <math alttext="\left\{\ket{\phi_{i}}\right\}" class="ltx_Math" display="inline" id="S3a.p1.3.m3.1"><semantics id="S3a.p1.3.m3.1a"><mrow 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id="S3a.p1.3.m3.1.1.1.1.1.cmml" xref="S3a.p1.3.m3.1.1.1.1">subscript</csymbol><ci id="S3a.p1.3.m3.1.1.1.1.2.cmml" xref="S3a.p1.3.m3.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p1.3.m3.1.1.1.1.3.cmml" xref="S3a.p1.3.m3.1.1.1.1.3">𝑖</ci></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S3a.p1.3.m3.1c">\left\{\ket{\phi_{i}}\right\}</annotation><annotation encoding="application/x-llamapun" id="S3a.p1.3.m3.1d">{ | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ }</annotation></semantics></math> is the electric field basis. By definition, a zero-mode follows</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx6"> <tbody id="S3.E22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle K\ket{\psi_{z}}=\sum_{i,n,\sigma}c_{i}H^{\sigma}_{n}\ket{\phi_{i% }}=0," class="ltx_Math" display="inline" id="S3.E22.m1.6"><semantics id="S3.E22.m1.6a"><mrow id="S3.E22.m1.6.6.1" xref="S3.E22.m1.6.6.1.1.cmml"><mrow id="S3.E22.m1.6.6.1.1" xref="S3.E22.m1.6.6.1.1.cmml"><mrow id="S3.E22.m1.6.6.1.1.2" xref="S3.E22.m1.6.6.1.1.2.cmml"><mi id="S3.E22.m1.6.6.1.1.2.2" xref="S3.E22.m1.6.6.1.1.2.2.cmml">K</mi><mo id="S3.E22.m1.6.6.1.1.2.1" xref="S3.E22.m1.6.6.1.1.2.1.cmml">⁢</mo><mrow id="S3.E22.m1.1.1a.3" xref="S3.E22.m1.1.1a.2.cmml"><mo id="S3.E22.m1.1.1a.3.1" stretchy="false" xref="S3.E22.m1.1.1a.2.1.cmml">|</mo><msub id="S3.E22.m1.1.1.1.1" xref="S3.E22.m1.1.1.1.1.cmml"><mi 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id="S3.E22.m1.6.6.1.1.4.2.3.2.2.cmml" xref="S3.E22.m1.6.6.1.1.4.2.3.2.2">𝐻</ci><ci id="S3.E22.m1.6.6.1.1.4.2.3.2.3.cmml" xref="S3.E22.m1.6.6.1.1.4.2.3.2.3">𝜎</ci></apply><ci id="S3.E22.m1.6.6.1.1.4.2.3.3.cmml" xref="S3.E22.m1.6.6.1.1.4.2.3.3">𝑛</ci></apply><apply id="S3.E22.m1.2.2a.2.cmml" xref="S3.E22.m1.2.2a.3"><csymbol cd="latexml" id="S3.E22.m1.2.2a.2.1.cmml" xref="S3.E22.m1.2.2a.3.1">ket</csymbol><apply id="S3.E22.m1.2.2.1.1.cmml" xref="S3.E22.m1.2.2.1.1"><csymbol cd="ambiguous" id="S3.E22.m1.2.2.1.1.1.cmml" xref="S3.E22.m1.2.2.1.1">subscript</csymbol><ci id="S3.E22.m1.2.2.1.1.2.cmml" xref="S3.E22.m1.2.2.1.1.2">italic-ϕ</ci><ci id="S3.E22.m1.2.2.1.1.3.cmml" xref="S3.E22.m1.2.2.1.1.3">𝑖</ci></apply></apply></apply></apply></apply><apply id="S3.E22.m1.6.6.1.1c.cmml" xref="S3.E22.m1.6.6.1"><eq id="S3.E22.m1.6.6.1.1.5.cmml" xref="S3.E22.m1.6.6.1.1.5"></eq><share href="https://arxiv.org/html/2403.08892v3#S3.E22.m1.6.6.1.1.4.cmml" id="S3.E22.m1.6.6.1.1d.cmml" xref="S3.E22.m1.6.6.1"></share><cn id="S3.E22.m1.6.6.1.1.6.cmml" type="integer" xref="S3.E22.m1.6.6.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E22.m1.6c">\displaystyle K\ket{\psi_{z}}=\sum_{i,n,\sigma}c_{i}H^{\sigma}_{n}\ket{\phi_{i% }}=0,</annotation><annotation encoding="application/x-llamapun" id="S3.E22.m1.6d">italic_K | start_ARG italic_ψ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_ARG ⟩ = ∑ start_POSTSUBSCRIPT italic_i , italic_n , italic_σ end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(22)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3a.p1.4">where <math alttext="\sigma\in\{+,-\}" class="ltx_Math" display="inline" id="S3a.p1.4.m1.2"><semantics id="S3a.p1.4.m1.2a"><mrow id="S3a.p1.4.m1.2.3" xref="S3a.p1.4.m1.2.3.cmml"><mi id="S3a.p1.4.m1.2.3.2" xref="S3a.p1.4.m1.2.3.2.cmml">σ</mi><mo id="S3a.p1.4.m1.2.3.1" xref="S3a.p1.4.m1.2.3.1.cmml">∈</mo><mrow id="S3a.p1.4.m1.2.3.3.2" xref="S3a.p1.4.m1.2.3.3.1.cmml"><mo id="S3a.p1.4.m1.2.3.3.2.1" stretchy="false" xref="S3a.p1.4.m1.2.3.3.1.cmml">{</mo><mo id="S3a.p1.4.m1.1.1" lspace="0em" rspace="0em" xref="S3a.p1.4.m1.1.1.cmml">+</mo><mo id="S3a.p1.4.m1.2.3.3.2.2" rspace="0em" xref="S3a.p1.4.m1.2.3.3.1.cmml">,</mo><mo id="S3a.p1.4.m1.2.2" lspace="0em" rspace="0em" xref="S3a.p1.4.m1.2.2.cmml">−</mo><mo id="S3a.p1.4.m1.2.3.3.2.3" stretchy="false" xref="S3a.p1.4.m1.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3a.p1.4.m1.2b"><apply id="S3a.p1.4.m1.2.3.cmml" xref="S3a.p1.4.m1.2.3"><in id="S3a.p1.4.m1.2.3.1.cmml" xref="S3a.p1.4.m1.2.3.1"></in><ci id="S3a.p1.4.m1.2.3.2.cmml" xref="S3a.p1.4.m1.2.3.2">𝜎</ci><set id="S3a.p1.4.m1.2.3.3.1.cmml" xref="S3a.p1.4.m1.2.3.3.2"><plus id="S3a.p1.4.m1.1.1.cmml" xref="S3a.p1.4.m1.1.1"></plus><minus id="S3a.p1.4.m1.2.2.cmml" xref="S3a.p1.4.m1.2.2"></minus></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p1.4.m1.2c">\sigma\in\{+,-\}</annotation><annotation encoding="application/x-llamapun" id="S3a.p1.4.m1.2d">italic_σ ∈ { + , - }</annotation></semantics></math>. This is very general and must be satisfied by <em class="ltx_emph ltx_font_italic" id="S3a.p1.4.1">all</em> zero-modes.</p> </div> <div class="ltx_para" id="S3a.p2"> <p class="ltx_p" id="S3a.p2.16">Spin-<math alttext="S" class="ltx_Math" display="inline" id="S3a.p2.1.m1.1"><semantics id="S3a.p2.1.m1.1a"><mi id="S3a.p2.1.m1.1.1" xref="S3a.p2.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S3a.p2.1.m1.1b"><ci id="S3a.p2.1.m1.1.1.cmml" xref="S3a.p2.1.m1.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.1.m1.1d">italic_S</annotation></semantics></math> TLMs have an especially simple structure, which we can use to find low-entropy zero-modes. If <math alttext="H^{\sigma}_{n}\ket{\phi_{i}}\neq 0" class="ltx_Math" display="inline" id="S3a.p2.2.m2.1"><semantics id="S3a.p2.2.m2.1a"><mrow id="S3a.p2.2.m2.1.2" xref="S3a.p2.2.m2.1.2.cmml"><mrow id="S3a.p2.2.m2.1.2.2" xref="S3a.p2.2.m2.1.2.2.cmml"><msubsup id="S3a.p2.2.m2.1.2.2.2" xref="S3a.p2.2.m2.1.2.2.2.cmml"><mi id="S3a.p2.2.m2.1.2.2.2.2.2" xref="S3a.p2.2.m2.1.2.2.2.2.2.cmml">H</mi><mi id="S3a.p2.2.m2.1.2.2.2.3" xref="S3a.p2.2.m2.1.2.2.2.3.cmml">n</mi><mi id="S3a.p2.2.m2.1.2.2.2.2.3" xref="S3a.p2.2.m2.1.2.2.2.2.3.cmml">σ</mi></msubsup><mo id="S3a.p2.2.m2.1.2.2.1" xref="S3a.p2.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S3a.p2.2.m2.1.1.3" xref="S3a.p2.2.m2.1.1.2.cmml"><mo id="S3a.p2.2.m2.1.1.3.1" stretchy="false" xref="S3a.p2.2.m2.1.1.2.1.cmml">|</mo><msub id="S3a.p2.2.m2.1.1.1.1" xref="S3a.p2.2.m2.1.1.1.1.cmml"><mi id="S3a.p2.2.m2.1.1.1.1.2" xref="S3a.p2.2.m2.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p2.2.m2.1.1.1.1.3" xref="S3a.p2.2.m2.1.1.1.1.3.cmml">i</mi></msub><mo id="S3a.p2.2.m2.1.1.3.2" stretchy="false" xref="S3a.p2.2.m2.1.1.2.1.cmml">⟩</mo></mrow></mrow><mo id="S3a.p2.2.m2.1.2.1" xref="S3a.p2.2.m2.1.2.1.cmml">≠</mo><mn id="S3a.p2.2.m2.1.2.3" xref="S3a.p2.2.m2.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3a.p2.2.m2.1b"><apply id="S3a.p2.2.m2.1.2.cmml" xref="S3a.p2.2.m2.1.2"><neq id="S3a.p2.2.m2.1.2.1.cmml" xref="S3a.p2.2.m2.1.2.1"></neq><apply id="S3a.p2.2.m2.1.2.2.cmml" xref="S3a.p2.2.m2.1.2.2"><times id="S3a.p2.2.m2.1.2.2.1.cmml" xref="S3a.p2.2.m2.1.2.2.1"></times><apply id="S3a.p2.2.m2.1.2.2.2.cmml" xref="S3a.p2.2.m2.1.2.2.2"><csymbol cd="ambiguous" id="S3a.p2.2.m2.1.2.2.2.1.cmml" xref="S3a.p2.2.m2.1.2.2.2">subscript</csymbol><apply id="S3a.p2.2.m2.1.2.2.2.2.cmml" xref="S3a.p2.2.m2.1.2.2.2"><csymbol cd="ambiguous" id="S3a.p2.2.m2.1.2.2.2.2.1.cmml" xref="S3a.p2.2.m2.1.2.2.2">superscript</csymbol><ci id="S3a.p2.2.m2.1.2.2.2.2.2.cmml" xref="S3a.p2.2.m2.1.2.2.2.2.2">𝐻</ci><ci id="S3a.p2.2.m2.1.2.2.2.2.3.cmml" xref="S3a.p2.2.m2.1.2.2.2.2.3">𝜎</ci></apply><ci id="S3a.p2.2.m2.1.2.2.2.3.cmml" xref="S3a.p2.2.m2.1.2.2.2.3">𝑛</ci></apply><apply id="S3a.p2.2.m2.1.1.2.cmml" xref="S3a.p2.2.m2.1.1.3"><csymbol cd="latexml" id="S3a.p2.2.m2.1.1.2.1.cmml" xref="S3a.p2.2.m2.1.1.3.1">ket</csymbol><apply id="S3a.p2.2.m2.1.1.1.1.cmml" xref="S3a.p2.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p2.2.m2.1.1.1.1.1.cmml" xref="S3a.p2.2.m2.1.1.1.1">subscript</csymbol><ci id="S3a.p2.2.m2.1.1.1.1.2.cmml" xref="S3a.p2.2.m2.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p2.2.m2.1.1.1.1.3.cmml" xref="S3a.p2.2.m2.1.1.1.1.3">𝑖</ci></apply></apply></apply><cn id="S3a.p2.2.m2.1.2.3.cmml" type="integer" xref="S3a.p2.2.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.2.m2.1c">H^{\sigma}_{n}\ket{\phi_{i}}\neq 0</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.2.m2.1d">italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ ≠ 0</annotation></semantics></math> then <math alttext="||H^{\sigma}_{n}\ket{\phi_{i}}||=1" class="ltx_Math" display="inline" id="S3a.p2.3.m3.2"><semantics id="S3a.p2.3.m3.2a"><mrow id="S3a.p2.3.m3.2.2" xref="S3a.p2.3.m3.2.2.cmml"><mrow id="S3a.p2.3.m3.2.2.1.1" xref="S3a.p2.3.m3.2.2.1.2.cmml"><mo id="S3a.p2.3.m3.2.2.1.1.2" stretchy="false" xref="S3a.p2.3.m3.2.2.1.2.1.cmml">‖</mo><mrow id="S3a.p2.3.m3.2.2.1.1.1" xref="S3a.p2.3.m3.2.2.1.1.1.cmml"><msubsup id="S3a.p2.3.m3.2.2.1.1.1.2" xref="S3a.p2.3.m3.2.2.1.1.1.2.cmml"><mi id="S3a.p2.3.m3.2.2.1.1.1.2.2.2" xref="S3a.p2.3.m3.2.2.1.1.1.2.2.2.cmml">H</mi><mi id="S3a.p2.3.m3.2.2.1.1.1.2.3" xref="S3a.p2.3.m3.2.2.1.1.1.2.3.cmml">n</mi><mi id="S3a.p2.3.m3.2.2.1.1.1.2.2.3" xref="S3a.p2.3.m3.2.2.1.1.1.2.2.3.cmml">σ</mi></msubsup><mo id="S3a.p2.3.m3.2.2.1.1.1.1" xref="S3a.p2.3.m3.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S3a.p2.3.m3.1.1.3" xref="S3a.p2.3.m3.1.1.2.cmml"><mo id="S3a.p2.3.m3.1.1.3.1" stretchy="false" xref="S3a.p2.3.m3.1.1.2.1.cmml">|</mo><msub id="S3a.p2.3.m3.1.1.1.1" xref="S3a.p2.3.m3.1.1.1.1.cmml"><mi id="S3a.p2.3.m3.1.1.1.1.2" xref="S3a.p2.3.m3.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p2.3.m3.1.1.1.1.3" xref="S3a.p2.3.m3.1.1.1.1.3.cmml">i</mi></msub><mo id="S3a.p2.3.m3.1.1.3.2" stretchy="false" xref="S3a.p2.3.m3.1.1.2.1.cmml">⟩</mo></mrow></mrow><mo id="S3a.p2.3.m3.2.2.1.1.3" stretchy="false" xref="S3a.p2.3.m3.2.2.1.2.1.cmml">‖</mo></mrow><mo id="S3a.p2.3.m3.2.2.2" xref="S3a.p2.3.m3.2.2.2.cmml">=</mo><mn id="S3a.p2.3.m3.2.2.3" xref="S3a.p2.3.m3.2.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3a.p2.3.m3.2b"><apply id="S3a.p2.3.m3.2.2.cmml" xref="S3a.p2.3.m3.2.2"><eq id="S3a.p2.3.m3.2.2.2.cmml" xref="S3a.p2.3.m3.2.2.2"></eq><apply id="S3a.p2.3.m3.2.2.1.2.cmml" xref="S3a.p2.3.m3.2.2.1.1"><csymbol cd="latexml" id="S3a.p2.3.m3.2.2.1.2.1.cmml" xref="S3a.p2.3.m3.2.2.1.1.2">norm</csymbol><apply id="S3a.p2.3.m3.2.2.1.1.1.cmml" xref="S3a.p2.3.m3.2.2.1.1.1"><times id="S3a.p2.3.m3.2.2.1.1.1.1.cmml" xref="S3a.p2.3.m3.2.2.1.1.1.1"></times><apply id="S3a.p2.3.m3.2.2.1.1.1.2.cmml" xref="S3a.p2.3.m3.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S3a.p2.3.m3.2.2.1.1.1.2.1.cmml" xref="S3a.p2.3.m3.2.2.1.1.1.2">subscript</csymbol><apply id="S3a.p2.3.m3.2.2.1.1.1.2.2.cmml" xref="S3a.p2.3.m3.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S3a.p2.3.m3.2.2.1.1.1.2.2.1.cmml" xref="S3a.p2.3.m3.2.2.1.1.1.2">superscript</csymbol><ci id="S3a.p2.3.m3.2.2.1.1.1.2.2.2.cmml" xref="S3a.p2.3.m3.2.2.1.1.1.2.2.2">𝐻</ci><ci id="S3a.p2.3.m3.2.2.1.1.1.2.2.3.cmml" xref="S3a.p2.3.m3.2.2.1.1.1.2.2.3">𝜎</ci></apply><ci id="S3a.p2.3.m3.2.2.1.1.1.2.3.cmml" xref="S3a.p2.3.m3.2.2.1.1.1.2.3">𝑛</ci></apply><apply id="S3a.p2.3.m3.1.1.2.cmml" xref="S3a.p2.3.m3.1.1.3"><csymbol cd="latexml" id="S3a.p2.3.m3.1.1.2.1.cmml" xref="S3a.p2.3.m3.1.1.3.1">ket</csymbol><apply id="S3a.p2.3.m3.1.1.1.1.cmml" xref="S3a.p2.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p2.3.m3.1.1.1.1.1.cmml" xref="S3a.p2.3.m3.1.1.1.1">subscript</csymbol><ci id="S3a.p2.3.m3.1.1.1.1.2.cmml" xref="S3a.p2.3.m3.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p2.3.m3.1.1.1.1.3.cmml" xref="S3a.p2.3.m3.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply><cn id="S3a.p2.3.m3.2.2.3.cmml" type="integer" xref="S3a.p2.3.m3.2.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.3.m3.2c">||H^{\sigma}_{n}\ket{\phi_{i}}||=1</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.3.m3.2d">| | italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ | | = 1</annotation></semantics></math> for all <math alttext="n" class="ltx_Math" display="inline" id="S3a.p2.4.m4.1"><semantics id="S3a.p2.4.m4.1a"><mi id="S3a.p2.4.m4.1.1" xref="S3a.p2.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3a.p2.4.m4.1b"><ci id="S3a.p2.4.m4.1.1.cmml" xref="S3a.p2.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.4.m4.1d">italic_n</annotation></semantics></math>, <math alttext="i" class="ltx_Math" display="inline" id="S3a.p2.5.m5.1"><semantics id="S3a.p2.5.m5.1a"><mi id="S3a.p2.5.m5.1.1" xref="S3a.p2.5.m5.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3a.p2.5.m5.1b"><ci id="S3a.p2.5.m5.1.1.cmml" xref="S3a.p2.5.m5.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.5.m5.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.5.m5.1d">italic_i</annotation></semantics></math> and <math alttext="\sigma" class="ltx_Math" display="inline" id="S3a.p2.6.m6.1"><semantics id="S3a.p2.6.m6.1a"><mi id="S3a.p2.6.m6.1.1" xref="S3a.p2.6.m6.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3a.p2.6.m6.1b"><ci id="S3a.p2.6.m6.1.1.cmml" xref="S3a.p2.6.m6.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.6.m6.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.6.m6.1d">italic_σ</annotation></semantics></math>. This suggests that we can look for solutions where each term in (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.E22" title="In III Low Entropy Zero-Modes in Truncated Link Models ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">22</span></a>) is canceled by exactly one other term in the sum. In other words, for each non-null term <math alttext="c_{i}H^{\sigma}_{n}\ket{\phi_{i}}" class="ltx_Math" display="inline" id="S3a.p2.7.m7.1"><semantics id="S3a.p2.7.m7.1a"><mrow id="S3a.p2.7.m7.1.2" xref="S3a.p2.7.m7.1.2.cmml"><msub id="S3a.p2.7.m7.1.2.2" xref="S3a.p2.7.m7.1.2.2.cmml"><mi id="S3a.p2.7.m7.1.2.2.2" xref="S3a.p2.7.m7.1.2.2.2.cmml">c</mi><mi id="S3a.p2.7.m7.1.2.2.3" xref="S3a.p2.7.m7.1.2.2.3.cmml">i</mi></msub><mo id="S3a.p2.7.m7.1.2.1" xref="S3a.p2.7.m7.1.2.1.cmml">⁢</mo><msubsup id="S3a.p2.7.m7.1.2.3" xref="S3a.p2.7.m7.1.2.3.cmml"><mi id="S3a.p2.7.m7.1.2.3.2.2" xref="S3a.p2.7.m7.1.2.3.2.2.cmml">H</mi><mi id="S3a.p2.7.m7.1.2.3.3" xref="S3a.p2.7.m7.1.2.3.3.cmml">n</mi><mi id="S3a.p2.7.m7.1.2.3.2.3" xref="S3a.p2.7.m7.1.2.3.2.3.cmml">σ</mi></msubsup><mo id="S3a.p2.7.m7.1.2.1a" xref="S3a.p2.7.m7.1.2.1.cmml">⁢</mo><mrow id="S3a.p2.7.m7.1.1.3" xref="S3a.p2.7.m7.1.1.2.cmml"><mo id="S3a.p2.7.m7.1.1.3.1" stretchy="false" xref="S3a.p2.7.m7.1.1.2.1.cmml">|</mo><msub id="S3a.p2.7.m7.1.1.1.1" xref="S3a.p2.7.m7.1.1.1.1.cmml"><mi id="S3a.p2.7.m7.1.1.1.1.2" xref="S3a.p2.7.m7.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p2.7.m7.1.1.1.1.3" xref="S3a.p2.7.m7.1.1.1.1.3.cmml">i</mi></msub><mo id="S3a.p2.7.m7.1.1.3.2" stretchy="false" xref="S3a.p2.7.m7.1.1.2.1.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3a.p2.7.m7.1b"><apply id="S3a.p2.7.m7.1.2.cmml" xref="S3a.p2.7.m7.1.2"><times id="S3a.p2.7.m7.1.2.1.cmml" xref="S3a.p2.7.m7.1.2.1"></times><apply id="S3a.p2.7.m7.1.2.2.cmml" xref="S3a.p2.7.m7.1.2.2"><csymbol cd="ambiguous" id="S3a.p2.7.m7.1.2.2.1.cmml" xref="S3a.p2.7.m7.1.2.2">subscript</csymbol><ci id="S3a.p2.7.m7.1.2.2.2.cmml" xref="S3a.p2.7.m7.1.2.2.2">𝑐</ci><ci id="S3a.p2.7.m7.1.2.2.3.cmml" xref="S3a.p2.7.m7.1.2.2.3">𝑖</ci></apply><apply id="S3a.p2.7.m7.1.2.3.cmml" xref="S3a.p2.7.m7.1.2.3"><csymbol cd="ambiguous" id="S3a.p2.7.m7.1.2.3.1.cmml" xref="S3a.p2.7.m7.1.2.3">subscript</csymbol><apply id="S3a.p2.7.m7.1.2.3.2.cmml" xref="S3a.p2.7.m7.1.2.3"><csymbol cd="ambiguous" id="S3a.p2.7.m7.1.2.3.2.1.cmml" xref="S3a.p2.7.m7.1.2.3">superscript</csymbol><ci id="S3a.p2.7.m7.1.2.3.2.2.cmml" xref="S3a.p2.7.m7.1.2.3.2.2">𝐻</ci><ci id="S3a.p2.7.m7.1.2.3.2.3.cmml" xref="S3a.p2.7.m7.1.2.3.2.3">𝜎</ci></apply><ci id="S3a.p2.7.m7.1.2.3.3.cmml" xref="S3a.p2.7.m7.1.2.3.3">𝑛</ci></apply><apply id="S3a.p2.7.m7.1.1.2.cmml" xref="S3a.p2.7.m7.1.1.3"><csymbol cd="latexml" id="S3a.p2.7.m7.1.1.2.1.cmml" xref="S3a.p2.7.m7.1.1.3.1">ket</csymbol><apply id="S3a.p2.7.m7.1.1.1.1.cmml" xref="S3a.p2.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p2.7.m7.1.1.1.1.1.cmml" xref="S3a.p2.7.m7.1.1.1.1">subscript</csymbol><ci id="S3a.p2.7.m7.1.1.1.1.2.cmml" xref="S3a.p2.7.m7.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p2.7.m7.1.1.1.1.3.cmml" xref="S3a.p2.7.m7.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.7.m7.1c">c_{i}H^{\sigma}_{n}\ket{\phi_{i}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.7.m7.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math>, there exists a term <math alttext="c_{k}H^{\sigma^{\prime}}_{n^{\prime}}\ket{\phi_{k}}" class="ltx_Math" display="inline" id="S3a.p2.8.m8.1"><semantics id="S3a.p2.8.m8.1a"><mrow id="S3a.p2.8.m8.1.2" xref="S3a.p2.8.m8.1.2.cmml"><msub id="S3a.p2.8.m8.1.2.2" xref="S3a.p2.8.m8.1.2.2.cmml"><mi id="S3a.p2.8.m8.1.2.2.2" xref="S3a.p2.8.m8.1.2.2.2.cmml">c</mi><mi id="S3a.p2.8.m8.1.2.2.3" xref="S3a.p2.8.m8.1.2.2.3.cmml">k</mi></msub><mo id="S3a.p2.8.m8.1.2.1" xref="S3a.p2.8.m8.1.2.1.cmml">⁢</mo><msubsup id="S3a.p2.8.m8.1.2.3" xref="S3a.p2.8.m8.1.2.3.cmml"><mi id="S3a.p2.8.m8.1.2.3.2.2" xref="S3a.p2.8.m8.1.2.3.2.2.cmml">H</mi><msup id="S3a.p2.8.m8.1.2.3.3" xref="S3a.p2.8.m8.1.2.3.3.cmml"><mi id="S3a.p2.8.m8.1.2.3.3.2" xref="S3a.p2.8.m8.1.2.3.3.2.cmml">n</mi><mo id="S3a.p2.8.m8.1.2.3.3.3" xref="S3a.p2.8.m8.1.2.3.3.3.cmml">′</mo></msup><msup id="S3a.p2.8.m8.1.2.3.2.3" xref="S3a.p2.8.m8.1.2.3.2.3.cmml"><mi id="S3a.p2.8.m8.1.2.3.2.3.2" xref="S3a.p2.8.m8.1.2.3.2.3.2.cmml">σ</mi><mo id="S3a.p2.8.m8.1.2.3.2.3.3" xref="S3a.p2.8.m8.1.2.3.2.3.3.cmml">′</mo></msup></msubsup><mo id="S3a.p2.8.m8.1.2.1a" xref="S3a.p2.8.m8.1.2.1.cmml">⁢</mo><mrow id="S3a.p2.8.m8.1.1.3" xref="S3a.p2.8.m8.1.1.2.cmml"><mo id="S3a.p2.8.m8.1.1.3.1" stretchy="false" xref="S3a.p2.8.m8.1.1.2.1.cmml">|</mo><msub id="S3a.p2.8.m8.1.1.1.1" xref="S3a.p2.8.m8.1.1.1.1.cmml"><mi id="S3a.p2.8.m8.1.1.1.1.2" xref="S3a.p2.8.m8.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p2.8.m8.1.1.1.1.3" xref="S3a.p2.8.m8.1.1.1.1.3.cmml">k</mi></msub><mo id="S3a.p2.8.m8.1.1.3.2" stretchy="false" xref="S3a.p2.8.m8.1.1.2.1.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3a.p2.8.m8.1b"><apply id="S3a.p2.8.m8.1.2.cmml" xref="S3a.p2.8.m8.1.2"><times id="S3a.p2.8.m8.1.2.1.cmml" xref="S3a.p2.8.m8.1.2.1"></times><apply id="S3a.p2.8.m8.1.2.2.cmml" xref="S3a.p2.8.m8.1.2.2"><csymbol cd="ambiguous" id="S3a.p2.8.m8.1.2.2.1.cmml" xref="S3a.p2.8.m8.1.2.2">subscript</csymbol><ci id="S3a.p2.8.m8.1.2.2.2.cmml" xref="S3a.p2.8.m8.1.2.2.2">𝑐</ci><ci id="S3a.p2.8.m8.1.2.2.3.cmml" xref="S3a.p2.8.m8.1.2.2.3">𝑘</ci></apply><apply id="S3a.p2.8.m8.1.2.3.cmml" xref="S3a.p2.8.m8.1.2.3"><csymbol cd="ambiguous" id="S3a.p2.8.m8.1.2.3.1.cmml" xref="S3a.p2.8.m8.1.2.3">subscript</csymbol><apply id="S3a.p2.8.m8.1.2.3.2.cmml" xref="S3a.p2.8.m8.1.2.3"><csymbol cd="ambiguous" id="S3a.p2.8.m8.1.2.3.2.1.cmml" xref="S3a.p2.8.m8.1.2.3">superscript</csymbol><ci id="S3a.p2.8.m8.1.2.3.2.2.cmml" xref="S3a.p2.8.m8.1.2.3.2.2">𝐻</ci><apply id="S3a.p2.8.m8.1.2.3.2.3.cmml" xref="S3a.p2.8.m8.1.2.3.2.3"><csymbol cd="ambiguous" id="S3a.p2.8.m8.1.2.3.2.3.1.cmml" xref="S3a.p2.8.m8.1.2.3.2.3">superscript</csymbol><ci id="S3a.p2.8.m8.1.2.3.2.3.2.cmml" xref="S3a.p2.8.m8.1.2.3.2.3.2">𝜎</ci><ci id="S3a.p2.8.m8.1.2.3.2.3.3.cmml" xref="S3a.p2.8.m8.1.2.3.2.3.3">′</ci></apply></apply><apply id="S3a.p2.8.m8.1.2.3.3.cmml" xref="S3a.p2.8.m8.1.2.3.3"><csymbol cd="ambiguous" id="S3a.p2.8.m8.1.2.3.3.1.cmml" xref="S3a.p2.8.m8.1.2.3.3">superscript</csymbol><ci id="S3a.p2.8.m8.1.2.3.3.2.cmml" xref="S3a.p2.8.m8.1.2.3.3.2">𝑛</ci><ci id="S3a.p2.8.m8.1.2.3.3.3.cmml" xref="S3a.p2.8.m8.1.2.3.3.3">′</ci></apply></apply><apply id="S3a.p2.8.m8.1.1.2.cmml" xref="S3a.p2.8.m8.1.1.3"><csymbol cd="latexml" id="S3a.p2.8.m8.1.1.2.1.cmml" xref="S3a.p2.8.m8.1.1.3.1">ket</csymbol><apply id="S3a.p2.8.m8.1.1.1.1.cmml" xref="S3a.p2.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p2.8.m8.1.1.1.1.1.cmml" xref="S3a.p2.8.m8.1.1.1.1">subscript</csymbol><ci id="S3a.p2.8.m8.1.1.1.1.2.cmml" xref="S3a.p2.8.m8.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p2.8.m8.1.1.1.1.3.cmml" xref="S3a.p2.8.m8.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.8.m8.1c">c_{k}H^{\sigma^{\prime}}_{n^{\prime}}\ket{\phi_{k}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.8.m8.1d">italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> with <math alttext="c_{k}=-c_{i}" class="ltx_Math" display="inline" id="S3a.p2.9.m9.1"><semantics id="S3a.p2.9.m9.1a"><mrow id="S3a.p2.9.m9.1.1" xref="S3a.p2.9.m9.1.1.cmml"><msub id="S3a.p2.9.m9.1.1.2" xref="S3a.p2.9.m9.1.1.2.cmml"><mi id="S3a.p2.9.m9.1.1.2.2" xref="S3a.p2.9.m9.1.1.2.2.cmml">c</mi><mi id="S3a.p2.9.m9.1.1.2.3" xref="S3a.p2.9.m9.1.1.2.3.cmml">k</mi></msub><mo id="S3a.p2.9.m9.1.1.1" xref="S3a.p2.9.m9.1.1.1.cmml">=</mo><mrow id="S3a.p2.9.m9.1.1.3" xref="S3a.p2.9.m9.1.1.3.cmml"><mo id="S3a.p2.9.m9.1.1.3a" xref="S3a.p2.9.m9.1.1.3.cmml">−</mo><msub id="S3a.p2.9.m9.1.1.3.2" xref="S3a.p2.9.m9.1.1.3.2.cmml"><mi id="S3a.p2.9.m9.1.1.3.2.2" xref="S3a.p2.9.m9.1.1.3.2.2.cmml">c</mi><mi id="S3a.p2.9.m9.1.1.3.2.3" xref="S3a.p2.9.m9.1.1.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3a.p2.9.m9.1b"><apply id="S3a.p2.9.m9.1.1.cmml" xref="S3a.p2.9.m9.1.1"><eq id="S3a.p2.9.m9.1.1.1.cmml" xref="S3a.p2.9.m9.1.1.1"></eq><apply id="S3a.p2.9.m9.1.1.2.cmml" xref="S3a.p2.9.m9.1.1.2"><csymbol cd="ambiguous" id="S3a.p2.9.m9.1.1.2.1.cmml" xref="S3a.p2.9.m9.1.1.2">subscript</csymbol><ci id="S3a.p2.9.m9.1.1.2.2.cmml" xref="S3a.p2.9.m9.1.1.2.2">𝑐</ci><ci id="S3a.p2.9.m9.1.1.2.3.cmml" xref="S3a.p2.9.m9.1.1.2.3">𝑘</ci></apply><apply id="S3a.p2.9.m9.1.1.3.cmml" xref="S3a.p2.9.m9.1.1.3"><minus id="S3a.p2.9.m9.1.1.3.1.cmml" xref="S3a.p2.9.m9.1.1.3"></minus><apply id="S3a.p2.9.m9.1.1.3.2.cmml" xref="S3a.p2.9.m9.1.1.3.2"><csymbol cd="ambiguous" id="S3a.p2.9.m9.1.1.3.2.1.cmml" xref="S3a.p2.9.m9.1.1.3.2">subscript</csymbol><ci id="S3a.p2.9.m9.1.1.3.2.2.cmml" xref="S3a.p2.9.m9.1.1.3.2.2">𝑐</ci><ci id="S3a.p2.9.m9.1.1.3.2.3.cmml" xref="S3a.p2.9.m9.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.9.m9.1c">c_{k}=-c_{i}</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.9.m9.1d">italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = - italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="H^{\sigma}_{n}\ket{\phi_{n}}=H^{\sigma^{\prime}}_{n^{\prime}}\ket{\phi_{k}}" class="ltx_Math" display="inline" id="S3a.p2.10.m10.2"><semantics id="S3a.p2.10.m10.2a"><mrow id="S3a.p2.10.m10.2.3" xref="S3a.p2.10.m10.2.3.cmml"><mrow id="S3a.p2.10.m10.2.3.2" xref="S3a.p2.10.m10.2.3.2.cmml"><msubsup id="S3a.p2.10.m10.2.3.2.2" xref="S3a.p2.10.m10.2.3.2.2.cmml"><mi id="S3a.p2.10.m10.2.3.2.2.2.2" xref="S3a.p2.10.m10.2.3.2.2.2.2.cmml">H</mi><mi id="S3a.p2.10.m10.2.3.2.2.3" xref="S3a.p2.10.m10.2.3.2.2.3.cmml">n</mi><mi id="S3a.p2.10.m10.2.3.2.2.2.3" xref="S3a.p2.10.m10.2.3.2.2.2.3.cmml">σ</mi></msubsup><mo id="S3a.p2.10.m10.2.3.2.1" xref="S3a.p2.10.m10.2.3.2.1.cmml">⁢</mo><mrow id="S3a.p2.10.m10.1.1.3" xref="S3a.p2.10.m10.1.1.2.cmml"><mo id="S3a.p2.10.m10.1.1.3.1" stretchy="false" xref="S3a.p2.10.m10.1.1.2.1.cmml">|</mo><msub id="S3a.p2.10.m10.1.1.1.1" xref="S3a.p2.10.m10.1.1.1.1.cmml"><mi id="S3a.p2.10.m10.1.1.1.1.2" xref="S3a.p2.10.m10.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p2.10.m10.1.1.1.1.3" xref="S3a.p2.10.m10.1.1.1.1.3.cmml">n</mi></msub><mo id="S3a.p2.10.m10.1.1.3.2" stretchy="false" xref="S3a.p2.10.m10.1.1.2.1.cmml">⟩</mo></mrow></mrow><mo id="S3a.p2.10.m10.2.3.1" xref="S3a.p2.10.m10.2.3.1.cmml">=</mo><mrow id="S3a.p2.10.m10.2.3.3" xref="S3a.p2.10.m10.2.3.3.cmml"><msubsup id="S3a.p2.10.m10.2.3.3.2" xref="S3a.p2.10.m10.2.3.3.2.cmml"><mi id="S3a.p2.10.m10.2.3.3.2.2.2" xref="S3a.p2.10.m10.2.3.3.2.2.2.cmml">H</mi><msup id="S3a.p2.10.m10.2.3.3.2.3" xref="S3a.p2.10.m10.2.3.3.2.3.cmml"><mi id="S3a.p2.10.m10.2.3.3.2.3.2" xref="S3a.p2.10.m10.2.3.3.2.3.2.cmml">n</mi><mo id="S3a.p2.10.m10.2.3.3.2.3.3" xref="S3a.p2.10.m10.2.3.3.2.3.3.cmml">′</mo></msup><msup id="S3a.p2.10.m10.2.3.3.2.2.3" xref="S3a.p2.10.m10.2.3.3.2.2.3.cmml"><mi id="S3a.p2.10.m10.2.3.3.2.2.3.2" xref="S3a.p2.10.m10.2.3.3.2.2.3.2.cmml">σ</mi><mo id="S3a.p2.10.m10.2.3.3.2.2.3.3" xref="S3a.p2.10.m10.2.3.3.2.2.3.3.cmml">′</mo></msup></msubsup><mo id="S3a.p2.10.m10.2.3.3.1" xref="S3a.p2.10.m10.2.3.3.1.cmml">⁢</mo><mrow id="S3a.p2.10.m10.2.2.3" xref="S3a.p2.10.m10.2.2.2.cmml"><mo id="S3a.p2.10.m10.2.2.3.1" stretchy="false" xref="S3a.p2.10.m10.2.2.2.1.cmml">|</mo><msub id="S3a.p2.10.m10.2.2.1.1" xref="S3a.p2.10.m10.2.2.1.1.cmml"><mi id="S3a.p2.10.m10.2.2.1.1.2" xref="S3a.p2.10.m10.2.2.1.1.2.cmml">ϕ</mi><mi id="S3a.p2.10.m10.2.2.1.1.3" xref="S3a.p2.10.m10.2.2.1.1.3.cmml">k</mi></msub><mo id="S3a.p2.10.m10.2.2.3.2" stretchy="false" xref="S3a.p2.10.m10.2.2.2.1.cmml">⟩</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3a.p2.10.m10.2b"><apply id="S3a.p2.10.m10.2.3.cmml" xref="S3a.p2.10.m10.2.3"><eq id="S3a.p2.10.m10.2.3.1.cmml" xref="S3a.p2.10.m10.2.3.1"></eq><apply id="S3a.p2.10.m10.2.3.2.cmml" xref="S3a.p2.10.m10.2.3.2"><times id="S3a.p2.10.m10.2.3.2.1.cmml" xref="S3a.p2.10.m10.2.3.2.1"></times><apply id="S3a.p2.10.m10.2.3.2.2.cmml" xref="S3a.p2.10.m10.2.3.2.2"><csymbol cd="ambiguous" id="S3a.p2.10.m10.2.3.2.2.1.cmml" xref="S3a.p2.10.m10.2.3.2.2">subscript</csymbol><apply id="S3a.p2.10.m10.2.3.2.2.2.cmml" xref="S3a.p2.10.m10.2.3.2.2"><csymbol cd="ambiguous" id="S3a.p2.10.m10.2.3.2.2.2.1.cmml" xref="S3a.p2.10.m10.2.3.2.2">superscript</csymbol><ci id="S3a.p2.10.m10.2.3.2.2.2.2.cmml" xref="S3a.p2.10.m10.2.3.2.2.2.2">𝐻</ci><ci id="S3a.p2.10.m10.2.3.2.2.2.3.cmml" xref="S3a.p2.10.m10.2.3.2.2.2.3">𝜎</ci></apply><ci id="S3a.p2.10.m10.2.3.2.2.3.cmml" xref="S3a.p2.10.m10.2.3.2.2.3">𝑛</ci></apply><apply id="S3a.p2.10.m10.1.1.2.cmml" xref="S3a.p2.10.m10.1.1.3"><csymbol cd="latexml" id="S3a.p2.10.m10.1.1.2.1.cmml" xref="S3a.p2.10.m10.1.1.3.1">ket</csymbol><apply id="S3a.p2.10.m10.1.1.1.1.cmml" xref="S3a.p2.10.m10.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p2.10.m10.1.1.1.1.1.cmml" xref="S3a.p2.10.m10.1.1.1.1">subscript</csymbol><ci id="S3a.p2.10.m10.1.1.1.1.2.cmml" xref="S3a.p2.10.m10.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p2.10.m10.1.1.1.1.3.cmml" xref="S3a.p2.10.m10.1.1.1.1.3">𝑛</ci></apply></apply></apply><apply id="S3a.p2.10.m10.2.3.3.cmml" xref="S3a.p2.10.m10.2.3.3"><times id="S3a.p2.10.m10.2.3.3.1.cmml" xref="S3a.p2.10.m10.2.3.3.1"></times><apply id="S3a.p2.10.m10.2.3.3.2.cmml" xref="S3a.p2.10.m10.2.3.3.2"><csymbol cd="ambiguous" id="S3a.p2.10.m10.2.3.3.2.1.cmml" xref="S3a.p2.10.m10.2.3.3.2">subscript</csymbol><apply id="S3a.p2.10.m10.2.3.3.2.2.cmml" xref="S3a.p2.10.m10.2.3.3.2"><csymbol cd="ambiguous" id="S3a.p2.10.m10.2.3.3.2.2.1.cmml" xref="S3a.p2.10.m10.2.3.3.2">superscript</csymbol><ci id="S3a.p2.10.m10.2.3.3.2.2.2.cmml" xref="S3a.p2.10.m10.2.3.3.2.2.2">𝐻</ci><apply id="S3a.p2.10.m10.2.3.3.2.2.3.cmml" xref="S3a.p2.10.m10.2.3.3.2.2.3"><csymbol cd="ambiguous" id="S3a.p2.10.m10.2.3.3.2.2.3.1.cmml" xref="S3a.p2.10.m10.2.3.3.2.2.3">superscript</csymbol><ci id="S3a.p2.10.m10.2.3.3.2.2.3.2.cmml" xref="S3a.p2.10.m10.2.3.3.2.2.3.2">𝜎</ci><ci id="S3a.p2.10.m10.2.3.3.2.2.3.3.cmml" xref="S3a.p2.10.m10.2.3.3.2.2.3.3">′</ci></apply></apply><apply id="S3a.p2.10.m10.2.3.3.2.3.cmml" xref="S3a.p2.10.m10.2.3.3.2.3"><csymbol cd="ambiguous" id="S3a.p2.10.m10.2.3.3.2.3.1.cmml" xref="S3a.p2.10.m10.2.3.3.2.3">superscript</csymbol><ci id="S3a.p2.10.m10.2.3.3.2.3.2.cmml" xref="S3a.p2.10.m10.2.3.3.2.3.2">𝑛</ci><ci id="S3a.p2.10.m10.2.3.3.2.3.3.cmml" xref="S3a.p2.10.m10.2.3.3.2.3.3">′</ci></apply></apply><apply id="S3a.p2.10.m10.2.2.2.cmml" xref="S3a.p2.10.m10.2.2.3"><csymbol cd="latexml" id="S3a.p2.10.m10.2.2.2.1.cmml" xref="S3a.p2.10.m10.2.2.3.1">ket</csymbol><apply id="S3a.p2.10.m10.2.2.1.1.cmml" xref="S3a.p2.10.m10.2.2.1.1"><csymbol cd="ambiguous" id="S3a.p2.10.m10.2.2.1.1.1.cmml" xref="S3a.p2.10.m10.2.2.1.1">subscript</csymbol><ci id="S3a.p2.10.m10.2.2.1.1.2.cmml" xref="S3a.p2.10.m10.2.2.1.1.2">italic-ϕ</ci><ci id="S3a.p2.10.m10.2.2.1.1.3.cmml" xref="S3a.p2.10.m10.2.2.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.10.m10.2c">H^{\sigma}_{n}\ket{\phi_{n}}=H^{\sigma^{\prime}}_{n^{\prime}}\ket{\phi_{k}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.10.m10.2d">italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG ⟩ = italic_H start_POSTSUPERSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> for some <math alttext="\sigma^{\prime},n^{\prime}" class="ltx_Math" display="inline" id="S3a.p2.11.m11.2"><semantics id="S3a.p2.11.m11.2a"><mrow id="S3a.p2.11.m11.2.2.2" xref="S3a.p2.11.m11.2.2.3.cmml"><msup id="S3a.p2.11.m11.1.1.1.1" xref="S3a.p2.11.m11.1.1.1.1.cmml"><mi id="S3a.p2.11.m11.1.1.1.1.2" xref="S3a.p2.11.m11.1.1.1.1.2.cmml">σ</mi><mo id="S3a.p2.11.m11.1.1.1.1.3" xref="S3a.p2.11.m11.1.1.1.1.3.cmml">′</mo></msup><mo id="S3a.p2.11.m11.2.2.2.3" xref="S3a.p2.11.m11.2.2.3.cmml">,</mo><msup id="S3a.p2.11.m11.2.2.2.2" xref="S3a.p2.11.m11.2.2.2.2.cmml"><mi id="S3a.p2.11.m11.2.2.2.2.2" xref="S3a.p2.11.m11.2.2.2.2.2.cmml">n</mi><mo id="S3a.p2.11.m11.2.2.2.2.3" xref="S3a.p2.11.m11.2.2.2.2.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3a.p2.11.m11.2b"><list id="S3a.p2.11.m11.2.2.3.cmml" xref="S3a.p2.11.m11.2.2.2"><apply id="S3a.p2.11.m11.1.1.1.1.cmml" xref="S3a.p2.11.m11.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p2.11.m11.1.1.1.1.1.cmml" xref="S3a.p2.11.m11.1.1.1.1">superscript</csymbol><ci id="S3a.p2.11.m11.1.1.1.1.2.cmml" xref="S3a.p2.11.m11.1.1.1.1.2">𝜎</ci><ci id="S3a.p2.11.m11.1.1.1.1.3.cmml" xref="S3a.p2.11.m11.1.1.1.1.3">′</ci></apply><apply id="S3a.p2.11.m11.2.2.2.2.cmml" xref="S3a.p2.11.m11.2.2.2.2"><csymbol cd="ambiguous" id="S3a.p2.11.m11.2.2.2.2.1.cmml" xref="S3a.p2.11.m11.2.2.2.2">superscript</csymbol><ci id="S3a.p2.11.m11.2.2.2.2.2.cmml" xref="S3a.p2.11.m11.2.2.2.2.2">𝑛</ci><ci id="S3a.p2.11.m11.2.2.2.2.3.cmml" xref="S3a.p2.11.m11.2.2.2.2.3">′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.11.m11.2c">\sigma^{\prime},n^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.11.m11.2d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. This introduces the minimal possible number of non-zero weights to cancel a given <math alttext="H^{\sigma}_{n}\ket{\phi_{i}}" class="ltx_Math" display="inline" id="S3a.p2.12.m12.1"><semantics id="S3a.p2.12.m12.1a"><mrow id="S3a.p2.12.m12.1.2" xref="S3a.p2.12.m12.1.2.cmml"><msubsup id="S3a.p2.12.m12.1.2.2" xref="S3a.p2.12.m12.1.2.2.cmml"><mi id="S3a.p2.12.m12.1.2.2.2.2" xref="S3a.p2.12.m12.1.2.2.2.2.cmml">H</mi><mi id="S3a.p2.12.m12.1.2.2.3" xref="S3a.p2.12.m12.1.2.2.3.cmml">n</mi><mi id="S3a.p2.12.m12.1.2.2.2.3" xref="S3a.p2.12.m12.1.2.2.2.3.cmml">σ</mi></msubsup><mo id="S3a.p2.12.m12.1.2.1" xref="S3a.p2.12.m12.1.2.1.cmml">⁢</mo><mrow id="S3a.p2.12.m12.1.1.3" xref="S3a.p2.12.m12.1.1.2.cmml"><mo id="S3a.p2.12.m12.1.1.3.1" stretchy="false" xref="S3a.p2.12.m12.1.1.2.1.cmml">|</mo><msub id="S3a.p2.12.m12.1.1.1.1" xref="S3a.p2.12.m12.1.1.1.1.cmml"><mi id="S3a.p2.12.m12.1.1.1.1.2" xref="S3a.p2.12.m12.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p2.12.m12.1.1.1.1.3" xref="S3a.p2.12.m12.1.1.1.1.3.cmml">i</mi></msub><mo id="S3a.p2.12.m12.1.1.3.2" stretchy="false" xref="S3a.p2.12.m12.1.1.2.1.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3a.p2.12.m12.1b"><apply id="S3a.p2.12.m12.1.2.cmml" xref="S3a.p2.12.m12.1.2"><times id="S3a.p2.12.m12.1.2.1.cmml" xref="S3a.p2.12.m12.1.2.1"></times><apply id="S3a.p2.12.m12.1.2.2.cmml" xref="S3a.p2.12.m12.1.2.2"><csymbol cd="ambiguous" id="S3a.p2.12.m12.1.2.2.1.cmml" xref="S3a.p2.12.m12.1.2.2">subscript</csymbol><apply id="S3a.p2.12.m12.1.2.2.2.cmml" xref="S3a.p2.12.m12.1.2.2"><csymbol cd="ambiguous" id="S3a.p2.12.m12.1.2.2.2.1.cmml" xref="S3a.p2.12.m12.1.2.2">superscript</csymbol><ci id="S3a.p2.12.m12.1.2.2.2.2.cmml" xref="S3a.p2.12.m12.1.2.2.2.2">𝐻</ci><ci id="S3a.p2.12.m12.1.2.2.2.3.cmml" xref="S3a.p2.12.m12.1.2.2.2.3">𝜎</ci></apply><ci id="S3a.p2.12.m12.1.2.2.3.cmml" xref="S3a.p2.12.m12.1.2.2.3">𝑛</ci></apply><apply id="S3a.p2.12.m12.1.1.2.cmml" xref="S3a.p2.12.m12.1.1.3"><csymbol cd="latexml" id="S3a.p2.12.m12.1.1.2.1.cmml" xref="S3a.p2.12.m12.1.1.3.1">ket</csymbol><apply id="S3a.p2.12.m12.1.1.1.1.cmml" xref="S3a.p2.12.m12.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p2.12.m12.1.1.1.1.1.cmml" xref="S3a.p2.12.m12.1.1.1.1">subscript</csymbol><ci id="S3a.p2.12.m12.1.1.1.1.2.cmml" xref="S3a.p2.12.m12.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p2.12.m12.1.1.1.1.3.cmml" xref="S3a.p2.12.m12.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.12.m12.1c">H^{\sigma}_{n}\ket{\phi_{i}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.12.m12.1d">italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math>. In TLMs, states <math alttext="\ket{\phi_{i}}" class="ltx_Math" display="inline" id="S3a.p2.13.m13.1"><semantics id="S3a.p2.13.m13.1a"><mrow id="S3a.p2.13.m13.1.1.3" xref="S3a.p2.13.m13.1.1.2.cmml"><mo id="S3a.p2.13.m13.1.1.3.1" stretchy="false" xref="S3a.p2.13.m13.1.1.2.1.cmml">|</mo><msub id="S3a.p2.13.m13.1.1.1.1" xref="S3a.p2.13.m13.1.1.1.1.cmml"><mi id="S3a.p2.13.m13.1.1.1.1.2" xref="S3a.p2.13.m13.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p2.13.m13.1.1.1.1.3" xref="S3a.p2.13.m13.1.1.1.1.3.cmml">i</mi></msub><mo id="S3a.p2.13.m13.1.1.3.2" stretchy="false" xref="S3a.p2.13.m13.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S3a.p2.13.m13.1b"><apply id="S3a.p2.13.m13.1.1.2.cmml" xref="S3a.p2.13.m13.1.1.3"><csymbol cd="latexml" id="S3a.p2.13.m13.1.1.2.1.cmml" xref="S3a.p2.13.m13.1.1.3.1">ket</csymbol><apply id="S3a.p2.13.m13.1.1.1.1.cmml" xref="S3a.p2.13.m13.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p2.13.m13.1.1.1.1.1.cmml" xref="S3a.p2.13.m13.1.1.1.1">subscript</csymbol><ci id="S3a.p2.13.m13.1.1.1.1.2.cmml" xref="S3a.p2.13.m13.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p2.13.m13.1.1.1.1.3.cmml" xref="S3a.p2.13.m13.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.13.m13.1c">\ket{\phi_{i}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.13.m13.1d">| start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math>, <math alttext="\ket{\phi_{k}}" class="ltx_Math" display="inline" id="S3a.p2.14.m14.1"><semantics id="S3a.p2.14.m14.1a"><mrow id="S3a.p2.14.m14.1.1.3" xref="S3a.p2.14.m14.1.1.2.cmml"><mo id="S3a.p2.14.m14.1.1.3.1" stretchy="false" xref="S3a.p2.14.m14.1.1.2.1.cmml">|</mo><msub id="S3a.p2.14.m14.1.1.1.1" xref="S3a.p2.14.m14.1.1.1.1.cmml"><mi id="S3a.p2.14.m14.1.1.1.1.2" xref="S3a.p2.14.m14.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p2.14.m14.1.1.1.1.3" xref="S3a.p2.14.m14.1.1.1.1.3.cmml">k</mi></msub><mo id="S3a.p2.14.m14.1.1.3.2" stretchy="false" xref="S3a.p2.14.m14.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S3a.p2.14.m14.1b"><apply id="S3a.p2.14.m14.1.1.2.cmml" xref="S3a.p2.14.m14.1.1.3"><csymbol cd="latexml" id="S3a.p2.14.m14.1.1.2.1.cmml" xref="S3a.p2.14.m14.1.1.3.1">ket</csymbol><apply id="S3a.p2.14.m14.1.1.1.1.cmml" xref="S3a.p2.14.m14.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p2.14.m14.1.1.1.1.1.cmml" xref="S3a.p2.14.m14.1.1.1.1">subscript</csymbol><ci id="S3a.p2.14.m14.1.1.1.1.2.cmml" xref="S3a.p2.14.m14.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p2.14.m14.1.1.1.1.3.cmml" xref="S3a.p2.14.m14.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.14.m14.1c">\ket{\phi_{k}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.14.m14.1d">| start_ARG italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> that obey <math alttext="H^{\sigma}_{n}\ket{\phi_{i}}=H^{\sigma^{\prime}}_{n^{\prime}}\ket{\phi_{k}}\neq 0" class="ltx_Math" display="inline" id="S3a.p2.15.m15.2"><semantics id="S3a.p2.15.m15.2a"><mrow id="S3a.p2.15.m15.2.3" xref="S3a.p2.15.m15.2.3.cmml"><mrow id="S3a.p2.15.m15.2.3.2" xref="S3a.p2.15.m15.2.3.2.cmml"><msubsup id="S3a.p2.15.m15.2.3.2.2" xref="S3a.p2.15.m15.2.3.2.2.cmml"><mi id="S3a.p2.15.m15.2.3.2.2.2.2" xref="S3a.p2.15.m15.2.3.2.2.2.2.cmml">H</mi><mi id="S3a.p2.15.m15.2.3.2.2.3" xref="S3a.p2.15.m15.2.3.2.2.3.cmml">n</mi><mi id="S3a.p2.15.m15.2.3.2.2.2.3" xref="S3a.p2.15.m15.2.3.2.2.2.3.cmml">σ</mi></msubsup><mo id="S3a.p2.15.m15.2.3.2.1" xref="S3a.p2.15.m15.2.3.2.1.cmml">⁢</mo><mrow id="S3a.p2.15.m15.1.1.3" xref="S3a.p2.15.m15.1.1.2.cmml"><mo id="S3a.p2.15.m15.1.1.3.1" stretchy="false" xref="S3a.p2.15.m15.1.1.2.1.cmml">|</mo><msub id="S3a.p2.15.m15.1.1.1.1" xref="S3a.p2.15.m15.1.1.1.1.cmml"><mi id="S3a.p2.15.m15.1.1.1.1.2" xref="S3a.p2.15.m15.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p2.15.m15.1.1.1.1.3" xref="S3a.p2.15.m15.1.1.1.1.3.cmml">i</mi></msub><mo id="S3a.p2.15.m15.1.1.3.2" stretchy="false" xref="S3a.p2.15.m15.1.1.2.1.cmml">⟩</mo></mrow></mrow><mo id="S3a.p2.15.m15.2.3.3" xref="S3a.p2.15.m15.2.3.3.cmml">=</mo><mrow id="S3a.p2.15.m15.2.3.4" xref="S3a.p2.15.m15.2.3.4.cmml"><msubsup id="S3a.p2.15.m15.2.3.4.2" xref="S3a.p2.15.m15.2.3.4.2.cmml"><mi id="S3a.p2.15.m15.2.3.4.2.2.2" xref="S3a.p2.15.m15.2.3.4.2.2.2.cmml">H</mi><msup id="S3a.p2.15.m15.2.3.4.2.3" xref="S3a.p2.15.m15.2.3.4.2.3.cmml"><mi id="S3a.p2.15.m15.2.3.4.2.3.2" xref="S3a.p2.15.m15.2.3.4.2.3.2.cmml">n</mi><mo id="S3a.p2.15.m15.2.3.4.2.3.3" xref="S3a.p2.15.m15.2.3.4.2.3.3.cmml">′</mo></msup><msup id="S3a.p2.15.m15.2.3.4.2.2.3" xref="S3a.p2.15.m15.2.3.4.2.2.3.cmml"><mi id="S3a.p2.15.m15.2.3.4.2.2.3.2" xref="S3a.p2.15.m15.2.3.4.2.2.3.2.cmml">σ</mi><mo id="S3a.p2.15.m15.2.3.4.2.2.3.3" xref="S3a.p2.15.m15.2.3.4.2.2.3.3.cmml">′</mo></msup></msubsup><mo id="S3a.p2.15.m15.2.3.4.1" xref="S3a.p2.15.m15.2.3.4.1.cmml">⁢</mo><mrow id="S3a.p2.15.m15.2.2.3" xref="S3a.p2.15.m15.2.2.2.cmml"><mo id="S3a.p2.15.m15.2.2.3.1" stretchy="false" xref="S3a.p2.15.m15.2.2.2.1.cmml">|</mo><msub id="S3a.p2.15.m15.2.2.1.1" xref="S3a.p2.15.m15.2.2.1.1.cmml"><mi id="S3a.p2.15.m15.2.2.1.1.2" xref="S3a.p2.15.m15.2.2.1.1.2.cmml">ϕ</mi><mi id="S3a.p2.15.m15.2.2.1.1.3" xref="S3a.p2.15.m15.2.2.1.1.3.cmml">k</mi></msub><mo id="S3a.p2.15.m15.2.2.3.2" stretchy="false" xref="S3a.p2.15.m15.2.2.2.1.cmml">⟩</mo></mrow></mrow><mo id="S3a.p2.15.m15.2.3.5" xref="S3a.p2.15.m15.2.3.5.cmml">≠</mo><mn id="S3a.p2.15.m15.2.3.6" xref="S3a.p2.15.m15.2.3.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3a.p2.15.m15.2b"><apply id="S3a.p2.15.m15.2.3.cmml" xref="S3a.p2.15.m15.2.3"><and id="S3a.p2.15.m15.2.3a.cmml" xref="S3a.p2.15.m15.2.3"></and><apply id="S3a.p2.15.m15.2.3b.cmml" xref="S3a.p2.15.m15.2.3"><eq id="S3a.p2.15.m15.2.3.3.cmml" xref="S3a.p2.15.m15.2.3.3"></eq><apply id="S3a.p2.15.m15.2.3.2.cmml" xref="S3a.p2.15.m15.2.3.2"><times id="S3a.p2.15.m15.2.3.2.1.cmml" xref="S3a.p2.15.m15.2.3.2.1"></times><apply id="S3a.p2.15.m15.2.3.2.2.cmml" xref="S3a.p2.15.m15.2.3.2.2"><csymbol cd="ambiguous" id="S3a.p2.15.m15.2.3.2.2.1.cmml" xref="S3a.p2.15.m15.2.3.2.2">subscript</csymbol><apply id="S3a.p2.15.m15.2.3.2.2.2.cmml" xref="S3a.p2.15.m15.2.3.2.2"><csymbol cd="ambiguous" id="S3a.p2.15.m15.2.3.2.2.2.1.cmml" xref="S3a.p2.15.m15.2.3.2.2">superscript</csymbol><ci id="S3a.p2.15.m15.2.3.2.2.2.2.cmml" xref="S3a.p2.15.m15.2.3.2.2.2.2">𝐻</ci><ci id="S3a.p2.15.m15.2.3.2.2.2.3.cmml" xref="S3a.p2.15.m15.2.3.2.2.2.3">𝜎</ci></apply><ci id="S3a.p2.15.m15.2.3.2.2.3.cmml" xref="S3a.p2.15.m15.2.3.2.2.3">𝑛</ci></apply><apply id="S3a.p2.15.m15.1.1.2.cmml" xref="S3a.p2.15.m15.1.1.3"><csymbol cd="latexml" id="S3a.p2.15.m15.1.1.2.1.cmml" xref="S3a.p2.15.m15.1.1.3.1">ket</csymbol><apply id="S3a.p2.15.m15.1.1.1.1.cmml" xref="S3a.p2.15.m15.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p2.15.m15.1.1.1.1.1.cmml" xref="S3a.p2.15.m15.1.1.1.1">subscript</csymbol><ci id="S3a.p2.15.m15.1.1.1.1.2.cmml" xref="S3a.p2.15.m15.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p2.15.m15.1.1.1.1.3.cmml" xref="S3a.p2.15.m15.1.1.1.1.3">𝑖</ci></apply></apply></apply><apply id="S3a.p2.15.m15.2.3.4.cmml" xref="S3a.p2.15.m15.2.3.4"><times id="S3a.p2.15.m15.2.3.4.1.cmml" xref="S3a.p2.15.m15.2.3.4.1"></times><apply id="S3a.p2.15.m15.2.3.4.2.cmml" xref="S3a.p2.15.m15.2.3.4.2"><csymbol cd="ambiguous" id="S3a.p2.15.m15.2.3.4.2.1.cmml" xref="S3a.p2.15.m15.2.3.4.2">subscript</csymbol><apply id="S3a.p2.15.m15.2.3.4.2.2.cmml" xref="S3a.p2.15.m15.2.3.4.2"><csymbol cd="ambiguous" id="S3a.p2.15.m15.2.3.4.2.2.1.cmml" xref="S3a.p2.15.m15.2.3.4.2">superscript</csymbol><ci id="S3a.p2.15.m15.2.3.4.2.2.2.cmml" xref="S3a.p2.15.m15.2.3.4.2.2.2">𝐻</ci><apply id="S3a.p2.15.m15.2.3.4.2.2.3.cmml" xref="S3a.p2.15.m15.2.3.4.2.2.3"><csymbol cd="ambiguous" id="S3a.p2.15.m15.2.3.4.2.2.3.1.cmml" xref="S3a.p2.15.m15.2.3.4.2.2.3">superscript</csymbol><ci id="S3a.p2.15.m15.2.3.4.2.2.3.2.cmml" xref="S3a.p2.15.m15.2.3.4.2.2.3.2">𝜎</ci><ci id="S3a.p2.15.m15.2.3.4.2.2.3.3.cmml" xref="S3a.p2.15.m15.2.3.4.2.2.3.3">′</ci></apply></apply><apply id="S3a.p2.15.m15.2.3.4.2.3.cmml" xref="S3a.p2.15.m15.2.3.4.2.3"><csymbol cd="ambiguous" id="S3a.p2.15.m15.2.3.4.2.3.1.cmml" xref="S3a.p2.15.m15.2.3.4.2.3">superscript</csymbol><ci id="S3a.p2.15.m15.2.3.4.2.3.2.cmml" xref="S3a.p2.15.m15.2.3.4.2.3.2">𝑛</ci><ci id="S3a.p2.15.m15.2.3.4.2.3.3.cmml" xref="S3a.p2.15.m15.2.3.4.2.3.3">′</ci></apply></apply><apply id="S3a.p2.15.m15.2.2.2.cmml" xref="S3a.p2.15.m15.2.2.3"><csymbol cd="latexml" id="S3a.p2.15.m15.2.2.2.1.cmml" xref="S3a.p2.15.m15.2.2.3.1">ket</csymbol><apply id="S3a.p2.15.m15.2.2.1.1.cmml" xref="S3a.p2.15.m15.2.2.1.1"><csymbol cd="ambiguous" id="S3a.p2.15.m15.2.2.1.1.1.cmml" xref="S3a.p2.15.m15.2.2.1.1">subscript</csymbol><ci id="S3a.p2.15.m15.2.2.1.1.2.cmml" xref="S3a.p2.15.m15.2.2.1.1.2">italic-ϕ</ci><ci id="S3a.p2.15.m15.2.2.1.1.3.cmml" xref="S3a.p2.15.m15.2.2.1.1.3">𝑘</ci></apply></apply></apply></apply><apply id="S3a.p2.15.m15.2.3c.cmml" xref="S3a.p2.15.m15.2.3"><neq id="S3a.p2.15.m15.2.3.5.cmml" xref="S3a.p2.15.m15.2.3.5"></neq><share href="https://arxiv.org/html/2403.08892v3#S3a.p2.15.m15.2.3.4.cmml" id="S3a.p2.15.m15.2.3d.cmml" xref="S3a.p2.15.m15.2.3"></share><cn id="S3a.p2.15.m15.2.3.6.cmml" type="integer" xref="S3a.p2.15.m15.2.3.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.15.m15.2c">H^{\sigma}_{n}\ket{\phi_{i}}=H^{\sigma^{\prime}}_{n^{\prime}}\ket{\phi_{k}}\neq 0</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.15.m15.2d">italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ = italic_H start_POSTSUPERSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ⟩ ≠ 0</annotation></semantics></math>, are related by <math alttext="\ket{\phi_{k}}=H^{-\sigma^{\prime}}_{n^{\prime}}H^{\sigma}_{n}\ket{\phi_{i}}=H% ^{\sigma}_{n}H^{-\sigma^{\prime}}_{n^{\prime}}\ket{\phi_{i}}" class="ltx_Math" display="inline" id="S3a.p2.16.m16.3"><semantics id="S3a.p2.16.m16.3a"><mrow id="S3a.p2.16.m16.3.4" xref="S3a.p2.16.m16.3.4.cmml"><mrow id="S3a.p2.16.m16.1.1.3" xref="S3a.p2.16.m16.1.1.2.cmml"><mo id="S3a.p2.16.m16.1.1.3.1" stretchy="false" xref="S3a.p2.16.m16.1.1.2.1.cmml">|</mo><msub id="S3a.p2.16.m16.1.1.1.1" xref="S3a.p2.16.m16.1.1.1.1.cmml"><mi id="S3a.p2.16.m16.1.1.1.1.2" xref="S3a.p2.16.m16.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p2.16.m16.1.1.1.1.3" xref="S3a.p2.16.m16.1.1.1.1.3.cmml">k</mi></msub><mo id="S3a.p2.16.m16.1.1.3.2" stretchy="false" xref="S3a.p2.16.m16.1.1.2.1.cmml">⟩</mo></mrow><mo id="S3a.p2.16.m16.3.4.2" xref="S3a.p2.16.m16.3.4.2.cmml">=</mo><mrow id="S3a.p2.16.m16.3.4.3" xref="S3a.p2.16.m16.3.4.3.cmml"><msubsup id="S3a.p2.16.m16.3.4.3.2" xref="S3a.p2.16.m16.3.4.3.2.cmml"><mi id="S3a.p2.16.m16.3.4.3.2.2.2" xref="S3a.p2.16.m16.3.4.3.2.2.2.cmml">H</mi><msup id="S3a.p2.16.m16.3.4.3.2.3" xref="S3a.p2.16.m16.3.4.3.2.3.cmml"><mi id="S3a.p2.16.m16.3.4.3.2.3.2" xref="S3a.p2.16.m16.3.4.3.2.3.2.cmml">n</mi><mo id="S3a.p2.16.m16.3.4.3.2.3.3" xref="S3a.p2.16.m16.3.4.3.2.3.3.cmml">′</mo></msup><mrow id="S3a.p2.16.m16.3.4.3.2.2.3" xref="S3a.p2.16.m16.3.4.3.2.2.3.cmml"><mo id="S3a.p2.16.m16.3.4.3.2.2.3a" xref="S3a.p2.16.m16.3.4.3.2.2.3.cmml">−</mo><msup id="S3a.p2.16.m16.3.4.3.2.2.3.2" xref="S3a.p2.16.m16.3.4.3.2.2.3.2.cmml"><mi id="S3a.p2.16.m16.3.4.3.2.2.3.2.2" xref="S3a.p2.16.m16.3.4.3.2.2.3.2.2.cmml">σ</mi><mo id="S3a.p2.16.m16.3.4.3.2.2.3.2.3" xref="S3a.p2.16.m16.3.4.3.2.2.3.2.3.cmml">′</mo></msup></mrow></msubsup><mo id="S3a.p2.16.m16.3.4.3.1" xref="S3a.p2.16.m16.3.4.3.1.cmml">⁢</mo><msubsup id="S3a.p2.16.m16.3.4.3.3" xref="S3a.p2.16.m16.3.4.3.3.cmml"><mi id="S3a.p2.16.m16.3.4.3.3.2.2" xref="S3a.p2.16.m16.3.4.3.3.2.2.cmml">H</mi><mi id="S3a.p2.16.m16.3.4.3.3.3" xref="S3a.p2.16.m16.3.4.3.3.3.cmml">n</mi><mi id="S3a.p2.16.m16.3.4.3.3.2.3" xref="S3a.p2.16.m16.3.4.3.3.2.3.cmml">σ</mi></msubsup><mo id="S3a.p2.16.m16.3.4.3.1a" xref="S3a.p2.16.m16.3.4.3.1.cmml">⁢</mo><mrow id="S3a.p2.16.m16.2.2.3" xref="S3a.p2.16.m16.2.2.2.cmml"><mo id="S3a.p2.16.m16.2.2.3.1" stretchy="false" xref="S3a.p2.16.m16.2.2.2.1.cmml">|</mo><msub id="S3a.p2.16.m16.2.2.1.1" xref="S3a.p2.16.m16.2.2.1.1.cmml"><mi id="S3a.p2.16.m16.2.2.1.1.2" xref="S3a.p2.16.m16.2.2.1.1.2.cmml">ϕ</mi><mi id="S3a.p2.16.m16.2.2.1.1.3" xref="S3a.p2.16.m16.2.2.1.1.3.cmml">i</mi></msub><mo id="S3a.p2.16.m16.2.2.3.2" stretchy="false" xref="S3a.p2.16.m16.2.2.2.1.cmml">⟩</mo></mrow></mrow><mo id="S3a.p2.16.m16.3.4.4" xref="S3a.p2.16.m16.3.4.4.cmml">=</mo><mrow id="S3a.p2.16.m16.3.4.5" xref="S3a.p2.16.m16.3.4.5.cmml"><msubsup id="S3a.p2.16.m16.3.4.5.2" xref="S3a.p2.16.m16.3.4.5.2.cmml"><mi id="S3a.p2.16.m16.3.4.5.2.2.2" xref="S3a.p2.16.m16.3.4.5.2.2.2.cmml">H</mi><mi id="S3a.p2.16.m16.3.4.5.2.3" xref="S3a.p2.16.m16.3.4.5.2.3.cmml">n</mi><mi id="S3a.p2.16.m16.3.4.5.2.2.3" xref="S3a.p2.16.m16.3.4.5.2.2.3.cmml">σ</mi></msubsup><mo id="S3a.p2.16.m16.3.4.5.1" xref="S3a.p2.16.m16.3.4.5.1.cmml">⁢</mo><msubsup id="S3a.p2.16.m16.3.4.5.3" xref="S3a.p2.16.m16.3.4.5.3.cmml"><mi id="S3a.p2.16.m16.3.4.5.3.2.2" xref="S3a.p2.16.m16.3.4.5.3.2.2.cmml">H</mi><msup id="S3a.p2.16.m16.3.4.5.3.3" xref="S3a.p2.16.m16.3.4.5.3.3.cmml"><mi id="S3a.p2.16.m16.3.4.5.3.3.2" xref="S3a.p2.16.m16.3.4.5.3.3.2.cmml">n</mi><mo id="S3a.p2.16.m16.3.4.5.3.3.3" xref="S3a.p2.16.m16.3.4.5.3.3.3.cmml">′</mo></msup><mrow id="S3a.p2.16.m16.3.4.5.3.2.3" 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xref="S3a.p2.16.m16.3.3.1.1"><csymbol cd="ambiguous" id="S3a.p2.16.m16.3.3.1.1.1.cmml" xref="S3a.p2.16.m16.3.3.1.1">subscript</csymbol><ci id="S3a.p2.16.m16.3.3.1.1.2.cmml" xref="S3a.p2.16.m16.3.3.1.1.2">italic-ϕ</ci><ci id="S3a.p2.16.m16.3.3.1.1.3.cmml" xref="S3a.p2.16.m16.3.3.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p2.16.m16.3c">\ket{\phi_{k}}=H^{-\sigma^{\prime}}_{n^{\prime}}H^{\sigma}_{n}\ket{\phi_{i}}=H% ^{\sigma}_{n}H^{-\sigma^{\prime}}_{n^{\prime}}\ket{\phi_{i}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p2.16.m16.3d">| start_ARG italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ⟩ = italic_H start_POSTSUPERSCRIPT - italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ = italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT - italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3a.p3"> <p class="ltx_p" id="S3a.p3.1">We then focus on two options to cancel a term <math alttext="c_{i}H^{\sigma}_{n}\ket{\phi_{i}}" class="ltx_Math" display="inline" id="S3a.p3.1.m1.1"><semantics id="S3a.p3.1.m1.1a"><mrow id="S3a.p3.1.m1.1.2" xref="S3a.p3.1.m1.1.2.cmml"><msub id="S3a.p3.1.m1.1.2.2" xref="S3a.p3.1.m1.1.2.2.cmml"><mi id="S3a.p3.1.m1.1.2.2.2" xref="S3a.p3.1.m1.1.2.2.2.cmml">c</mi><mi id="S3a.p3.1.m1.1.2.2.3" xref="S3a.p3.1.m1.1.2.2.3.cmml">i</mi></msub><mo id="S3a.p3.1.m1.1.2.1" xref="S3a.p3.1.m1.1.2.1.cmml">⁢</mo><msubsup id="S3a.p3.1.m1.1.2.3" xref="S3a.p3.1.m1.1.2.3.cmml"><mi id="S3a.p3.1.m1.1.2.3.2.2" xref="S3a.p3.1.m1.1.2.3.2.2.cmml">H</mi><mi id="S3a.p3.1.m1.1.2.3.3" xref="S3a.p3.1.m1.1.2.3.3.cmml">n</mi><mi id="S3a.p3.1.m1.1.2.3.2.3" xref="S3a.p3.1.m1.1.2.3.2.3.cmml">σ</mi></msubsup><mo id="S3a.p3.1.m1.1.2.1a" xref="S3a.p3.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3a.p3.1.m1.1.1.3" xref="S3a.p3.1.m1.1.1.2.cmml"><mo id="S3a.p3.1.m1.1.1.3.1" stretchy="false" xref="S3a.p3.1.m1.1.1.2.1.cmml">|</mo><msub id="S3a.p3.1.m1.1.1.1.1" xref="S3a.p3.1.m1.1.1.1.1.cmml"><mi id="S3a.p3.1.m1.1.1.1.1.2" xref="S3a.p3.1.m1.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p3.1.m1.1.1.1.1.3" xref="S3a.p3.1.m1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3a.p3.1.m1.1.1.3.2" stretchy="false" xref="S3a.p3.1.m1.1.1.2.1.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3a.p3.1.m1.1b"><apply id="S3a.p3.1.m1.1.2.cmml" xref="S3a.p3.1.m1.1.2"><times id="S3a.p3.1.m1.1.2.1.cmml" xref="S3a.p3.1.m1.1.2.1"></times><apply id="S3a.p3.1.m1.1.2.2.cmml" xref="S3a.p3.1.m1.1.2.2"><csymbol cd="ambiguous" id="S3a.p3.1.m1.1.2.2.1.cmml" xref="S3a.p3.1.m1.1.2.2">subscript</csymbol><ci id="S3a.p3.1.m1.1.2.2.2.cmml" xref="S3a.p3.1.m1.1.2.2.2">𝑐</ci><ci id="S3a.p3.1.m1.1.2.2.3.cmml" xref="S3a.p3.1.m1.1.2.2.3">𝑖</ci></apply><apply id="S3a.p3.1.m1.1.2.3.cmml" xref="S3a.p3.1.m1.1.2.3"><csymbol cd="ambiguous" id="S3a.p3.1.m1.1.2.3.1.cmml" xref="S3a.p3.1.m1.1.2.3">subscript</csymbol><apply id="S3a.p3.1.m1.1.2.3.2.cmml" xref="S3a.p3.1.m1.1.2.3"><csymbol cd="ambiguous" id="S3a.p3.1.m1.1.2.3.2.1.cmml" xref="S3a.p3.1.m1.1.2.3">superscript</csymbol><ci id="S3a.p3.1.m1.1.2.3.2.2.cmml" xref="S3a.p3.1.m1.1.2.3.2.2">𝐻</ci><ci id="S3a.p3.1.m1.1.2.3.2.3.cmml" xref="S3a.p3.1.m1.1.2.3.2.3">𝜎</ci></apply><ci id="S3a.p3.1.m1.1.2.3.3.cmml" xref="S3a.p3.1.m1.1.2.3.3">𝑛</ci></apply><apply id="S3a.p3.1.m1.1.1.2.cmml" xref="S3a.p3.1.m1.1.1.3"><csymbol cd="latexml" id="S3a.p3.1.m1.1.1.2.1.cmml" xref="S3a.p3.1.m1.1.1.3.1">ket</csymbol><apply id="S3a.p3.1.m1.1.1.1.1.cmml" xref="S3a.p3.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p3.1.m1.1.1.1.1.1.cmml" xref="S3a.p3.1.m1.1.1.1.1">subscript</csymbol><ci id="S3a.p3.1.m1.1.1.1.1.2.cmml" xref="S3a.p3.1.m1.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p3.1.m1.1.1.1.1.3.cmml" xref="S3a.p3.1.m1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p3.1.m1.1c">c_{i}H^{\sigma}_{n}\ket{\phi_{i}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p3.1.m1.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> in Eq. (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.E22" title="In III Low Entropy Zero-Modes in Truncated Link Models ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">22</span></a>) for TLMs:</p> <ol class="ltx_enumerate" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.2"><math alttext="\ket{\phi_{i}}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.1.m1.1"><semantics id="S3.I1.i1.p1.1.m1.1a"><mrow id="S3.I1.i1.p1.1.m1.1.1.3" xref="S3.I1.i1.p1.1.m1.1.1.2.cmml"><mo id="S3.I1.i1.p1.1.m1.1.1.3.1" stretchy="false" xref="S3.I1.i1.p1.1.m1.1.1.2.1.cmml">|</mo><msub id="S3.I1.i1.p1.1.m1.1.1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.1.1.cmml"><mi id="S3.I1.i1.p1.1.m1.1.1.1.1.2" xref="S3.I1.i1.p1.1.m1.1.1.1.1.2.cmml">ϕ</mi><mi id="S3.I1.i1.p1.1.m1.1.1.1.1.3" xref="S3.I1.i1.p1.1.m1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.I1.i1.p1.1.m1.1.1.3.2" stretchy="false" xref="S3.I1.i1.p1.1.m1.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.1.m1.1b"><apply id="S3.I1.i1.p1.1.m1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3"><csymbol cd="latexml" id="S3.I1.i1.p1.1.m1.1.1.2.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.1">ket</csymbol><apply id="S3.I1.i1.p1.1.m1.1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.1.m1.1.1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S3.I1.i1.p1.1.m1.1.1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1.1.2">italic-ϕ</ci><ci id="S3.I1.i1.p1.1.m1.1.1.1.1.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.1.m1.1c">\ket{\phi_{i}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.1.m1.1d">| start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> is already annihilated by <math alttext="H^{\sigma}_{n}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.2.m2.1"><semantics id="S3.I1.i1.p1.2.m2.1a"><msubsup id="S3.I1.i1.p1.2.m2.1.1" xref="S3.I1.i1.p1.2.m2.1.1.cmml"><mi id="S3.I1.i1.p1.2.m2.1.1.2.2" xref="S3.I1.i1.p1.2.m2.1.1.2.2.cmml">H</mi><mi id="S3.I1.i1.p1.2.m2.1.1.3" xref="S3.I1.i1.p1.2.m2.1.1.3.cmml">n</mi><mi id="S3.I1.i1.p1.2.m2.1.1.2.3" xref="S3.I1.i1.p1.2.m2.1.1.2.3.cmml">σ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.2.m2.1b"><apply id="S3.I1.i1.p1.2.m2.1.1.cmml" xref="S3.I1.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.2.m2.1.1.1.cmml" xref="S3.I1.i1.p1.2.m2.1.1">subscript</csymbol><apply id="S3.I1.i1.p1.2.m2.1.1.2.cmml" xref="S3.I1.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.2.m2.1.1.2.1.cmml" xref="S3.I1.i1.p1.2.m2.1.1">superscript</csymbol><ci id="S3.I1.i1.p1.2.m2.1.1.2.2.cmml" xref="S3.I1.i1.p1.2.m2.1.1.2.2">𝐻</ci><ci id="S3.I1.i1.p1.2.m2.1.1.2.3.cmml" xref="S3.I1.i1.p1.2.m2.1.1.2.3">𝜎</ci></apply><ci id="S3.I1.i1.p1.2.m2.1.1.3.cmml" xref="S3.I1.i1.p1.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.2.m2.1c">H^{\sigma}_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.2.m2.1d">italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>;</p> </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S3.I1.i2.p1"> <p class="ltx_p" id="S3.I1.i2.p1.3">The zero-mode also includes the term <math alttext="-c_{i}\ket{\phi_{k}}=-c_{i}H^{-\sigma^{\prime}}_{n^{\prime}}H^{\sigma}_{n}\ket% {\phi_{i}}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.1.m1.2"><semantics id="S3.I1.i2.p1.1.m1.2a"><mrow id="S3.I1.i2.p1.1.m1.2.3" xref="S3.I1.i2.p1.1.m1.2.3.cmml"><mrow id="S3.I1.i2.p1.1.m1.2.3.2" xref="S3.I1.i2.p1.1.m1.2.3.2.cmml"><mo id="S3.I1.i2.p1.1.m1.2.3.2a" xref="S3.I1.i2.p1.1.m1.2.3.2.cmml">−</mo><mrow id="S3.I1.i2.p1.1.m1.2.3.2.2" xref="S3.I1.i2.p1.1.m1.2.3.2.2.cmml"><msub id="S3.I1.i2.p1.1.m1.2.3.2.2.2" xref="S3.I1.i2.p1.1.m1.2.3.2.2.2.cmml"><mi id="S3.I1.i2.p1.1.m1.2.3.2.2.2.2" xref="S3.I1.i2.p1.1.m1.2.3.2.2.2.2.cmml">c</mi><mi id="S3.I1.i2.p1.1.m1.2.3.2.2.2.3" xref="S3.I1.i2.p1.1.m1.2.3.2.2.2.3.cmml">i</mi></msub><mo id="S3.I1.i2.p1.1.m1.2.3.2.2.1" xref="S3.I1.i2.p1.1.m1.2.3.2.2.1.cmml">⁢</mo><mrow id="S3.I1.i2.p1.1.m1.1.1.3" xref="S3.I1.i2.p1.1.m1.1.1.2.cmml"><mo id="S3.I1.i2.p1.1.m1.1.1.3.1" stretchy="false" xref="S3.I1.i2.p1.1.m1.1.1.2.1.cmml">|</mo><msub id="S3.I1.i2.p1.1.m1.1.1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.1.1.cmml"><mi id="S3.I1.i2.p1.1.m1.1.1.1.1.2" xref="S3.I1.i2.p1.1.m1.1.1.1.1.2.cmml">ϕ</mi><mi id="S3.I1.i2.p1.1.m1.1.1.1.1.3" xref="S3.I1.i2.p1.1.m1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.I1.i2.p1.1.m1.1.1.3.2" stretchy="false" xref="S3.I1.i2.p1.1.m1.1.1.2.1.cmml">⟩</mo></mrow></mrow></mrow><mo id="S3.I1.i2.p1.1.m1.2.3.1" xref="S3.I1.i2.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S3.I1.i2.p1.1.m1.2.3.3" xref="S3.I1.i2.p1.1.m1.2.3.3.cmml"><mo id="S3.I1.i2.p1.1.m1.2.3.3a" xref="S3.I1.i2.p1.1.m1.2.3.3.cmml">−</mo><mrow id="S3.I1.i2.p1.1.m1.2.3.3.2" xref="S3.I1.i2.p1.1.m1.2.3.3.2.cmml"><msub id="S3.I1.i2.p1.1.m1.2.3.3.2.2" xref="S3.I1.i2.p1.1.m1.2.3.3.2.2.cmml"><mi id="S3.I1.i2.p1.1.m1.2.3.3.2.2.2" xref="S3.I1.i2.p1.1.m1.2.3.3.2.2.2.cmml">c</mi><mi id="S3.I1.i2.p1.1.m1.2.3.3.2.2.3" xref="S3.I1.i2.p1.1.m1.2.3.3.2.2.3.cmml">i</mi></msub><mo id="S3.I1.i2.p1.1.m1.2.3.3.2.1" xref="S3.I1.i2.p1.1.m1.2.3.3.2.1.cmml">⁢</mo><msubsup id="S3.I1.i2.p1.1.m1.2.3.3.2.3" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.cmml"><mi id="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.2" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.2.cmml">H</mi><msup id="S3.I1.i2.p1.1.m1.2.3.3.2.3.3" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.3.cmml"><mi id="S3.I1.i2.p1.1.m1.2.3.3.2.3.3.2" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.3.2.cmml">n</mi><mo id="S3.I1.i2.p1.1.m1.2.3.3.2.3.3.3" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.3.3.cmml">′</mo></msup><mrow id="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.cmml"><mo id="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3a" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.cmml">−</mo><msup id="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.2" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.2.cmml"><mi id="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.2.2" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.2.2.cmml">σ</mi><mo id="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.2.3" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.2.3.cmml">′</mo></msup></mrow></msubsup><mo id="S3.I1.i2.p1.1.m1.2.3.3.2.1a" xref="S3.I1.i2.p1.1.m1.2.3.3.2.1.cmml">⁢</mo><msubsup id="S3.I1.i2.p1.1.m1.2.3.3.2.4" xref="S3.I1.i2.p1.1.m1.2.3.3.2.4.cmml"><mi id="S3.I1.i2.p1.1.m1.2.3.3.2.4.2.2" xref="S3.I1.i2.p1.1.m1.2.3.3.2.4.2.2.cmml">H</mi><mi id="S3.I1.i2.p1.1.m1.2.3.3.2.4.3" 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xref="S3.I1.i2.p1.1.m1.2.3.2"><minus id="S3.I1.i2.p1.1.m1.2.3.2.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.2"></minus><apply id="S3.I1.i2.p1.1.m1.2.3.2.2.cmml" xref="S3.I1.i2.p1.1.m1.2.3.2.2"><times id="S3.I1.i2.p1.1.m1.2.3.2.2.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.2.2.1"></times><apply id="S3.I1.i2.p1.1.m1.2.3.2.2.2.cmml" xref="S3.I1.i2.p1.1.m1.2.3.2.2.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.2.3.2.2.2.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.2.2.2">subscript</csymbol><ci id="S3.I1.i2.p1.1.m1.2.3.2.2.2.2.cmml" xref="S3.I1.i2.p1.1.m1.2.3.2.2.2.2">𝑐</ci><ci id="S3.I1.i2.p1.1.m1.2.3.2.2.2.3.cmml" xref="S3.I1.i2.p1.1.m1.2.3.2.2.2.3">𝑖</ci></apply><apply id="S3.I1.i2.p1.1.m1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3"><csymbol cd="latexml" id="S3.I1.i2.p1.1.m1.1.1.2.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.1">ket</csymbol><apply id="S3.I1.i2.p1.1.m1.1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.1.1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S3.I1.i2.p1.1.m1.1.1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.1.2">italic-ϕ</ci><ci id="S3.I1.i2.p1.1.m1.1.1.1.1.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply><apply id="S3.I1.i2.p1.1.m1.2.3.3.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3"><minus id="S3.I1.i2.p1.1.m1.2.3.3.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3"></minus><apply id="S3.I1.i2.p1.1.m1.2.3.3.2.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2"><times id="S3.I1.i2.p1.1.m1.2.3.3.2.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.1"></times><apply id="S3.I1.i2.p1.1.m1.2.3.3.2.2.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.2.3.3.2.2.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.2">subscript</csymbol><ci id="S3.I1.i2.p1.1.m1.2.3.3.2.2.2.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.2.2">𝑐</ci><ci id="S3.I1.i2.p1.1.m1.2.3.3.2.2.3.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.2.3">𝑖</ci></apply><apply id="S3.I1.i2.p1.1.m1.2.3.3.2.3.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.2.3.3.2.3.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3">subscript</csymbol><apply id="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3">superscript</csymbol><ci id="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.2.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.2">𝐻</ci><apply id="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3"><minus id="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3"></minus><apply id="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.2.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.2.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.2">superscript</csymbol><ci id="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.2.2.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.2.2">𝜎</ci><ci id="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.2.3.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.2.3.2.3">′</ci></apply></apply></apply><apply id="S3.I1.i2.p1.1.m1.2.3.3.2.3.3.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.2.3.3.2.3.3.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.3">superscript</csymbol><ci id="S3.I1.i2.p1.1.m1.2.3.3.2.3.3.2.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.3.2">𝑛</ci><ci id="S3.I1.i2.p1.1.m1.2.3.3.2.3.3.3.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.3.3.3">′</ci></apply></apply><apply id="S3.I1.i2.p1.1.m1.2.3.3.2.4.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.4"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.2.3.3.2.4.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.4">subscript</csymbol><apply id="S3.I1.i2.p1.1.m1.2.3.3.2.4.2.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.4"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.2.3.3.2.4.2.1.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.4">superscript</csymbol><ci id="S3.I1.i2.p1.1.m1.2.3.3.2.4.2.2.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.4.2.2">𝐻</ci><ci id="S3.I1.i2.p1.1.m1.2.3.3.2.4.2.3.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.4.2.3">𝜎</ci></apply><ci id="S3.I1.i2.p1.1.m1.2.3.3.2.4.3.cmml" xref="S3.I1.i2.p1.1.m1.2.3.3.2.4.3">𝑛</ci></apply><apply id="S3.I1.i2.p1.1.m1.2.2.2.cmml" xref="S3.I1.i2.p1.1.m1.2.2.3"><csymbol cd="latexml" id="S3.I1.i2.p1.1.m1.2.2.2.1.cmml" xref="S3.I1.i2.p1.1.m1.2.2.3.1">ket</csymbol><apply id="S3.I1.i2.p1.1.m1.2.2.1.1.cmml" xref="S3.I1.i2.p1.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.2.2.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.2.2.1.1">subscript</csymbol><ci id="S3.I1.i2.p1.1.m1.2.2.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.2.2.1.1.2">italic-ϕ</ci><ci id="S3.I1.i2.p1.1.m1.2.2.1.1.3.cmml" xref="S3.I1.i2.p1.1.m1.2.2.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.1.m1.2c">-c_{i}\ket{\phi_{k}}=-c_{i}H^{-\sigma^{\prime}}_{n^{\prime}}H^{\sigma}_{n}\ket% {\phi_{i}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.1.m1.2d">- italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ⟩ = - italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT - italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> with some <math alttext="\sigma^{\prime}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.2.m2.1"><semantics id="S3.I1.i2.p1.2.m2.1a"><msup id="S3.I1.i2.p1.2.m2.1.1" xref="S3.I1.i2.p1.2.m2.1.1.cmml"><mi id="S3.I1.i2.p1.2.m2.1.1.2" xref="S3.I1.i2.p1.2.m2.1.1.2.cmml">σ</mi><mo id="S3.I1.i2.p1.2.m2.1.1.3" xref="S3.I1.i2.p1.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.2.m2.1b"><apply id="S3.I1.i2.p1.2.m2.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.2.m2.1.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1">superscript</csymbol><ci id="S3.I1.i2.p1.2.m2.1.1.2.cmml" xref="S3.I1.i2.p1.2.m2.1.1.2">𝜎</ci><ci id="S3.I1.i2.p1.2.m2.1.1.3.cmml" xref="S3.I1.i2.p1.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.2.m2.1c">\sigma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.2.m2.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="n^{\prime}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.3.m3.1"><semantics id="S3.I1.i2.p1.3.m3.1a"><msup id="S3.I1.i2.p1.3.m3.1.1" xref="S3.I1.i2.p1.3.m3.1.1.cmml"><mi id="S3.I1.i2.p1.3.m3.1.1.2" xref="S3.I1.i2.p1.3.m3.1.1.2.cmml">n</mi><mo id="S3.I1.i2.p1.3.m3.1.1.3" xref="S3.I1.i2.p1.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.3.m3.1b"><apply id="S3.I1.i2.p1.3.m3.1.1.cmml" xref="S3.I1.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.3.m3.1.1.1.cmml" xref="S3.I1.i2.p1.3.m3.1.1">superscript</csymbol><ci id="S3.I1.i2.p1.3.m3.1.1.2.cmml" xref="S3.I1.i2.p1.3.m3.1.1.2">𝑛</ci><ci id="S3.I1.i2.p1.3.m3.1.1.3.cmml" xref="S3.I1.i2.p1.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.3.m3.1c">n^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.3.m3.1d">italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </li> </ol> </div> <div class="ltx_para" id="S3a.p4"> <p class="ltx_p" id="S3a.p4.8">The second option not only assures that <math alttext="H^{\sigma}_{n}\ket{\phi_{i}}-H^{\sigma^{\prime}}_{n^{\prime}}\ket{\phi_{k}}=0" class="ltx_Math" display="inline" id="S3a.p4.1.m1.2"><semantics id="S3a.p4.1.m1.2a"><mrow id="S3a.p4.1.m1.2.3" xref="S3a.p4.1.m1.2.3.cmml"><mrow id="S3a.p4.1.m1.2.3.2" xref="S3a.p4.1.m1.2.3.2.cmml"><mrow id="S3a.p4.1.m1.2.3.2.2" xref="S3a.p4.1.m1.2.3.2.2.cmml"><msubsup id="S3a.p4.1.m1.2.3.2.2.2" xref="S3a.p4.1.m1.2.3.2.2.2.cmml"><mi id="S3a.p4.1.m1.2.3.2.2.2.2.2" xref="S3a.p4.1.m1.2.3.2.2.2.2.2.cmml">H</mi><mi id="S3a.p4.1.m1.2.3.2.2.2.3" xref="S3a.p4.1.m1.2.3.2.2.2.3.cmml">n</mi><mi id="S3a.p4.1.m1.2.3.2.2.2.2.3" xref="S3a.p4.1.m1.2.3.2.2.2.2.3.cmml">σ</mi></msubsup><mo id="S3a.p4.1.m1.2.3.2.2.1" xref="S3a.p4.1.m1.2.3.2.2.1.cmml">⁢</mo><mrow id="S3a.p4.1.m1.1.1.3" xref="S3a.p4.1.m1.1.1.2.cmml"><mo id="S3a.p4.1.m1.1.1.3.1" stretchy="false" xref="S3a.p4.1.m1.1.1.2.1.cmml">|</mo><msub id="S3a.p4.1.m1.1.1.1.1" xref="S3a.p4.1.m1.1.1.1.1.cmml"><mi id="S3a.p4.1.m1.1.1.1.1.2" xref="S3a.p4.1.m1.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p4.1.m1.1.1.1.1.3" xref="S3a.p4.1.m1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3a.p4.1.m1.1.1.3.2" stretchy="false" xref="S3a.p4.1.m1.1.1.2.1.cmml">⟩</mo></mrow></mrow><mo id="S3a.p4.1.m1.2.3.2.1" xref="S3a.p4.1.m1.2.3.2.1.cmml">−</mo><mrow id="S3a.p4.1.m1.2.3.2.3" xref="S3a.p4.1.m1.2.3.2.3.cmml"><msubsup id="S3a.p4.1.m1.2.3.2.3.2" xref="S3a.p4.1.m1.2.3.2.3.2.cmml"><mi id="S3a.p4.1.m1.2.3.2.3.2.2.2" xref="S3a.p4.1.m1.2.3.2.3.2.2.2.cmml">H</mi><msup id="S3a.p4.1.m1.2.3.2.3.2.3" xref="S3a.p4.1.m1.2.3.2.3.2.3.cmml"><mi id="S3a.p4.1.m1.2.3.2.3.2.3.2" xref="S3a.p4.1.m1.2.3.2.3.2.3.2.cmml">n</mi><mo id="S3a.p4.1.m1.2.3.2.3.2.3.3" xref="S3a.p4.1.m1.2.3.2.3.2.3.3.cmml">′</mo></msup><msup id="S3a.p4.1.m1.2.3.2.3.2.2.3" xref="S3a.p4.1.m1.2.3.2.3.2.2.3.cmml"><mi id="S3a.p4.1.m1.2.3.2.3.2.2.3.2" xref="S3a.p4.1.m1.2.3.2.3.2.2.3.2.cmml">σ</mi><mo id="S3a.p4.1.m1.2.3.2.3.2.2.3.3" xref="S3a.p4.1.m1.2.3.2.3.2.2.3.3.cmml">′</mo></msup></msubsup><mo id="S3a.p4.1.m1.2.3.2.3.1" xref="S3a.p4.1.m1.2.3.2.3.1.cmml">⁢</mo><mrow id="S3a.p4.1.m1.2.2.3" xref="S3a.p4.1.m1.2.2.2.cmml"><mo id="S3a.p4.1.m1.2.2.3.1" stretchy="false" xref="S3a.p4.1.m1.2.2.2.1.cmml">|</mo><msub id="S3a.p4.1.m1.2.2.1.1" xref="S3a.p4.1.m1.2.2.1.1.cmml"><mi id="S3a.p4.1.m1.2.2.1.1.2" xref="S3a.p4.1.m1.2.2.1.1.2.cmml">ϕ</mi><mi id="S3a.p4.1.m1.2.2.1.1.3" xref="S3a.p4.1.m1.2.2.1.1.3.cmml">k</mi></msub><mo id="S3a.p4.1.m1.2.2.3.2" stretchy="false" xref="S3a.p4.1.m1.2.2.2.1.cmml">⟩</mo></mrow></mrow></mrow><mo id="S3a.p4.1.m1.2.3.1" xref="S3a.p4.1.m1.2.3.1.cmml">=</mo><mn id="S3a.p4.1.m1.2.3.3" xref="S3a.p4.1.m1.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3a.p4.1.m1.2b"><apply id="S3a.p4.1.m1.2.3.cmml" xref="S3a.p4.1.m1.2.3"><eq id="S3a.p4.1.m1.2.3.1.cmml" xref="S3a.p4.1.m1.2.3.1"></eq><apply id="S3a.p4.1.m1.2.3.2.cmml" xref="S3a.p4.1.m1.2.3.2"><minus id="S3a.p4.1.m1.2.3.2.1.cmml" xref="S3a.p4.1.m1.2.3.2.1"></minus><apply id="S3a.p4.1.m1.2.3.2.2.cmml" xref="S3a.p4.1.m1.2.3.2.2"><times id="S3a.p4.1.m1.2.3.2.2.1.cmml" xref="S3a.p4.1.m1.2.3.2.2.1"></times><apply id="S3a.p4.1.m1.2.3.2.2.2.cmml" xref="S3a.p4.1.m1.2.3.2.2.2"><csymbol cd="ambiguous" id="S3a.p4.1.m1.2.3.2.2.2.1.cmml" xref="S3a.p4.1.m1.2.3.2.2.2">subscript</csymbol><apply id="S3a.p4.1.m1.2.3.2.2.2.2.cmml" xref="S3a.p4.1.m1.2.3.2.2.2"><csymbol cd="ambiguous" id="S3a.p4.1.m1.2.3.2.2.2.2.1.cmml" xref="S3a.p4.1.m1.2.3.2.2.2">superscript</csymbol><ci id="S3a.p4.1.m1.2.3.2.2.2.2.2.cmml" xref="S3a.p4.1.m1.2.3.2.2.2.2.2">𝐻</ci><ci id="S3a.p4.1.m1.2.3.2.2.2.2.3.cmml" xref="S3a.p4.1.m1.2.3.2.2.2.2.3">𝜎</ci></apply><ci id="S3a.p4.1.m1.2.3.2.2.2.3.cmml" xref="S3a.p4.1.m1.2.3.2.2.2.3">𝑛</ci></apply><apply id="S3a.p4.1.m1.1.1.2.cmml" xref="S3a.p4.1.m1.1.1.3"><csymbol cd="latexml" id="S3a.p4.1.m1.1.1.2.1.cmml" xref="S3a.p4.1.m1.1.1.3.1">ket</csymbol><apply id="S3a.p4.1.m1.1.1.1.1.cmml" xref="S3a.p4.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p4.1.m1.1.1.1.1.1.cmml" xref="S3a.p4.1.m1.1.1.1.1">subscript</csymbol><ci id="S3a.p4.1.m1.1.1.1.1.2.cmml" xref="S3a.p4.1.m1.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p4.1.m1.1.1.1.1.3.cmml" xref="S3a.p4.1.m1.1.1.1.1.3">𝑖</ci></apply></apply></apply><apply id="S3a.p4.1.m1.2.3.2.3.cmml" xref="S3a.p4.1.m1.2.3.2.3"><times id="S3a.p4.1.m1.2.3.2.3.1.cmml" xref="S3a.p4.1.m1.2.3.2.3.1"></times><apply id="S3a.p4.1.m1.2.3.2.3.2.cmml" xref="S3a.p4.1.m1.2.3.2.3.2"><csymbol cd="ambiguous" id="S3a.p4.1.m1.2.3.2.3.2.1.cmml" xref="S3a.p4.1.m1.2.3.2.3.2">subscript</csymbol><apply id="S3a.p4.1.m1.2.3.2.3.2.2.cmml" xref="S3a.p4.1.m1.2.3.2.3.2"><csymbol cd="ambiguous" id="S3a.p4.1.m1.2.3.2.3.2.2.1.cmml" xref="S3a.p4.1.m1.2.3.2.3.2">superscript</csymbol><ci id="S3a.p4.1.m1.2.3.2.3.2.2.2.cmml" xref="S3a.p4.1.m1.2.3.2.3.2.2.2">𝐻</ci><apply id="S3a.p4.1.m1.2.3.2.3.2.2.3.cmml" xref="S3a.p4.1.m1.2.3.2.3.2.2.3"><csymbol cd="ambiguous" id="S3a.p4.1.m1.2.3.2.3.2.2.3.1.cmml" xref="S3a.p4.1.m1.2.3.2.3.2.2.3">superscript</csymbol><ci id="S3a.p4.1.m1.2.3.2.3.2.2.3.2.cmml" xref="S3a.p4.1.m1.2.3.2.3.2.2.3.2">𝜎</ci><ci id="S3a.p4.1.m1.2.3.2.3.2.2.3.3.cmml" xref="S3a.p4.1.m1.2.3.2.3.2.2.3.3">′</ci></apply></apply><apply id="S3a.p4.1.m1.2.3.2.3.2.3.cmml" xref="S3a.p4.1.m1.2.3.2.3.2.3"><csymbol cd="ambiguous" id="S3a.p4.1.m1.2.3.2.3.2.3.1.cmml" xref="S3a.p4.1.m1.2.3.2.3.2.3">superscript</csymbol><ci id="S3a.p4.1.m1.2.3.2.3.2.3.2.cmml" xref="S3a.p4.1.m1.2.3.2.3.2.3.2">𝑛</ci><ci id="S3a.p4.1.m1.2.3.2.3.2.3.3.cmml" xref="S3a.p4.1.m1.2.3.2.3.2.3.3">′</ci></apply></apply><apply id="S3a.p4.1.m1.2.2.2.cmml" xref="S3a.p4.1.m1.2.2.3"><csymbol cd="latexml" id="S3a.p4.1.m1.2.2.2.1.cmml" xref="S3a.p4.1.m1.2.2.3.1">ket</csymbol><apply id="S3a.p4.1.m1.2.2.1.1.cmml" xref="S3a.p4.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S3a.p4.1.m1.2.2.1.1.1.cmml" xref="S3a.p4.1.m1.2.2.1.1">subscript</csymbol><ci id="S3a.p4.1.m1.2.2.1.1.2.cmml" xref="S3a.p4.1.m1.2.2.1.1.2">italic-ϕ</ci><ci id="S3a.p4.1.m1.2.2.1.1.3.cmml" xref="S3a.p4.1.m1.2.2.1.1.3">𝑘</ci></apply></apply></apply></apply><cn id="S3a.p4.1.m1.2.3.3.cmml" type="integer" xref="S3a.p4.1.m1.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p4.1.m1.2c">H^{\sigma}_{n}\ket{\phi_{i}}-H^{\sigma^{\prime}}_{n^{\prime}}\ket{\phi_{k}}=0</annotation><annotation encoding="application/x-llamapun" id="S3a.p4.1.m1.2d">italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ - italic_H start_POSTSUPERSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ⟩ = 0</annotation></semantics></math>, but also that <math alttext="H^{-\sigma^{\prime}}_{n^{\prime}}\ket{\phi_{i}}-H^{-\sigma}_{n}\ket{\phi_{k}}=0" class="ltx_Math" display="inline" id="S3a.p4.2.m2.2"><semantics id="S3a.p4.2.m2.2a"><mrow id="S3a.p4.2.m2.2.3" xref="S3a.p4.2.m2.2.3.cmml"><mrow id="S3a.p4.2.m2.2.3.2" xref="S3a.p4.2.m2.2.3.2.cmml"><mrow id="S3a.p4.2.m2.2.3.2.2" xref="S3a.p4.2.m2.2.3.2.2.cmml"><msubsup id="S3a.p4.2.m2.2.3.2.2.2" xref="S3a.p4.2.m2.2.3.2.2.2.cmml"><mi id="S3a.p4.2.m2.2.3.2.2.2.2.2" xref="S3a.p4.2.m2.2.3.2.2.2.2.2.cmml">H</mi><msup id="S3a.p4.2.m2.2.3.2.2.2.3" xref="S3a.p4.2.m2.2.3.2.2.2.3.cmml"><mi id="S3a.p4.2.m2.2.3.2.2.2.3.2" xref="S3a.p4.2.m2.2.3.2.2.2.3.2.cmml">n</mi><mo id="S3a.p4.2.m2.2.3.2.2.2.3.3" xref="S3a.p4.2.m2.2.3.2.2.2.3.3.cmml">′</mo></msup><mrow id="S3a.p4.2.m2.2.3.2.2.2.2.3" xref="S3a.p4.2.m2.2.3.2.2.2.2.3.cmml"><mo id="S3a.p4.2.m2.2.3.2.2.2.2.3a" xref="S3a.p4.2.m2.2.3.2.2.2.2.3.cmml">−</mo><msup id="S3a.p4.2.m2.2.3.2.2.2.2.3.2" xref="S3a.p4.2.m2.2.3.2.2.2.2.3.2.cmml"><mi id="S3a.p4.2.m2.2.3.2.2.2.2.3.2.2" xref="S3a.p4.2.m2.2.3.2.2.2.2.3.2.2.cmml">σ</mi><mo id="S3a.p4.2.m2.2.3.2.2.2.2.3.2.3" xref="S3a.p4.2.m2.2.3.2.2.2.2.3.2.3.cmml">′</mo></msup></mrow></msubsup><mo id="S3a.p4.2.m2.2.3.2.2.1" xref="S3a.p4.2.m2.2.3.2.2.1.cmml">⁢</mo><mrow id="S3a.p4.2.m2.1.1.3" xref="S3a.p4.2.m2.1.1.2.cmml"><mo id="S3a.p4.2.m2.1.1.3.1" stretchy="false" xref="S3a.p4.2.m2.1.1.2.1.cmml">|</mo><msub id="S3a.p4.2.m2.1.1.1.1" xref="S3a.p4.2.m2.1.1.1.1.cmml"><mi id="S3a.p4.2.m2.1.1.1.1.2" xref="S3a.p4.2.m2.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p4.2.m2.1.1.1.1.3" xref="S3a.p4.2.m2.1.1.1.1.3.cmml">i</mi></msub><mo id="S3a.p4.2.m2.1.1.3.2" stretchy="false" xref="S3a.p4.2.m2.1.1.2.1.cmml">⟩</mo></mrow></mrow><mo id="S3a.p4.2.m2.2.3.2.1" xref="S3a.p4.2.m2.2.3.2.1.cmml">−</mo><mrow id="S3a.p4.2.m2.2.3.2.3" xref="S3a.p4.2.m2.2.3.2.3.cmml"><msubsup id="S3a.p4.2.m2.2.3.2.3.2" xref="S3a.p4.2.m2.2.3.2.3.2.cmml"><mi id="S3a.p4.2.m2.2.3.2.3.2.2.2" xref="S3a.p4.2.m2.2.3.2.3.2.2.2.cmml">H</mi><mi id="S3a.p4.2.m2.2.3.2.3.2.3" xref="S3a.p4.2.m2.2.3.2.3.2.3.cmml">n</mi><mrow id="S3a.p4.2.m2.2.3.2.3.2.2.3" xref="S3a.p4.2.m2.2.3.2.3.2.2.3.cmml"><mo id="S3a.p4.2.m2.2.3.2.3.2.2.3a" xref="S3a.p4.2.m2.2.3.2.3.2.2.3.cmml">−</mo><mi id="S3a.p4.2.m2.2.3.2.3.2.2.3.2" xref="S3a.p4.2.m2.2.3.2.3.2.2.3.2.cmml">σ</mi></mrow></msubsup><mo id="S3a.p4.2.m2.2.3.2.3.1" xref="S3a.p4.2.m2.2.3.2.3.1.cmml">⁢</mo><mrow id="S3a.p4.2.m2.2.2.3" xref="S3a.p4.2.m2.2.2.2.cmml"><mo id="S3a.p4.2.m2.2.2.3.1" stretchy="false" xref="S3a.p4.2.m2.2.2.2.1.cmml">|</mo><msub id="S3a.p4.2.m2.2.2.1.1" xref="S3a.p4.2.m2.2.2.1.1.cmml"><mi id="S3a.p4.2.m2.2.2.1.1.2" xref="S3a.p4.2.m2.2.2.1.1.2.cmml">ϕ</mi><mi id="S3a.p4.2.m2.2.2.1.1.3" xref="S3a.p4.2.m2.2.2.1.1.3.cmml">k</mi></msub><mo id="S3a.p4.2.m2.2.2.3.2" stretchy="false" xref="S3a.p4.2.m2.2.2.2.1.cmml">⟩</mo></mrow></mrow></mrow><mo id="S3a.p4.2.m2.2.3.1" xref="S3a.p4.2.m2.2.3.1.cmml">=</mo><mn id="S3a.p4.2.m2.2.3.3" xref="S3a.p4.2.m2.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3a.p4.2.m2.2b"><apply id="S3a.p4.2.m2.2.3.cmml" xref="S3a.p4.2.m2.2.3"><eq id="S3a.p4.2.m2.2.3.1.cmml" xref="S3a.p4.2.m2.2.3.1"></eq><apply id="S3a.p4.2.m2.2.3.2.cmml" xref="S3a.p4.2.m2.2.3.2"><minus id="S3a.p4.2.m2.2.3.2.1.cmml" xref="S3a.p4.2.m2.2.3.2.1"></minus><apply id="S3a.p4.2.m2.2.3.2.2.cmml" xref="S3a.p4.2.m2.2.3.2.2"><times id="S3a.p4.2.m2.2.3.2.2.1.cmml" xref="S3a.p4.2.m2.2.3.2.2.1"></times><apply id="S3a.p4.2.m2.2.3.2.2.2.cmml" xref="S3a.p4.2.m2.2.3.2.2.2"><csymbol cd="ambiguous" id="S3a.p4.2.m2.2.3.2.2.2.1.cmml" xref="S3a.p4.2.m2.2.3.2.2.2">subscript</csymbol><apply id="S3a.p4.2.m2.2.3.2.2.2.2.cmml" xref="S3a.p4.2.m2.2.3.2.2.2"><csymbol cd="ambiguous" id="S3a.p4.2.m2.2.3.2.2.2.2.1.cmml" xref="S3a.p4.2.m2.2.3.2.2.2">superscript</csymbol><ci id="S3a.p4.2.m2.2.3.2.2.2.2.2.cmml" xref="S3a.p4.2.m2.2.3.2.2.2.2.2">𝐻</ci><apply id="S3a.p4.2.m2.2.3.2.2.2.2.3.cmml" xref="S3a.p4.2.m2.2.3.2.2.2.2.3"><minus id="S3a.p4.2.m2.2.3.2.2.2.2.3.1.cmml" xref="S3a.p4.2.m2.2.3.2.2.2.2.3"></minus><apply id="S3a.p4.2.m2.2.3.2.2.2.2.3.2.cmml" xref="S3a.p4.2.m2.2.3.2.2.2.2.3.2"><csymbol cd="ambiguous" id="S3a.p4.2.m2.2.3.2.2.2.2.3.2.1.cmml" xref="S3a.p4.2.m2.2.3.2.2.2.2.3.2">superscript</csymbol><ci id="S3a.p4.2.m2.2.3.2.2.2.2.3.2.2.cmml" xref="S3a.p4.2.m2.2.3.2.2.2.2.3.2.2">𝜎</ci><ci id="S3a.p4.2.m2.2.3.2.2.2.2.3.2.3.cmml" xref="S3a.p4.2.m2.2.3.2.2.2.2.3.2.3">′</ci></apply></apply></apply><apply id="S3a.p4.2.m2.2.3.2.2.2.3.cmml" xref="S3a.p4.2.m2.2.3.2.2.2.3"><csymbol cd="ambiguous" id="S3a.p4.2.m2.2.3.2.2.2.3.1.cmml" xref="S3a.p4.2.m2.2.3.2.2.2.3">superscript</csymbol><ci id="S3a.p4.2.m2.2.3.2.2.2.3.2.cmml" xref="S3a.p4.2.m2.2.3.2.2.2.3.2">𝑛</ci><ci id="S3a.p4.2.m2.2.3.2.2.2.3.3.cmml" xref="S3a.p4.2.m2.2.3.2.2.2.3.3">′</ci></apply></apply><apply id="S3a.p4.2.m2.1.1.2.cmml" xref="S3a.p4.2.m2.1.1.3"><csymbol cd="latexml" id="S3a.p4.2.m2.1.1.2.1.cmml" xref="S3a.p4.2.m2.1.1.3.1">ket</csymbol><apply id="S3a.p4.2.m2.1.1.1.1.cmml" xref="S3a.p4.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p4.2.m2.1.1.1.1.1.cmml" xref="S3a.p4.2.m2.1.1.1.1">subscript</csymbol><ci id="S3a.p4.2.m2.1.1.1.1.2.cmml" xref="S3a.p4.2.m2.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p4.2.m2.1.1.1.1.3.cmml" xref="S3a.p4.2.m2.1.1.1.1.3">𝑖</ci></apply></apply></apply><apply id="S3a.p4.2.m2.2.3.2.3.cmml" xref="S3a.p4.2.m2.2.3.2.3"><times id="S3a.p4.2.m2.2.3.2.3.1.cmml" xref="S3a.p4.2.m2.2.3.2.3.1"></times><apply id="S3a.p4.2.m2.2.3.2.3.2.cmml" xref="S3a.p4.2.m2.2.3.2.3.2"><csymbol cd="ambiguous" id="S3a.p4.2.m2.2.3.2.3.2.1.cmml" xref="S3a.p4.2.m2.2.3.2.3.2">subscript</csymbol><apply id="S3a.p4.2.m2.2.3.2.3.2.2.cmml" xref="S3a.p4.2.m2.2.3.2.3.2"><csymbol cd="ambiguous" id="S3a.p4.2.m2.2.3.2.3.2.2.1.cmml" xref="S3a.p4.2.m2.2.3.2.3.2">superscript</csymbol><ci id="S3a.p4.2.m2.2.3.2.3.2.2.2.cmml" xref="S3a.p4.2.m2.2.3.2.3.2.2.2">𝐻</ci><apply id="S3a.p4.2.m2.2.3.2.3.2.2.3.cmml" xref="S3a.p4.2.m2.2.3.2.3.2.2.3"><minus id="S3a.p4.2.m2.2.3.2.3.2.2.3.1.cmml" xref="S3a.p4.2.m2.2.3.2.3.2.2.3"></minus><ci id="S3a.p4.2.m2.2.3.2.3.2.2.3.2.cmml" xref="S3a.p4.2.m2.2.3.2.3.2.2.3.2">𝜎</ci></apply></apply><ci id="S3a.p4.2.m2.2.3.2.3.2.3.cmml" xref="S3a.p4.2.m2.2.3.2.3.2.3">𝑛</ci></apply><apply id="S3a.p4.2.m2.2.2.2.cmml" xref="S3a.p4.2.m2.2.2.3"><csymbol cd="latexml" id="S3a.p4.2.m2.2.2.2.1.cmml" xref="S3a.p4.2.m2.2.2.3.1">ket</csymbol><apply id="S3a.p4.2.m2.2.2.1.1.cmml" xref="S3a.p4.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S3a.p4.2.m2.2.2.1.1.1.cmml" xref="S3a.p4.2.m2.2.2.1.1">subscript</csymbol><ci id="S3a.p4.2.m2.2.2.1.1.2.cmml" xref="S3a.p4.2.m2.2.2.1.1.2">italic-ϕ</ci><ci id="S3a.p4.2.m2.2.2.1.1.3.cmml" xref="S3a.p4.2.m2.2.2.1.1.3">𝑘</ci></apply></apply></apply></apply><cn id="S3a.p4.2.m2.2.3.3.cmml" type="integer" xref="S3a.p4.2.m2.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p4.2.m2.2c">H^{-\sigma^{\prime}}_{n^{\prime}}\ket{\phi_{i}}-H^{-\sigma}_{n}\ket{\phi_{k}}=0</annotation><annotation encoding="application/x-llamapun" id="S3a.p4.2.m2.2d">italic_H start_POSTSUPERSCRIPT - italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ - italic_H start_POSTSUPERSCRIPT - italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ⟩ = 0</annotation></semantics></math>. This means that the contributions <math alttext="c_{i}H^{-\sigma^{\prime}}_{n^{\prime}}\ket{\phi_{i}}" class="ltx_Math" display="inline" id="S3a.p4.3.m3.1"><semantics id="S3a.p4.3.m3.1a"><mrow id="S3a.p4.3.m3.1.2" xref="S3a.p4.3.m3.1.2.cmml"><msub id="S3a.p4.3.m3.1.2.2" xref="S3a.p4.3.m3.1.2.2.cmml"><mi id="S3a.p4.3.m3.1.2.2.2" xref="S3a.p4.3.m3.1.2.2.2.cmml">c</mi><mi id="S3a.p4.3.m3.1.2.2.3" xref="S3a.p4.3.m3.1.2.2.3.cmml">i</mi></msub><mo id="S3a.p4.3.m3.1.2.1" xref="S3a.p4.3.m3.1.2.1.cmml">⁢</mo><msubsup id="S3a.p4.3.m3.1.2.3" xref="S3a.p4.3.m3.1.2.3.cmml"><mi id="S3a.p4.3.m3.1.2.3.2.2" xref="S3a.p4.3.m3.1.2.3.2.2.cmml">H</mi><msup id="S3a.p4.3.m3.1.2.3.3" xref="S3a.p4.3.m3.1.2.3.3.cmml"><mi id="S3a.p4.3.m3.1.2.3.3.2" xref="S3a.p4.3.m3.1.2.3.3.2.cmml">n</mi><mo id="S3a.p4.3.m3.1.2.3.3.3" xref="S3a.p4.3.m3.1.2.3.3.3.cmml">′</mo></msup><mrow id="S3a.p4.3.m3.1.2.3.2.3" xref="S3a.p4.3.m3.1.2.3.2.3.cmml"><mo id="S3a.p4.3.m3.1.2.3.2.3a" xref="S3a.p4.3.m3.1.2.3.2.3.cmml">−</mo><msup id="S3a.p4.3.m3.1.2.3.2.3.2" xref="S3a.p4.3.m3.1.2.3.2.3.2.cmml"><mi id="S3a.p4.3.m3.1.2.3.2.3.2.2" xref="S3a.p4.3.m3.1.2.3.2.3.2.2.cmml">σ</mi><mo id="S3a.p4.3.m3.1.2.3.2.3.2.3" xref="S3a.p4.3.m3.1.2.3.2.3.2.3.cmml">′</mo></msup></mrow></msubsup><mo id="S3a.p4.3.m3.1.2.1a" xref="S3a.p4.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3a.p4.3.m3.1.1.3" xref="S3a.p4.3.m3.1.1.2.cmml"><mo id="S3a.p4.3.m3.1.1.3.1" stretchy="false" xref="S3a.p4.3.m3.1.1.2.1.cmml">|</mo><msub id="S3a.p4.3.m3.1.1.1.1" xref="S3a.p4.3.m3.1.1.1.1.cmml"><mi id="S3a.p4.3.m3.1.1.1.1.2" xref="S3a.p4.3.m3.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p4.3.m3.1.1.1.1.3" xref="S3a.p4.3.m3.1.1.1.1.3.cmml">i</mi></msub><mo id="S3a.p4.3.m3.1.1.3.2" stretchy="false" xref="S3a.p4.3.m3.1.1.2.1.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3a.p4.3.m3.1b"><apply id="S3a.p4.3.m3.1.2.cmml" xref="S3a.p4.3.m3.1.2"><times id="S3a.p4.3.m3.1.2.1.cmml" xref="S3a.p4.3.m3.1.2.1"></times><apply id="S3a.p4.3.m3.1.2.2.cmml" xref="S3a.p4.3.m3.1.2.2"><csymbol cd="ambiguous" id="S3a.p4.3.m3.1.2.2.1.cmml" xref="S3a.p4.3.m3.1.2.2">subscript</csymbol><ci id="S3a.p4.3.m3.1.2.2.2.cmml" xref="S3a.p4.3.m3.1.2.2.2">𝑐</ci><ci id="S3a.p4.3.m3.1.2.2.3.cmml" xref="S3a.p4.3.m3.1.2.2.3">𝑖</ci></apply><apply id="S3a.p4.3.m3.1.2.3.cmml" xref="S3a.p4.3.m3.1.2.3"><csymbol cd="ambiguous" id="S3a.p4.3.m3.1.2.3.1.cmml" xref="S3a.p4.3.m3.1.2.3">subscript</csymbol><apply id="S3a.p4.3.m3.1.2.3.2.cmml" xref="S3a.p4.3.m3.1.2.3"><csymbol cd="ambiguous" id="S3a.p4.3.m3.1.2.3.2.1.cmml" xref="S3a.p4.3.m3.1.2.3">superscript</csymbol><ci id="S3a.p4.3.m3.1.2.3.2.2.cmml" xref="S3a.p4.3.m3.1.2.3.2.2">𝐻</ci><apply id="S3a.p4.3.m3.1.2.3.2.3.cmml" xref="S3a.p4.3.m3.1.2.3.2.3"><minus id="S3a.p4.3.m3.1.2.3.2.3.1.cmml" xref="S3a.p4.3.m3.1.2.3.2.3"></minus><apply id="S3a.p4.3.m3.1.2.3.2.3.2.cmml" xref="S3a.p4.3.m3.1.2.3.2.3.2"><csymbol cd="ambiguous" id="S3a.p4.3.m3.1.2.3.2.3.2.1.cmml" xref="S3a.p4.3.m3.1.2.3.2.3.2">superscript</csymbol><ci id="S3a.p4.3.m3.1.2.3.2.3.2.2.cmml" xref="S3a.p4.3.m3.1.2.3.2.3.2.2">𝜎</ci><ci id="S3a.p4.3.m3.1.2.3.2.3.2.3.cmml" xref="S3a.p4.3.m3.1.2.3.2.3.2.3">′</ci></apply></apply></apply><apply id="S3a.p4.3.m3.1.2.3.3.cmml" xref="S3a.p4.3.m3.1.2.3.3"><csymbol cd="ambiguous" id="S3a.p4.3.m3.1.2.3.3.1.cmml" xref="S3a.p4.3.m3.1.2.3.3">superscript</csymbol><ci id="S3a.p4.3.m3.1.2.3.3.2.cmml" xref="S3a.p4.3.m3.1.2.3.3.2">𝑛</ci><ci id="S3a.p4.3.m3.1.2.3.3.3.cmml" xref="S3a.p4.3.m3.1.2.3.3.3">′</ci></apply></apply><apply id="S3a.p4.3.m3.1.1.2.cmml" xref="S3a.p4.3.m3.1.1.3"><csymbol cd="latexml" id="S3a.p4.3.m3.1.1.2.1.cmml" xref="S3a.p4.3.m3.1.1.3.1">ket</csymbol><apply id="S3a.p4.3.m3.1.1.1.1.cmml" xref="S3a.p4.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p4.3.m3.1.1.1.1.1.cmml" xref="S3a.p4.3.m3.1.1.1.1">subscript</csymbol><ci id="S3a.p4.3.m3.1.1.1.1.2.cmml" xref="S3a.p4.3.m3.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p4.3.m3.1.1.1.1.3.cmml" xref="S3a.p4.3.m3.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p4.3.m3.1c">c_{i}H^{-\sigma^{\prime}}_{n^{\prime}}\ket{\phi_{i}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p4.3.m3.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT - italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> and <math alttext="c_{k}H^{-\sigma}_{n}\ket{\phi_{k}}" class="ltx_Math" display="inline" id="S3a.p4.4.m4.1"><semantics id="S3a.p4.4.m4.1a"><mrow id="S3a.p4.4.m4.1.2" xref="S3a.p4.4.m4.1.2.cmml"><msub id="S3a.p4.4.m4.1.2.2" xref="S3a.p4.4.m4.1.2.2.cmml"><mi id="S3a.p4.4.m4.1.2.2.2" xref="S3a.p4.4.m4.1.2.2.2.cmml">c</mi><mi id="S3a.p4.4.m4.1.2.2.3" xref="S3a.p4.4.m4.1.2.2.3.cmml">k</mi></msub><mo id="S3a.p4.4.m4.1.2.1" xref="S3a.p4.4.m4.1.2.1.cmml">⁢</mo><msubsup id="S3a.p4.4.m4.1.2.3" xref="S3a.p4.4.m4.1.2.3.cmml"><mi id="S3a.p4.4.m4.1.2.3.2.2" xref="S3a.p4.4.m4.1.2.3.2.2.cmml">H</mi><mi id="S3a.p4.4.m4.1.2.3.3" xref="S3a.p4.4.m4.1.2.3.3.cmml">n</mi><mrow id="S3a.p4.4.m4.1.2.3.2.3" xref="S3a.p4.4.m4.1.2.3.2.3.cmml"><mo id="S3a.p4.4.m4.1.2.3.2.3a" xref="S3a.p4.4.m4.1.2.3.2.3.cmml">−</mo><mi id="S3a.p4.4.m4.1.2.3.2.3.2" xref="S3a.p4.4.m4.1.2.3.2.3.2.cmml">σ</mi></mrow></msubsup><mo id="S3a.p4.4.m4.1.2.1a" xref="S3a.p4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S3a.p4.4.m4.1.1.3" xref="S3a.p4.4.m4.1.1.2.cmml"><mo id="S3a.p4.4.m4.1.1.3.1" stretchy="false" xref="S3a.p4.4.m4.1.1.2.1.cmml">|</mo><msub id="S3a.p4.4.m4.1.1.1.1" xref="S3a.p4.4.m4.1.1.1.1.cmml"><mi id="S3a.p4.4.m4.1.1.1.1.2" xref="S3a.p4.4.m4.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p4.4.m4.1.1.1.1.3" xref="S3a.p4.4.m4.1.1.1.1.3.cmml">k</mi></msub><mo id="S3a.p4.4.m4.1.1.3.2" stretchy="false" xref="S3a.p4.4.m4.1.1.2.1.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3a.p4.4.m4.1b"><apply id="S3a.p4.4.m4.1.2.cmml" xref="S3a.p4.4.m4.1.2"><times id="S3a.p4.4.m4.1.2.1.cmml" xref="S3a.p4.4.m4.1.2.1"></times><apply id="S3a.p4.4.m4.1.2.2.cmml" xref="S3a.p4.4.m4.1.2.2"><csymbol cd="ambiguous" id="S3a.p4.4.m4.1.2.2.1.cmml" xref="S3a.p4.4.m4.1.2.2">subscript</csymbol><ci id="S3a.p4.4.m4.1.2.2.2.cmml" xref="S3a.p4.4.m4.1.2.2.2">𝑐</ci><ci id="S3a.p4.4.m4.1.2.2.3.cmml" xref="S3a.p4.4.m4.1.2.2.3">𝑘</ci></apply><apply id="S3a.p4.4.m4.1.2.3.cmml" xref="S3a.p4.4.m4.1.2.3"><csymbol cd="ambiguous" id="S3a.p4.4.m4.1.2.3.1.cmml" xref="S3a.p4.4.m4.1.2.3">subscript</csymbol><apply id="S3a.p4.4.m4.1.2.3.2.cmml" xref="S3a.p4.4.m4.1.2.3"><csymbol cd="ambiguous" id="S3a.p4.4.m4.1.2.3.2.1.cmml" xref="S3a.p4.4.m4.1.2.3">superscript</csymbol><ci id="S3a.p4.4.m4.1.2.3.2.2.cmml" xref="S3a.p4.4.m4.1.2.3.2.2">𝐻</ci><apply id="S3a.p4.4.m4.1.2.3.2.3.cmml" xref="S3a.p4.4.m4.1.2.3.2.3"><minus id="S3a.p4.4.m4.1.2.3.2.3.1.cmml" xref="S3a.p4.4.m4.1.2.3.2.3"></minus><ci id="S3a.p4.4.m4.1.2.3.2.3.2.cmml" xref="S3a.p4.4.m4.1.2.3.2.3.2">𝜎</ci></apply></apply><ci id="S3a.p4.4.m4.1.2.3.3.cmml" xref="S3a.p4.4.m4.1.2.3.3">𝑛</ci></apply><apply id="S3a.p4.4.m4.1.1.2.cmml" xref="S3a.p4.4.m4.1.1.3"><csymbol cd="latexml" id="S3a.p4.4.m4.1.1.2.1.cmml" xref="S3a.p4.4.m4.1.1.3.1">ket</csymbol><apply id="S3a.p4.4.m4.1.1.1.1.cmml" xref="S3a.p4.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p4.4.m4.1.1.1.1.1.cmml" xref="S3a.p4.4.m4.1.1.1.1">subscript</csymbol><ci id="S3a.p4.4.m4.1.1.1.1.2.cmml" xref="S3a.p4.4.m4.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p4.4.m4.1.1.1.1.3.cmml" xref="S3a.p4.4.m4.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p4.4.m4.1c">c_{k}H^{-\sigma}_{n}\ket{\phi_{k}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p4.4.m4.1d">italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT - italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> also cancel each other. What remains are the terms <math alttext="c_{i}H^{-\sigma}_{n}\ket{\phi_{i}}" class="ltx_Math" display="inline" id="S3a.p4.5.m5.1"><semantics id="S3a.p4.5.m5.1a"><mrow id="S3a.p4.5.m5.1.2" xref="S3a.p4.5.m5.1.2.cmml"><msub id="S3a.p4.5.m5.1.2.2" xref="S3a.p4.5.m5.1.2.2.cmml"><mi id="S3a.p4.5.m5.1.2.2.2" xref="S3a.p4.5.m5.1.2.2.2.cmml">c</mi><mi id="S3a.p4.5.m5.1.2.2.3" xref="S3a.p4.5.m5.1.2.2.3.cmml">i</mi></msub><mo id="S3a.p4.5.m5.1.2.1" xref="S3a.p4.5.m5.1.2.1.cmml">⁢</mo><msubsup id="S3a.p4.5.m5.1.2.3" xref="S3a.p4.5.m5.1.2.3.cmml"><mi id="S3a.p4.5.m5.1.2.3.2.2" xref="S3a.p4.5.m5.1.2.3.2.2.cmml">H</mi><mi id="S3a.p4.5.m5.1.2.3.3" xref="S3a.p4.5.m5.1.2.3.3.cmml">n</mi><mrow id="S3a.p4.5.m5.1.2.3.2.3" xref="S3a.p4.5.m5.1.2.3.2.3.cmml"><mo id="S3a.p4.5.m5.1.2.3.2.3a" xref="S3a.p4.5.m5.1.2.3.2.3.cmml">−</mo><mi id="S3a.p4.5.m5.1.2.3.2.3.2" xref="S3a.p4.5.m5.1.2.3.2.3.2.cmml">σ</mi></mrow></msubsup><mo id="S3a.p4.5.m5.1.2.1a" xref="S3a.p4.5.m5.1.2.1.cmml">⁢</mo><mrow 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id="S3a.p4.8.m8.1.2.2.cmml" xref="S3a.p4.8.m8.1.2.2"><csymbol cd="ambiguous" id="S3a.p4.8.m8.1.2.2.1.cmml" xref="S3a.p4.8.m8.1.2.2">subscript</csymbol><ci id="S3a.p4.8.m8.1.2.2.2.cmml" xref="S3a.p4.8.m8.1.2.2.2">𝑐</ci><ci id="S3a.p4.8.m8.1.2.2.3.cmml" xref="S3a.p4.8.m8.1.2.2.3">𝑘</ci></apply><apply id="S3a.p4.8.m8.1.2.3.cmml" xref="S3a.p4.8.m8.1.2.3"><csymbol cd="ambiguous" id="S3a.p4.8.m8.1.2.3.1.cmml" xref="S3a.p4.8.m8.1.2.3">subscript</csymbol><apply id="S3a.p4.8.m8.1.2.3.2.cmml" xref="S3a.p4.8.m8.1.2.3"><csymbol cd="ambiguous" id="S3a.p4.8.m8.1.2.3.2.1.cmml" xref="S3a.p4.8.m8.1.2.3">superscript</csymbol><ci id="S3a.p4.8.m8.1.2.3.2.2.cmml" xref="S3a.p4.8.m8.1.2.3.2.2">𝐻</ci><ci id="S3a.p4.8.m8.1.2.3.2.3.cmml" xref="S3a.p4.8.m8.1.2.3.2.3">𝜎</ci></apply><ci id="S3a.p4.8.m8.1.2.3.3.cmml" xref="S3a.p4.8.m8.1.2.3.3">𝑛</ci></apply><apply id="S3a.p4.8.m8.1.1.2.cmml" xref="S3a.p4.8.m8.1.1.3"><csymbol cd="latexml" id="S3a.p4.8.m8.1.1.2.1.cmml" xref="S3a.p4.8.m8.1.1.3.1">ket</csymbol><apply id="S3a.p4.8.m8.1.1.1.1.cmml" xref="S3a.p4.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p4.8.m8.1.1.1.1.1.cmml" xref="S3a.p4.8.m8.1.1.1.1">subscript</csymbol><ci id="S3a.p4.8.m8.1.1.1.1.2.cmml" xref="S3a.p4.8.m8.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p4.8.m8.1.1.1.1.3.cmml" xref="S3a.p4.8.m8.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p4.8.m8.1c">c_{k}H^{\sigma}_{n}\ket{\phi_{k}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p4.8.m8.1d">italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> as well as all terms that act on different sites. These also need to be canceled to get a zero-mode. If it is possible to find a set of states that manage to cancel all contributions in this fashion, this state is a good candidate to be a zero-mode with anomalously low entropy.</p> </div> <div class="ltx_para" id="S3a.p5"> <p class="ltx_p" id="S3a.p5.8">Canceling the remaining contributions, could, in principle, be achieved by considering any of these states and acting upon them with another kinetic operator. For example, <math alttext="H^{\sigma}_{n}\ket{\phi_{k}}" class="ltx_Math" display="inline" id="S3a.p5.1.m1.1"><semantics id="S3a.p5.1.m1.1a"><mrow id="S3a.p5.1.m1.1.2" xref="S3a.p5.1.m1.1.2.cmml"><msubsup id="S3a.p5.1.m1.1.2.2" xref="S3a.p5.1.m1.1.2.2.cmml"><mi id="S3a.p5.1.m1.1.2.2.2.2" xref="S3a.p5.1.m1.1.2.2.2.2.cmml">H</mi><mi id="S3a.p5.1.m1.1.2.2.3" xref="S3a.p5.1.m1.1.2.2.3.cmml">n</mi><mi id="S3a.p5.1.m1.1.2.2.2.3" xref="S3a.p5.1.m1.1.2.2.2.3.cmml">σ</mi></msubsup><mo id="S3a.p5.1.m1.1.2.1" xref="S3a.p5.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3a.p5.1.m1.1.1.3" xref="S3a.p5.1.m1.1.1.2.cmml"><mo id="S3a.p5.1.m1.1.1.3.1" stretchy="false" xref="S3a.p5.1.m1.1.1.2.1.cmml">|</mo><msub id="S3a.p5.1.m1.1.1.1.1" xref="S3a.p5.1.m1.1.1.1.1.cmml"><mi id="S3a.p5.1.m1.1.1.1.1.2" xref="S3a.p5.1.m1.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p5.1.m1.1.1.1.1.3" xref="S3a.p5.1.m1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3a.p5.1.m1.1.1.3.2" stretchy="false" 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xref="S3a.p5.2.m2.1.2.3">superscript</csymbol><ci id="S3a.p5.2.m2.1.2.3.2.2.cmml" xref="S3a.p5.2.m2.1.2.3.2.2">𝐻</ci><ci id="S3a.p5.2.m2.1.2.3.2.3.cmml" xref="S3a.p5.2.m2.1.2.3.2.3">𝜎</ci></apply><ci id="S3a.p5.2.m2.1.2.3.3.cmml" xref="S3a.p5.2.m2.1.2.3.3">𝑛</ci></apply><apply id="S3a.p5.2.m2.1.1.2.cmml" xref="S3a.p5.2.m2.1.1.3"><csymbol cd="latexml" id="S3a.p5.2.m2.1.1.2.1.cmml" xref="S3a.p5.2.m2.1.1.3.1">ket</csymbol><apply id="S3a.p5.2.m2.1.1.1.1.cmml" xref="S3a.p5.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p5.2.m2.1.1.1.1.1.cmml" xref="S3a.p5.2.m2.1.1.1.1">subscript</csymbol><ci id="S3a.p5.2.m2.1.1.1.1.2.cmml" xref="S3a.p5.2.m2.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p5.2.m2.1.1.1.1.3.cmml" xref="S3a.p5.2.m2.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p5.2.m2.1c">H^{\sigma^{\prime\prime}}_{n^{\prime\prime}}H^{\sigma}_{n}\ket{\phi_{k}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p5.2.m2.1d">italic_H start_POSTSUPERSCRIPT italic_σ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math>, for arbitrary choices of <math alttext="n^{\prime\prime}" class="ltx_Math" display="inline" id="S3a.p5.3.m3.1"><semantics id="S3a.p5.3.m3.1a"><msup id="S3a.p5.3.m3.1.1" xref="S3a.p5.3.m3.1.1.cmml"><mi id="S3a.p5.3.m3.1.1.2" xref="S3a.p5.3.m3.1.1.2.cmml">n</mi><mo id="S3a.p5.3.m3.1.1.3" xref="S3a.p5.3.m3.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S3a.p5.3.m3.1b"><apply id="S3a.p5.3.m3.1.1.cmml" xref="S3a.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S3a.p5.3.m3.1.1.1.cmml" xref="S3a.p5.3.m3.1.1">superscript</csymbol><ci id="S3a.p5.3.m3.1.1.2.cmml" xref="S3a.p5.3.m3.1.1.2">𝑛</ci><ci id="S3a.p5.3.m3.1.1.3.cmml" xref="S3a.p5.3.m3.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p5.3.m3.1c">n^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S3a.p5.3.m3.1d">italic_n start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\sigma^{\prime\prime}" class="ltx_Math" display="inline" id="S3a.p5.4.m4.1"><semantics id="S3a.p5.4.m4.1a"><msup id="S3a.p5.4.m4.1.1" xref="S3a.p5.4.m4.1.1.cmml"><mi id="S3a.p5.4.m4.1.1.2" xref="S3a.p5.4.m4.1.1.2.cmml">σ</mi><mo id="S3a.p5.4.m4.1.1.3" xref="S3a.p5.4.m4.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S3a.p5.4.m4.1b"><apply id="S3a.p5.4.m4.1.1.cmml" xref="S3a.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S3a.p5.4.m4.1.1.1.cmml" xref="S3a.p5.4.m4.1.1">superscript</csymbol><ci id="S3a.p5.4.m4.1.1.2.cmml" xref="S3a.p5.4.m4.1.1.2">𝜎</ci><ci id="S3a.p5.4.m4.1.1.3.cmml" xref="S3a.p5.4.m4.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p5.4.m4.1c">\sigma^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S3a.p5.4.m4.1d">italic_σ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> that do not annihilate the state. To arrive at the zero-modes in arbitrary integer spin, as described in the main text, we choose to cancel them using the same operators as before, so with the state <math alttext="\ket{\phi_{l}}=H^{-\sigma^{\prime}}_{n^{\prime}}H^{\sigma}_{n}\ket{\phi_{k}}" class="ltx_Math" display="inline" id="S3a.p5.5.m5.2"><semantics id="S3a.p5.5.m5.2a"><mrow id="S3a.p5.5.m5.2.3" xref="S3a.p5.5.m5.2.3.cmml"><mrow id="S3a.p5.5.m5.1.1.3" xref="S3a.p5.5.m5.1.1.2.cmml"><mo id="S3a.p5.5.m5.1.1.3.1" stretchy="false" xref="S3a.p5.5.m5.1.1.2.1.cmml">|</mo><msub id="S3a.p5.5.m5.1.1.1.1" xref="S3a.p5.5.m5.1.1.1.1.cmml"><mi id="S3a.p5.5.m5.1.1.1.1.2" xref="S3a.p5.5.m5.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p5.5.m5.1.1.1.1.3" xref="S3a.p5.5.m5.1.1.1.1.3.cmml">l</mi></msub><mo id="S3a.p5.5.m5.1.1.3.2" stretchy="false" xref="S3a.p5.5.m5.1.1.2.1.cmml">⟩</mo></mrow><mo id="S3a.p5.5.m5.2.3.1" xref="S3a.p5.5.m5.2.3.1.cmml">=</mo><mrow id="S3a.p5.5.m5.2.3.2" xref="S3a.p5.5.m5.2.3.2.cmml"><msubsup id="S3a.p5.5.m5.2.3.2.2" xref="S3a.p5.5.m5.2.3.2.2.cmml"><mi id="S3a.p5.5.m5.2.3.2.2.2.2" xref="S3a.p5.5.m5.2.3.2.2.2.2.cmml">H</mi><msup id="S3a.p5.5.m5.2.3.2.2.3" xref="S3a.p5.5.m5.2.3.2.2.3.cmml"><mi id="S3a.p5.5.m5.2.3.2.2.3.2" xref="S3a.p5.5.m5.2.3.2.2.3.2.cmml">n</mi><mo id="S3a.p5.5.m5.2.3.2.2.3.3" xref="S3a.p5.5.m5.2.3.2.2.3.3.cmml">′</mo></msup><mrow id="S3a.p5.5.m5.2.3.2.2.2.3" xref="S3a.p5.5.m5.2.3.2.2.2.3.cmml"><mo id="S3a.p5.5.m5.2.3.2.2.2.3a" xref="S3a.p5.5.m5.2.3.2.2.2.3.cmml">−</mo><msup id="S3a.p5.5.m5.2.3.2.2.2.3.2" xref="S3a.p5.5.m5.2.3.2.2.2.3.2.cmml"><mi id="S3a.p5.5.m5.2.3.2.2.2.3.2.2" xref="S3a.p5.5.m5.2.3.2.2.2.3.2.2.cmml">σ</mi><mo id="S3a.p5.5.m5.2.3.2.2.2.3.2.3" xref="S3a.p5.5.m5.2.3.2.2.2.3.2.3.cmml">′</mo></msup></mrow></msubsup><mo id="S3a.p5.5.m5.2.3.2.1" xref="S3a.p5.5.m5.2.3.2.1.cmml">⁢</mo><msubsup id="S3a.p5.5.m5.2.3.2.3" xref="S3a.p5.5.m5.2.3.2.3.cmml"><mi id="S3a.p5.5.m5.2.3.2.3.2.2" xref="S3a.p5.5.m5.2.3.2.3.2.2.cmml">H</mi><mi id="S3a.p5.5.m5.2.3.2.3.3" xref="S3a.p5.5.m5.2.3.2.3.3.cmml">n</mi><mi id="S3a.p5.5.m5.2.3.2.3.2.3" xref="S3a.p5.5.m5.2.3.2.3.2.3.cmml">σ</mi></msubsup><mo id="S3a.p5.5.m5.2.3.2.1a" xref="S3a.p5.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S3a.p5.5.m5.2.2.3" xref="S3a.p5.5.m5.2.2.2.cmml"><mo id="S3a.p5.5.m5.2.2.3.1" stretchy="false" xref="S3a.p5.5.m5.2.2.2.1.cmml">|</mo><msub id="S3a.p5.5.m5.2.2.1.1" xref="S3a.p5.5.m5.2.2.1.1.cmml"><mi id="S3a.p5.5.m5.2.2.1.1.2" xref="S3a.p5.5.m5.2.2.1.1.2.cmml">ϕ</mi><mi id="S3a.p5.5.m5.2.2.1.1.3" xref="S3a.p5.5.m5.2.2.1.1.3.cmml">k</mi></msub><mo id="S3a.p5.5.m5.2.2.3.2" stretchy="false" xref="S3a.p5.5.m5.2.2.2.1.cmml">⟩</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3a.p5.5.m5.2b"><apply id="S3a.p5.5.m5.2.3.cmml" xref="S3a.p5.5.m5.2.3"><eq id="S3a.p5.5.m5.2.3.1.cmml" xref="S3a.p5.5.m5.2.3.1"></eq><apply id="S3a.p5.5.m5.1.1.2.cmml" xref="S3a.p5.5.m5.1.1.3"><csymbol cd="latexml" id="S3a.p5.5.m5.1.1.2.1.cmml" xref="S3a.p5.5.m5.1.1.3.1">ket</csymbol><apply id="S3a.p5.5.m5.1.1.1.1.cmml" xref="S3a.p5.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S3a.p5.5.m5.1.1.1.1.1.cmml" xref="S3a.p5.5.m5.1.1.1.1">subscript</csymbol><ci id="S3a.p5.5.m5.1.1.1.1.2.cmml" xref="S3a.p5.5.m5.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p5.5.m5.1.1.1.1.3.cmml" xref="S3a.p5.5.m5.1.1.1.1.3">𝑙</ci></apply></apply><apply id="S3a.p5.5.m5.2.3.2.cmml" xref="S3a.p5.5.m5.2.3.2"><times id="S3a.p5.5.m5.2.3.2.1.cmml" xref="S3a.p5.5.m5.2.3.2.1"></times><apply id="S3a.p5.5.m5.2.3.2.2.cmml" xref="S3a.p5.5.m5.2.3.2.2"><csymbol cd="ambiguous" id="S3a.p5.5.m5.2.3.2.2.1.cmml" xref="S3a.p5.5.m5.2.3.2.2">subscript</csymbol><apply id="S3a.p5.5.m5.2.3.2.2.2.cmml" xref="S3a.p5.5.m5.2.3.2.2"><csymbol cd="ambiguous" id="S3a.p5.5.m5.2.3.2.2.2.1.cmml" xref="S3a.p5.5.m5.2.3.2.2">superscript</csymbol><ci id="S3a.p5.5.m5.2.3.2.2.2.2.cmml" xref="S3a.p5.5.m5.2.3.2.2.2.2">𝐻</ci><apply id="S3a.p5.5.m5.2.3.2.2.2.3.cmml" xref="S3a.p5.5.m5.2.3.2.2.2.3"><minus id="S3a.p5.5.m5.2.3.2.2.2.3.1.cmml" xref="S3a.p5.5.m5.2.3.2.2.2.3"></minus><apply id="S3a.p5.5.m5.2.3.2.2.2.3.2.cmml" xref="S3a.p5.5.m5.2.3.2.2.2.3.2"><csymbol cd="ambiguous" id="S3a.p5.5.m5.2.3.2.2.2.3.2.1.cmml" xref="S3a.p5.5.m5.2.3.2.2.2.3.2">superscript</csymbol><ci id="S3a.p5.5.m5.2.3.2.2.2.3.2.2.cmml" xref="S3a.p5.5.m5.2.3.2.2.2.3.2.2">𝜎</ci><ci id="S3a.p5.5.m5.2.3.2.2.2.3.2.3.cmml" xref="S3a.p5.5.m5.2.3.2.2.2.3.2.3">′</ci></apply></apply></apply><apply id="S3a.p5.5.m5.2.3.2.2.3.cmml" xref="S3a.p5.5.m5.2.3.2.2.3"><csymbol cd="ambiguous" id="S3a.p5.5.m5.2.3.2.2.3.1.cmml" xref="S3a.p5.5.m5.2.3.2.2.3">superscript</csymbol><ci id="S3a.p5.5.m5.2.3.2.2.3.2.cmml" xref="S3a.p5.5.m5.2.3.2.2.3.2">𝑛</ci><ci id="S3a.p5.5.m5.2.3.2.2.3.3.cmml" xref="S3a.p5.5.m5.2.3.2.2.3.3">′</ci></apply></apply><apply id="S3a.p5.5.m5.2.3.2.3.cmml" xref="S3a.p5.5.m5.2.3.2.3"><csymbol cd="ambiguous" id="S3a.p5.5.m5.2.3.2.3.1.cmml" xref="S3a.p5.5.m5.2.3.2.3">subscript</csymbol><apply id="S3a.p5.5.m5.2.3.2.3.2.cmml" xref="S3a.p5.5.m5.2.3.2.3"><csymbol cd="ambiguous" id="S3a.p5.5.m5.2.3.2.3.2.1.cmml" xref="S3a.p5.5.m5.2.3.2.3">superscript</csymbol><ci id="S3a.p5.5.m5.2.3.2.3.2.2.cmml" xref="S3a.p5.5.m5.2.3.2.3.2.2">𝐻</ci><ci id="S3a.p5.5.m5.2.3.2.3.2.3.cmml" xref="S3a.p5.5.m5.2.3.2.3.2.3">𝜎</ci></apply><ci id="S3a.p5.5.m5.2.3.2.3.3.cmml" xref="S3a.p5.5.m5.2.3.2.3.3">𝑛</ci></apply><apply id="S3a.p5.5.m5.2.2.2.cmml" xref="S3a.p5.5.m5.2.2.3"><csymbol cd="latexml" id="S3a.p5.5.m5.2.2.2.1.cmml" xref="S3a.p5.5.m5.2.2.3.1">ket</csymbol><apply id="S3a.p5.5.m5.2.2.1.1.cmml" xref="S3a.p5.5.m5.2.2.1.1"><csymbol cd="ambiguous" id="S3a.p5.5.m5.2.2.1.1.1.cmml" xref="S3a.p5.5.m5.2.2.1.1">subscript</csymbol><ci id="S3a.p5.5.m5.2.2.1.1.2.cmml" xref="S3a.p5.5.m5.2.2.1.1.2">italic-ϕ</ci><ci id="S3a.p5.5.m5.2.2.1.1.3.cmml" xref="S3a.p5.5.m5.2.2.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p5.5.m5.2c">\ket{\phi_{l}}=H^{-\sigma^{\prime}}_{n^{\prime}}H^{\sigma}_{n}\ket{\phi_{k}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p5.5.m5.2d">| start_ARG italic_ϕ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG ⟩ = italic_H start_POSTSUPERSCRIPT - italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math>. This cancels two contributions at the same time, <math alttext="H^{\sigma}_{n}\ket{\phi_{k}}" class="ltx_Math" display="inline" id="S3a.p5.6.m6.1"><semantics id="S3a.p5.6.m6.1a"><mrow id="S3a.p5.6.m6.1.2" xref="S3a.p5.6.m6.1.2.cmml"><msubsup id="S3a.p5.6.m6.1.2.2" xref="S3a.p5.6.m6.1.2.2.cmml"><mi id="S3a.p5.6.m6.1.2.2.2.2" xref="S3a.p5.6.m6.1.2.2.2.2.cmml">H</mi><mi id="S3a.p5.6.m6.1.2.2.3" xref="S3a.p5.6.m6.1.2.2.3.cmml">n</mi><mi id="S3a.p5.6.m6.1.2.2.2.3" xref="S3a.p5.6.m6.1.2.2.2.3.cmml">σ</mi></msubsup><mo id="S3a.p5.6.m6.1.2.1" xref="S3a.p5.6.m6.1.2.1.cmml">⁢</mo><mrow id="S3a.p5.6.m6.1.1.3" xref="S3a.p5.6.m6.1.1.2.cmml"><mo id="S3a.p5.6.m6.1.1.3.1" stretchy="false" xref="S3a.p5.6.m6.1.1.2.1.cmml">|</mo><msub id="S3a.p5.6.m6.1.1.1.1" xref="S3a.p5.6.m6.1.1.1.1.cmml"><mi id="S3a.p5.6.m6.1.1.1.1.2" xref="S3a.p5.6.m6.1.1.1.1.2.cmml">ϕ</mi><mi id="S3a.p5.6.m6.1.1.1.1.3" xref="S3a.p5.6.m6.1.1.1.1.3.cmml">k</mi></msub><mo id="S3a.p5.6.m6.1.1.3.2" stretchy="false" 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id="S3a.p5.7.m7.1.1.1.1.2.cmml" xref="S3a.p5.7.m7.1.1.1.1.2">italic-ϕ</ci><ci id="S3a.p5.7.m7.1.1.1.1.3.cmml" xref="S3a.p5.7.m7.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p5.7.m7.1c">H^{-\sigma^{\prime}}_{n^{\prime}}\ket{\phi_{k}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p5.7.m7.1d">italic_H start_POSTSUPERSCRIPT - italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math>, further reducing the number of states that need to be included. We then build chains of such states, where both ends are annihilated by the remaining kinetic operators. This way, we build the <math alttext="2\times 1" class="ltx_Math" display="inline" id="S3a.p5.8.m8.1"><semantics id="S3a.p5.8.m8.1a"><mrow id="S3a.p5.8.m8.1.1" xref="S3a.p5.8.m8.1.1.cmml"><mn id="S3a.p5.8.m8.1.1.2" xref="S3a.p5.8.m8.1.1.2.cmml">2</mn><mo id="S3a.p5.8.m8.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3a.p5.8.m8.1.1.1.cmml">×</mo><mn id="S3a.p5.8.m8.1.1.3" xref="S3a.p5.8.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3a.p5.8.m8.1b"><apply id="S3a.p5.8.m8.1.1.cmml" xref="S3a.p5.8.m8.1.1"><times id="S3a.p5.8.m8.1.1.1.cmml" xref="S3a.p5.8.m8.1.1.1"></times><cn id="S3a.p5.8.m8.1.1.2.cmml" type="integer" xref="S3a.p5.8.m8.1.1.2">2</cn><cn id="S3a.p5.8.m8.1.1.3.cmml" type="integer" xref="S3a.p5.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p5.8.m8.1c">2\times 1</annotation><annotation encoding="application/x-llamapun" id="S3a.p5.8.m8.1d">2 × 1</annotation></semantics></math> zero-mode state that can be tiled according to the product defined in the following section.</p> </div> <div class="ltx_para" id="S3a.p6"> <p class="ltx_p" id="S3a.p6.1">All analytically constructed zero-mode scars that have been derived for the <math alttext="S=1/2" class="ltx_Math" display="inline" id="S3a.p6.1.m1.1"><semantics id="S3a.p6.1.m1.1a"><mrow id="S3a.p6.1.m1.1.1" xref="S3a.p6.1.m1.1.1.cmml"><mi id="S3a.p6.1.m1.1.1.2" xref="S3a.p6.1.m1.1.1.2.cmml">S</mi><mo id="S3a.p6.1.m1.1.1.1" xref="S3a.p6.1.m1.1.1.1.cmml">=</mo><mrow id="S3a.p6.1.m1.1.1.3" xref="S3a.p6.1.m1.1.1.3.cmml"><mn id="S3a.p6.1.m1.1.1.3.2" xref="S3a.p6.1.m1.1.1.3.2.cmml">1</mn><mo id="S3a.p6.1.m1.1.1.3.1" xref="S3a.p6.1.m1.1.1.3.1.cmml">/</mo><mn id="S3a.p6.1.m1.1.1.3.3" xref="S3a.p6.1.m1.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3a.p6.1.m1.1b"><apply id="S3a.p6.1.m1.1.1.cmml" xref="S3a.p6.1.m1.1.1"><eq id="S3a.p6.1.m1.1.1.1.cmml" xref="S3a.p6.1.m1.1.1.1"></eq><ci id="S3a.p6.1.m1.1.1.2.cmml" xref="S3a.p6.1.m1.1.1.2">𝑆</ci><apply id="S3a.p6.1.m1.1.1.3.cmml" xref="S3a.p6.1.m1.1.1.3"><divide id="S3a.p6.1.m1.1.1.3.1.cmml" xref="S3a.p6.1.m1.1.1.3.1"></divide><cn id="S3a.p6.1.m1.1.1.3.2.cmml" type="integer" xref="S3a.p6.1.m1.1.1.3.2">1</cn><cn id="S3a.p6.1.m1.1.1.3.3.cmml" type="integer" xref="S3a.p6.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3a.p6.1.m1.1c">S=1/2</annotation><annotation encoding="application/x-llamapun" id="S3a.p6.1.m1.1d">italic_S = 1 / 2</annotation></semantics></math> QLM <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#bib.bib32" title="">32</a>]</cite> have this structure and therefore an equal magnitude of weights for all contributing electric field basis states.</p> </div> <div class="ltx_para" id="S3a.p7"> <p class="ltx_p" id="S3a.p7.1">The outlined strategy can be used to construct more zero-mode scars in systems that are not accessible through ED. For other formulations, like QLMs, it will still be true that states that cancel each other in (<a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S3.E22" title="In III Low Entropy Zero-Modes in Truncated Link Models ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">22</span></a>) differ by two applications of a kinetic operator. This insight was crucial to understanding the low entropy states observed here for TLMs and may prove useful in the analysis of other systems.</p> </div> </section> <section class="ltx_section" id="S4a"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span>The tiling product</h2> <div class="ltx_para" id="S4a.p1"> <p class="ltx_p" id="S4a.p1.1">There are multiple options for generalizing scars found in small systems to arbitrarily large lattices. The most straightforward approach is to apply tensor products in the electric field basis. This is only possible if the links at the boundary agree and the resulting system still follows Gauss’ law. We generalize this to the <em class="ltx_emph ltx_font_italic" id="S4a.p1.1.1">tiling product</em> <math alttext="\odot" class="ltx_Math" display="inline" id="S4a.p1.1.m1.1"><semantics id="S4a.p1.1.m1.1a"><mo id="S4a.p1.1.m1.1.1" xref="S4a.p1.1.m1.1.1.cmml">⊙</mo><annotation-xml encoding="MathML-Content" id="S4a.p1.1.m1.1b"><csymbol cd="latexml" id="S4a.p1.1.m1.1.1.cmml" xref="S4a.p1.1.m1.1.1">direct-product</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4a.p1.1.m1.1c">\odot</annotation><annotation encoding="application/x-llamapun" id="S4a.p1.1.m1.1d">⊙</annotation></semantics></math>, which acts like a tensor product on the dual representation instead. It is defined as</p> <table class="ltx_equation ltx_eqn_table" id="S4.E23"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\ket{M}\odot\ket{M^{\prime}}=\ket{MM^{\prime}}," class="ltx_Math" display="block" id="S4.E23.m1.4"><semantics id="S4.E23.m1.4a"><mrow id="S4.E23.m1.4.4.1" xref="S4.E23.m1.4.4.1.1.cmml"><mrow id="S4.E23.m1.4.4.1.1" xref="S4.E23.m1.4.4.1.1.cmml"><mrow id="S4.E23.m1.4.4.1.1.2" xref="S4.E23.m1.4.4.1.1.2.cmml"><mrow id="S4.E23.m1.1.1.3" xref="S4.E23.m1.1.1.2.cmml"><mo id="S4.E23.m1.1.1.3.1" stretchy="false" xref="S4.E23.m1.1.1.2.1.cmml">|</mo><mi id="S4.E23.m1.1.1.1.1" xref="S4.E23.m1.1.1.1.1.cmml">M</mi><mo id="S4.E23.m1.1.1.3.2" rspace="0.055em" stretchy="false" xref="S4.E23.m1.1.1.2.1.cmml">⟩</mo></mrow><mo id="S4.E23.m1.4.4.1.1.2.1" rspace="0.222em" xref="S4.E23.m1.4.4.1.1.2.1.cmml">⊙</mo><mrow id="S4.E23.m1.2.2.3" xref="S4.E23.m1.2.2.2.cmml"><mo id="S4.E23.m1.2.2.3.1" stretchy="false" xref="S4.E23.m1.2.2.2.1.cmml">|</mo><msup id="S4.E23.m1.2.2.1.1" xref="S4.E23.m1.2.2.1.1.cmml"><mi id="S4.E23.m1.2.2.1.1.2" xref="S4.E23.m1.2.2.1.1.2.cmml">M</mi><mo id="S4.E23.m1.2.2.1.1.3" xref="S4.E23.m1.2.2.1.1.3.cmml">′</mo></msup><mo id="S4.E23.m1.2.2.3.2" stretchy="false" xref="S4.E23.m1.2.2.2.1.cmml">⟩</mo></mrow></mrow><mo id="S4.E23.m1.4.4.1.1.1" xref="S4.E23.m1.4.4.1.1.1.cmml">=</mo><mrow id="S4.E23.m1.3.3.3" xref="S4.E23.m1.3.3.2.cmml"><mo id="S4.E23.m1.3.3.3.1" stretchy="false" xref="S4.E23.m1.3.3.2.1.cmml">|</mo><mrow id="S4.E23.m1.3.3.1.1" xref="S4.E23.m1.3.3.1.1.cmml"><mi id="S4.E23.m1.3.3.1.1.2" xref="S4.E23.m1.3.3.1.1.2.cmml">M</mi><mo id="S4.E23.m1.3.3.1.1.1" xref="S4.E23.m1.3.3.1.1.1.cmml">⁢</mo><msup id="S4.E23.m1.3.3.1.1.3" xref="S4.E23.m1.3.3.1.1.3.cmml"><mi id="S4.E23.m1.3.3.1.1.3.2" xref="S4.E23.m1.3.3.1.1.3.2.cmml">M</mi><mo id="S4.E23.m1.3.3.1.1.3.3" xref="S4.E23.m1.3.3.1.1.3.3.cmml">′</mo></msup></mrow><mo id="S4.E23.m1.3.3.3.2" stretchy="false" xref="S4.E23.m1.3.3.2.1.cmml">⟩</mo></mrow></mrow><mo id="S4.E23.m1.4.4.1.2" xref="S4.E23.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E23.m1.4b"><apply id="S4.E23.m1.4.4.1.1.cmml" xref="S4.E23.m1.4.4.1"><eq id="S4.E23.m1.4.4.1.1.1.cmml" xref="S4.E23.m1.4.4.1.1.1"></eq><apply id="S4.E23.m1.4.4.1.1.2.cmml" xref="S4.E23.m1.4.4.1.1.2"><csymbol cd="latexml" id="S4.E23.m1.4.4.1.1.2.1.cmml" xref="S4.E23.m1.4.4.1.1.2.1">direct-product</csymbol><apply id="S4.E23.m1.1.1.2.cmml" xref="S4.E23.m1.1.1.3"><csymbol cd="latexml" id="S4.E23.m1.1.1.2.1.cmml" xref="S4.E23.m1.1.1.3.1">ket</csymbol><ci id="S4.E23.m1.1.1.1.1.cmml" xref="S4.E23.m1.1.1.1.1">𝑀</ci></apply><apply id="S4.E23.m1.2.2.2.cmml" xref="S4.E23.m1.2.2.3"><csymbol cd="latexml" id="S4.E23.m1.2.2.2.1.cmml" xref="S4.E23.m1.2.2.3.1">ket</csymbol><apply id="S4.E23.m1.2.2.1.1.cmml" xref="S4.E23.m1.2.2.1.1"><csymbol cd="ambiguous" id="S4.E23.m1.2.2.1.1.1.cmml" xref="S4.E23.m1.2.2.1.1">superscript</csymbol><ci id="S4.E23.m1.2.2.1.1.2.cmml" xref="S4.E23.m1.2.2.1.1.2">𝑀</ci><ci id="S4.E23.m1.2.2.1.1.3.cmml" xref="S4.E23.m1.2.2.1.1.3">′</ci></apply></apply></apply><apply id="S4.E23.m1.3.3.2.cmml" xref="S4.E23.m1.3.3.3"><csymbol cd="latexml" id="S4.E23.m1.3.3.2.1.cmml" xref="S4.E23.m1.3.3.3.1">ket</csymbol><apply id="S4.E23.m1.3.3.1.1.cmml" xref="S4.E23.m1.3.3.1.1"><times id="S4.E23.m1.3.3.1.1.1.cmml" xref="S4.E23.m1.3.3.1.1.1"></times><ci id="S4.E23.m1.3.3.1.1.2.cmml" xref="S4.E23.m1.3.3.1.1.2">𝑀</ci><apply id="S4.E23.m1.3.3.1.1.3.cmml" xref="S4.E23.m1.3.3.1.1.3"><csymbol cd="ambiguous" id="S4.E23.m1.3.3.1.1.3.1.cmml" xref="S4.E23.m1.3.3.1.1.3">superscript</csymbol><ci id="S4.E23.m1.3.3.1.1.3.2.cmml" xref="S4.E23.m1.3.3.1.1.3.2">𝑀</ci><ci id="S4.E23.m1.3.3.1.1.3.3.cmml" xref="S4.E23.m1.3.3.1.1.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E23.m1.4c">\ket{M}\odot\ket{M^{\prime}}=\ket{MM^{\prime}},</annotation><annotation encoding="application/x-llamapun" id="S4.E23.m1.4d">| start_ARG italic_M end_ARG ⟩ ⊙ | start_ARG italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG ⟩ = | start_ARG italic_M italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG ⟩ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(23)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4a.p1.7">i.e. the matrices of height variables are concatenated. This is not the tensor product, since the values of the boundary links are added. If the concatenation of <math alttext="M" class="ltx_Math" display="inline" id="S4a.p1.2.m1.1"><semantics id="S4a.p1.2.m1.1a"><mi id="S4a.p1.2.m1.1.1" xref="S4a.p1.2.m1.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="S4a.p1.2.m1.1b"><ci id="S4a.p1.2.m1.1.1.cmml" xref="S4a.p1.2.m1.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4a.p1.2.m1.1c">M</annotation><annotation encoding="application/x-llamapun" id="S4a.p1.2.m1.1d">italic_M</annotation></semantics></math> and <math alttext="M^{\prime}" class="ltx_Math" display="inline" id="S4a.p1.3.m2.1"><semantics id="S4a.p1.3.m2.1a"><msup id="S4a.p1.3.m2.1.1" xref="S4a.p1.3.m2.1.1.cmml"><mi id="S4a.p1.3.m2.1.1.2" xref="S4a.p1.3.m2.1.1.2.cmml">M</mi><mo id="S4a.p1.3.m2.1.1.3" xref="S4a.p1.3.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4a.p1.3.m2.1b"><apply id="S4a.p1.3.m2.1.1.cmml" xref="S4a.p1.3.m2.1.1"><csymbol cd="ambiguous" id="S4a.p1.3.m2.1.1.1.cmml" xref="S4a.p1.3.m2.1.1">superscript</csymbol><ci id="S4a.p1.3.m2.1.1.2.cmml" xref="S4a.p1.3.m2.1.1.2">𝑀</ci><ci id="S4a.p1.3.m2.1.1.3.cmml" xref="S4a.p1.3.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4a.p1.3.m2.1c">M^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4a.p1.3.m2.1d">italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> would give rise to an invalid gauge field configuration, i.e. some neighboring height variables satisfy <math alttext="|h_{i}-h_{j}|>S" class="ltx_Math" display="inline" id="S4a.p1.4.m3.1"><semantics id="S4a.p1.4.m3.1a"><mrow id="S4a.p1.4.m3.1.1" xref="S4a.p1.4.m3.1.1.cmml"><mrow id="S4a.p1.4.m3.1.1.1.1" xref="S4a.p1.4.m3.1.1.1.2.cmml"><mo id="S4a.p1.4.m3.1.1.1.1.2" stretchy="false" xref="S4a.p1.4.m3.1.1.1.2.1.cmml">|</mo><mrow id="S4a.p1.4.m3.1.1.1.1.1" xref="S4a.p1.4.m3.1.1.1.1.1.cmml"><msub id="S4a.p1.4.m3.1.1.1.1.1.2" xref="S4a.p1.4.m3.1.1.1.1.1.2.cmml"><mi id="S4a.p1.4.m3.1.1.1.1.1.2.2" xref="S4a.p1.4.m3.1.1.1.1.1.2.2.cmml">h</mi><mi id="S4a.p1.4.m3.1.1.1.1.1.2.3" xref="S4a.p1.4.m3.1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S4a.p1.4.m3.1.1.1.1.1.1" xref="S4a.p1.4.m3.1.1.1.1.1.1.cmml">−</mo><msub id="S4a.p1.4.m3.1.1.1.1.1.3" xref="S4a.p1.4.m3.1.1.1.1.1.3.cmml"><mi id="S4a.p1.4.m3.1.1.1.1.1.3.2" xref="S4a.p1.4.m3.1.1.1.1.1.3.2.cmml">h</mi><mi id="S4a.p1.4.m3.1.1.1.1.1.3.3" xref="S4a.p1.4.m3.1.1.1.1.1.3.3.cmml">j</mi></msub></mrow><mo id="S4a.p1.4.m3.1.1.1.1.3" stretchy="false" xref="S4a.p1.4.m3.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4a.p1.4.m3.1.1.2" xref="S4a.p1.4.m3.1.1.2.cmml">></mo><mi id="S4a.p1.4.m3.1.1.3" xref="S4a.p1.4.m3.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S4a.p1.4.m3.1b"><apply id="S4a.p1.4.m3.1.1.cmml" xref="S4a.p1.4.m3.1.1"><gt id="S4a.p1.4.m3.1.1.2.cmml" xref="S4a.p1.4.m3.1.1.2"></gt><apply id="S4a.p1.4.m3.1.1.1.2.cmml" xref="S4a.p1.4.m3.1.1.1.1"><abs id="S4a.p1.4.m3.1.1.1.2.1.cmml" xref="S4a.p1.4.m3.1.1.1.1.2"></abs><apply id="S4a.p1.4.m3.1.1.1.1.1.cmml" xref="S4a.p1.4.m3.1.1.1.1.1"><minus id="S4a.p1.4.m3.1.1.1.1.1.1.cmml" xref="S4a.p1.4.m3.1.1.1.1.1.1"></minus><apply id="S4a.p1.4.m3.1.1.1.1.1.2.cmml" xref="S4a.p1.4.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4a.p1.4.m3.1.1.1.1.1.2.1.cmml" xref="S4a.p1.4.m3.1.1.1.1.1.2">subscript</csymbol><ci id="S4a.p1.4.m3.1.1.1.1.1.2.2.cmml" xref="S4a.p1.4.m3.1.1.1.1.1.2.2">ℎ</ci><ci id="S4a.p1.4.m3.1.1.1.1.1.2.3.cmml" xref="S4a.p1.4.m3.1.1.1.1.1.2.3">𝑖</ci></apply><apply id="S4a.p1.4.m3.1.1.1.1.1.3.cmml" xref="S4a.p1.4.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4a.p1.4.m3.1.1.1.1.1.3.1.cmml" xref="S4a.p1.4.m3.1.1.1.1.1.3">subscript</csymbol><ci id="S4a.p1.4.m3.1.1.1.1.1.3.2.cmml" xref="S4a.p1.4.m3.1.1.1.1.1.3.2">ℎ</ci><ci id="S4a.p1.4.m3.1.1.1.1.1.3.3.cmml" xref="S4a.p1.4.m3.1.1.1.1.1.3.3">𝑗</ci></apply></apply></apply><ci id="S4a.p1.4.m3.1.1.3.cmml" xref="S4a.p1.4.m3.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4a.p1.4.m3.1c">|h_{i}-h_{j}|>S</annotation><annotation encoding="application/x-llamapun" id="S4a.p1.4.m3.1d">| italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | > italic_S</annotation></semantics></math>, we will say that <math alttext="\ket{M}" class="ltx_Math" display="inline" id="S4a.p1.5.m4.1"><semantics id="S4a.p1.5.m4.1a"><mrow id="S4a.p1.5.m4.1.1.3" xref="S4a.p1.5.m4.1.1.2.cmml"><mo id="S4a.p1.5.m4.1.1.3.1" stretchy="false" xref="S4a.p1.5.m4.1.1.2.1.cmml">|</mo><mi id="S4a.p1.5.m4.1.1.1.1" xref="S4a.p1.5.m4.1.1.1.1.cmml">M</mi><mo id="S4a.p1.5.m4.1.1.3.2" stretchy="false" xref="S4a.p1.5.m4.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4a.p1.5.m4.1b"><apply id="S4a.p1.5.m4.1.1.2.cmml" xref="S4a.p1.5.m4.1.1.3"><csymbol cd="latexml" id="S4a.p1.5.m4.1.1.2.1.cmml" xref="S4a.p1.5.m4.1.1.3.1">ket</csymbol><ci id="S4a.p1.5.m4.1.1.1.1.cmml" xref="S4a.p1.5.m4.1.1.1.1">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4a.p1.5.m4.1c">\ket{M}</annotation><annotation encoding="application/x-llamapun" id="S4a.p1.5.m4.1d">| start_ARG italic_M end_ARG ⟩</annotation></semantics></math> and <math alttext="\ket{M^{\prime}}" class="ltx_Math" display="inline" id="S4a.p1.6.m5.1"><semantics id="S4a.p1.6.m5.1a"><mrow id="S4a.p1.6.m5.1.1.3" xref="S4a.p1.6.m5.1.1.2.cmml"><mo id="S4a.p1.6.m5.1.1.3.1" stretchy="false" xref="S4a.p1.6.m5.1.1.2.1.cmml">|</mo><msup id="S4a.p1.6.m5.1.1.1.1" xref="S4a.p1.6.m5.1.1.1.1.cmml"><mi id="S4a.p1.6.m5.1.1.1.1.2" xref="S4a.p1.6.m5.1.1.1.1.2.cmml">M</mi><mo id="S4a.p1.6.m5.1.1.1.1.3" xref="S4a.p1.6.m5.1.1.1.1.3.cmml">′</mo></msup><mo id="S4a.p1.6.m5.1.1.3.2" stretchy="false" xref="S4a.p1.6.m5.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4a.p1.6.m5.1b"><apply id="S4a.p1.6.m5.1.1.2.cmml" xref="S4a.p1.6.m5.1.1.3"><csymbol cd="latexml" id="S4a.p1.6.m5.1.1.2.1.cmml" xref="S4a.p1.6.m5.1.1.3.1">ket</csymbol><apply id="S4a.p1.6.m5.1.1.1.1.cmml" xref="S4a.p1.6.m5.1.1.1.1"><csymbol cd="ambiguous" id="S4a.p1.6.m5.1.1.1.1.1.cmml" xref="S4a.p1.6.m5.1.1.1.1">superscript</csymbol><ci id="S4a.p1.6.m5.1.1.1.1.2.cmml" xref="S4a.p1.6.m5.1.1.1.1.2">𝑀</ci><ci id="S4a.p1.6.m5.1.1.1.1.3.cmml" xref="S4a.p1.6.m5.1.1.1.1.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4a.p1.6.m5.1c">\ket{M^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4a.p1.6.m5.1d">| start_ARG italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG ⟩</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S4a.p1.7.1">do not tile</em> and define their tilling product to give the null state. More generally, we are interested in tiling products of the form <math alttext="\left(\sum_{a}c_{a}\ket{M_{a}}\right)\odot\left(\sum_{b}c_{b}\ket{M^{\prime}_{% b}}\right)" class="ltx_Math" display="inline" id="S4a.p1.7.m6.4"><semantics id="S4a.p1.7.m6.4a"><mrow id="S4a.p1.7.m6.4.4" xref="S4a.p1.7.m6.4.4.cmml"><mrow id="S4a.p1.7.m6.3.3.1.1" xref="S4a.p1.7.m6.3.3.1.1.1.cmml"><mo id="S4a.p1.7.m6.3.3.1.1.2" xref="S4a.p1.7.m6.3.3.1.1.1.cmml">(</mo><mrow id="S4a.p1.7.m6.3.3.1.1.1" xref="S4a.p1.7.m6.3.3.1.1.1.cmml"><msub id="S4a.p1.7.m6.3.3.1.1.1.1" xref="S4a.p1.7.m6.3.3.1.1.1.1.cmml"><mo id="S4a.p1.7.m6.3.3.1.1.1.1.2" lspace="0em" xref="S4a.p1.7.m6.3.3.1.1.1.1.2.cmml">∑</mo><mi id="S4a.p1.7.m6.3.3.1.1.1.1.3" xref="S4a.p1.7.m6.3.3.1.1.1.1.3.cmml">a</mi></msub><mrow id="S4a.p1.7.m6.3.3.1.1.1.2" xref="S4a.p1.7.m6.3.3.1.1.1.2.cmml"><msub id="S4a.p1.7.m6.3.3.1.1.1.2.2" xref="S4a.p1.7.m6.3.3.1.1.1.2.2.cmml"><mi id="S4a.p1.7.m6.3.3.1.1.1.2.2.2" xref="S4a.p1.7.m6.3.3.1.1.1.2.2.2.cmml">c</mi><mi id="S4a.p1.7.m6.3.3.1.1.1.2.2.3" xref="S4a.p1.7.m6.3.3.1.1.1.2.2.3.cmml">a</mi></msub><mo id="S4a.p1.7.m6.3.3.1.1.1.2.1" xref="S4a.p1.7.m6.3.3.1.1.1.2.1.cmml">⁢</mo><mrow id="S4a.p1.7.m6.1.1.3" xref="S4a.p1.7.m6.1.1.2.cmml"><mo id="S4a.p1.7.m6.1.1.3.1" stretchy="false" xref="S4a.p1.7.m6.1.1.2.1.cmml">|</mo><msub id="S4a.p1.7.m6.1.1.1.1" xref="S4a.p1.7.m6.1.1.1.1.cmml"><mi id="S4a.p1.7.m6.1.1.1.1.2" xref="S4a.p1.7.m6.1.1.1.1.2.cmml">M</mi><mi id="S4a.p1.7.m6.1.1.1.1.3" xref="S4a.p1.7.m6.1.1.1.1.3.cmml">a</mi></msub><mo id="S4a.p1.7.m6.1.1.3.2" stretchy="false" xref="S4a.p1.7.m6.1.1.2.1.cmml">⟩</mo></mrow></mrow></mrow><mo id="S4a.p1.7.m6.3.3.1.1.3" rspace="0.055em" xref="S4a.p1.7.m6.3.3.1.1.1.cmml">)</mo></mrow><mo id="S4a.p1.7.m6.4.4.3" rspace="0.222em" xref="S4a.p1.7.m6.4.4.3.cmml">⊙</mo><mrow id="S4a.p1.7.m6.4.4.2.1" xref="S4a.p1.7.m6.4.4.2.1.1.cmml"><mo id="S4a.p1.7.m6.4.4.2.1.2" xref="S4a.p1.7.m6.4.4.2.1.1.cmml">(</mo><mrow id="S4a.p1.7.m6.4.4.2.1.1" xref="S4a.p1.7.m6.4.4.2.1.1.cmml"><msub id="S4a.p1.7.m6.4.4.2.1.1.1" xref="S4a.p1.7.m6.4.4.2.1.1.1.cmml"><mo id="S4a.p1.7.m6.4.4.2.1.1.1.2" lspace="0em" xref="S4a.p1.7.m6.4.4.2.1.1.1.2.cmml">∑</mo><mi id="S4a.p1.7.m6.4.4.2.1.1.1.3" xref="S4a.p1.7.m6.4.4.2.1.1.1.3.cmml">b</mi></msub><mrow id="S4a.p1.7.m6.4.4.2.1.1.2" xref="S4a.p1.7.m6.4.4.2.1.1.2.cmml"><msub id="S4a.p1.7.m6.4.4.2.1.1.2.2" xref="S4a.p1.7.m6.4.4.2.1.1.2.2.cmml"><mi id="S4a.p1.7.m6.4.4.2.1.1.2.2.2" xref="S4a.p1.7.m6.4.4.2.1.1.2.2.2.cmml">c</mi><mi id="S4a.p1.7.m6.4.4.2.1.1.2.2.3" xref="S4a.p1.7.m6.4.4.2.1.1.2.2.3.cmml">b</mi></msub><mo id="S4a.p1.7.m6.4.4.2.1.1.2.1" xref="S4a.p1.7.m6.4.4.2.1.1.2.1.cmml">⁢</mo><mrow id="S4a.p1.7.m6.2.2.3" xref="S4a.p1.7.m6.2.2.2.cmml"><mo id="S4a.p1.7.m6.2.2.3.1" stretchy="false" xref="S4a.p1.7.m6.2.2.2.1.cmml">|</mo><msubsup id="S4a.p1.7.m6.2.2.1.1" xref="S4a.p1.7.m6.2.2.1.1.cmml"><mi id="S4a.p1.7.m6.2.2.1.1.2.2" xref="S4a.p1.7.m6.2.2.1.1.2.2.cmml">M</mi><mi id="S4a.p1.7.m6.2.2.1.1.3" xref="S4a.p1.7.m6.2.2.1.1.3.cmml">b</mi><mo id="S4a.p1.7.m6.2.2.1.1.2.3" xref="S4a.p1.7.m6.2.2.1.1.2.3.cmml">′</mo></msubsup><mo id="S4a.p1.7.m6.2.2.3.2" stretchy="false" xref="S4a.p1.7.m6.2.2.2.1.cmml">⟩</mo></mrow></mrow></mrow><mo id="S4a.p1.7.m6.4.4.2.1.3" xref="S4a.p1.7.m6.4.4.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4a.p1.7.m6.4b"><apply id="S4a.p1.7.m6.4.4.cmml" xref="S4a.p1.7.m6.4.4"><csymbol cd="latexml" id="S4a.p1.7.m6.4.4.3.cmml" xref="S4a.p1.7.m6.4.4.3">direct-product</csymbol><apply id="S4a.p1.7.m6.3.3.1.1.1.cmml" xref="S4a.p1.7.m6.3.3.1.1"><apply id="S4a.p1.7.m6.3.3.1.1.1.1.cmml" xref="S4a.p1.7.m6.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4a.p1.7.m6.3.3.1.1.1.1.1.cmml" xref="S4a.p1.7.m6.3.3.1.1.1.1">subscript</csymbol><sum id="S4a.p1.7.m6.3.3.1.1.1.1.2.cmml" xref="S4a.p1.7.m6.3.3.1.1.1.1.2"></sum><ci id="S4a.p1.7.m6.3.3.1.1.1.1.3.cmml" xref="S4a.p1.7.m6.3.3.1.1.1.1.3">𝑎</ci></apply><apply id="S4a.p1.7.m6.3.3.1.1.1.2.cmml" xref="S4a.p1.7.m6.3.3.1.1.1.2"><times id="S4a.p1.7.m6.3.3.1.1.1.2.1.cmml" xref="S4a.p1.7.m6.3.3.1.1.1.2.1"></times><apply id="S4a.p1.7.m6.3.3.1.1.1.2.2.cmml" xref="S4a.p1.7.m6.3.3.1.1.1.2.2"><csymbol cd="ambiguous" id="S4a.p1.7.m6.3.3.1.1.1.2.2.1.cmml" xref="S4a.p1.7.m6.3.3.1.1.1.2.2">subscript</csymbol><ci id="S4a.p1.7.m6.3.3.1.1.1.2.2.2.cmml" xref="S4a.p1.7.m6.3.3.1.1.1.2.2.2">𝑐</ci><ci id="S4a.p1.7.m6.3.3.1.1.1.2.2.3.cmml" xref="S4a.p1.7.m6.3.3.1.1.1.2.2.3">𝑎</ci></apply><apply id="S4a.p1.7.m6.1.1.2.cmml" xref="S4a.p1.7.m6.1.1.3"><csymbol cd="latexml" id="S4a.p1.7.m6.1.1.2.1.cmml" xref="S4a.p1.7.m6.1.1.3.1">ket</csymbol><apply id="S4a.p1.7.m6.1.1.1.1.cmml" xref="S4a.p1.7.m6.1.1.1.1"><csymbol cd="ambiguous" id="S4a.p1.7.m6.1.1.1.1.1.cmml" xref="S4a.p1.7.m6.1.1.1.1">subscript</csymbol><ci id="S4a.p1.7.m6.1.1.1.1.2.cmml" xref="S4a.p1.7.m6.1.1.1.1.2">𝑀</ci><ci id="S4a.p1.7.m6.1.1.1.1.3.cmml" xref="S4a.p1.7.m6.1.1.1.1.3">𝑎</ci></apply></apply></apply></apply><apply 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cd="latexml" id="S4a.p1.7.m6.2.2.2.1.cmml" xref="S4a.p1.7.m6.2.2.3.1">ket</csymbol><apply id="S4a.p1.7.m6.2.2.1.1.cmml" xref="S4a.p1.7.m6.2.2.1.1"><csymbol cd="ambiguous" id="S4a.p1.7.m6.2.2.1.1.1.cmml" xref="S4a.p1.7.m6.2.2.1.1">subscript</csymbol><apply id="S4a.p1.7.m6.2.2.1.1.2.cmml" xref="S4a.p1.7.m6.2.2.1.1"><csymbol cd="ambiguous" id="S4a.p1.7.m6.2.2.1.1.2.1.cmml" xref="S4a.p1.7.m6.2.2.1.1">superscript</csymbol><ci id="S4a.p1.7.m6.2.2.1.1.2.2.cmml" xref="S4a.p1.7.m6.2.2.1.1.2.2">𝑀</ci><ci id="S4a.p1.7.m6.2.2.1.1.2.3.cmml" xref="S4a.p1.7.m6.2.2.1.1.2.3">′</ci></apply><ci id="S4a.p1.7.m6.2.2.1.1.3.cmml" xref="S4a.p1.7.m6.2.2.1.1.3">𝑏</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4a.p1.7.m6.4c">\left(\sum_{a}c_{a}\ket{M_{a}}\right)\odot\left(\sum_{b}c_{b}\ket{M^{\prime}_{% b}}\right)</annotation><annotation encoding="application/x-llamapun" id="S4a.p1.7.m6.4d">( ∑ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT | start_ARG italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_ARG ⟩ ) ⊙ ( ∑ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT | start_ARG italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_ARG ⟩ )</annotation></semantics></math>. We set this product to result in the null state if <em class="ltx_emph ltx_font_italic" id="S4a.p1.7.2">any</em> state in the left-hand side does not tile with <em class="ltx_emph ltx_font_italic" id="S4a.p1.7.3">any</em> state in the right side, otherwise, we just apply the distributive law. In other words, the tiling is distributive if <em class="ltx_emph ltx_font_italic" id="S4a.p1.7.4">all</em> the states in one entry tile with <em class="ltx_emph ltx_font_italic" id="S4a.p1.7.5">all</em> the states in the other.</p> </div> <div class="ltx_para" id="S4a.p2"> <p class="ltx_p" id="S4a.p2.3">For example, states that only contain height variables <math alttext="h_{i}" class="ltx_Math" display="inline" id="S4a.p2.1.m1.1"><semantics id="S4a.p2.1.m1.1a"><msub id="S4a.p2.1.m1.1.1" xref="S4a.p2.1.m1.1.1.cmml"><mi id="S4a.p2.1.m1.1.1.2" xref="S4a.p2.1.m1.1.1.2.cmml">h</mi><mi id="S4a.p2.1.m1.1.1.3" xref="S4a.p2.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4a.p2.1.m1.1b"><apply id="S4a.p2.1.m1.1.1.cmml" xref="S4a.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4a.p2.1.m1.1.1.1.cmml" xref="S4a.p2.1.m1.1.1">subscript</csymbol><ci id="S4a.p2.1.m1.1.1.2.cmml" xref="S4a.p2.1.m1.1.1.2">ℎ</ci><ci id="S4a.p2.1.m1.1.1.3.cmml" xref="S4a.p2.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4a.p2.1.m1.1c">h_{i}</annotation><annotation encoding="application/x-llamapun" id="S4a.p2.1.m1.1d">italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\max_{i}(h_{i})-\min_{i}(h_{i})\leq S" class="ltx_Math" display="inline" id="S4a.p2.2.m2.4"><semantics id="S4a.p2.2.m2.4a"><mrow id="S4a.p2.2.m2.4.4" xref="S4a.p2.2.m2.4.4.cmml"><mrow id="S4a.p2.2.m2.4.4.4" xref="S4a.p2.2.m2.4.4.4.cmml"><mrow id="S4a.p2.2.m2.2.2.2.2.2" xref="S4a.p2.2.m2.2.2.2.2.3.cmml"><msub id="S4a.p2.2.m2.1.1.1.1.1.1" xref="S4a.p2.2.m2.1.1.1.1.1.1.cmml"><mi id="S4a.p2.2.m2.1.1.1.1.1.1.2" xref="S4a.p2.2.m2.1.1.1.1.1.1.2.cmml">max</mi><mi id="S4a.p2.2.m2.1.1.1.1.1.1.3" xref="S4a.p2.2.m2.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4a.p2.2.m2.2.2.2.2.2a" xref="S4a.p2.2.m2.2.2.2.2.3.cmml">⁡</mo><mrow id="S4a.p2.2.m2.2.2.2.2.2.2" xref="S4a.p2.2.m2.2.2.2.2.3.cmml"><mo id="S4a.p2.2.m2.2.2.2.2.2.2.2" stretchy="false" xref="S4a.p2.2.m2.2.2.2.2.3.cmml">(</mo><msub id="S4a.p2.2.m2.2.2.2.2.2.2.1" xref="S4a.p2.2.m2.2.2.2.2.2.2.1.cmml"><mi id="S4a.p2.2.m2.2.2.2.2.2.2.1.2" xref="S4a.p2.2.m2.2.2.2.2.2.2.1.2.cmml">h</mi><mi id="S4a.p2.2.m2.2.2.2.2.2.2.1.3" xref="S4a.p2.2.m2.2.2.2.2.2.2.1.3.cmml">i</mi></msub><mo id="S4a.p2.2.m2.2.2.2.2.2.2.3" stretchy="false" xref="S4a.p2.2.m2.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4a.p2.2.m2.4.4.4.5" xref="S4a.p2.2.m2.4.4.4.5.cmml">−</mo><mrow id="S4a.p2.2.m2.4.4.4.4.2" xref="S4a.p2.2.m2.4.4.4.4.3.cmml"><msub id="S4a.p2.2.m2.3.3.3.3.1.1" xref="S4a.p2.2.m2.3.3.3.3.1.1.cmml"><mi id="S4a.p2.2.m2.3.3.3.3.1.1.2" xref="S4a.p2.2.m2.3.3.3.3.1.1.2.cmml">min</mi><mi id="S4a.p2.2.m2.3.3.3.3.1.1.3" xref="S4a.p2.2.m2.3.3.3.3.1.1.3.cmml">i</mi></msub><mo id="S4a.p2.2.m2.4.4.4.4.2a" xref="S4a.p2.2.m2.4.4.4.4.3.cmml">⁡</mo><mrow id="S4a.p2.2.m2.4.4.4.4.2.2" xref="S4a.p2.2.m2.4.4.4.4.3.cmml"><mo id="S4a.p2.2.m2.4.4.4.4.2.2.2" stretchy="false" xref="S4a.p2.2.m2.4.4.4.4.3.cmml">(</mo><msub id="S4a.p2.2.m2.4.4.4.4.2.2.1" xref="S4a.p2.2.m2.4.4.4.4.2.2.1.cmml"><mi id="S4a.p2.2.m2.4.4.4.4.2.2.1.2" xref="S4a.p2.2.m2.4.4.4.4.2.2.1.2.cmml">h</mi><mi id="S4a.p2.2.m2.4.4.4.4.2.2.1.3" xref="S4a.p2.2.m2.4.4.4.4.2.2.1.3.cmml">i</mi></msub><mo id="S4a.p2.2.m2.4.4.4.4.2.2.3" stretchy="false" xref="S4a.p2.2.m2.4.4.4.4.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4a.p2.2.m2.4.4.5" xref="S4a.p2.2.m2.4.4.5.cmml">≤</mo><mi id="S4a.p2.2.m2.4.4.6" xref="S4a.p2.2.m2.4.4.6.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S4a.p2.2.m2.4b"><apply id="S4a.p2.2.m2.4.4.cmml" xref="S4a.p2.2.m2.4.4"><leq id="S4a.p2.2.m2.4.4.5.cmml" xref="S4a.p2.2.m2.4.4.5"></leq><apply id="S4a.p2.2.m2.4.4.4.cmml" xref="S4a.p2.2.m2.4.4.4"><minus id="S4a.p2.2.m2.4.4.4.5.cmml" xref="S4a.p2.2.m2.4.4.4.5"></minus><apply id="S4a.p2.2.m2.2.2.2.2.3.cmml" xref="S4a.p2.2.m2.2.2.2.2.2"><apply id="S4a.p2.2.m2.1.1.1.1.1.1.cmml" xref="S4a.p2.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4a.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="S4a.p2.2.m2.1.1.1.1.1.1">subscript</csymbol><max id="S4a.p2.2.m2.1.1.1.1.1.1.2.cmml" xref="S4a.p2.2.m2.1.1.1.1.1.1.2"></max><ci id="S4a.p2.2.m2.1.1.1.1.1.1.3.cmml" xref="S4a.p2.2.m2.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S4a.p2.2.m2.2.2.2.2.2.2.1.cmml" xref="S4a.p2.2.m2.2.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4a.p2.2.m2.2.2.2.2.2.2.1.1.cmml" xref="S4a.p2.2.m2.2.2.2.2.2.2.1">subscript</csymbol><ci id="S4a.p2.2.m2.2.2.2.2.2.2.1.2.cmml" xref="S4a.p2.2.m2.2.2.2.2.2.2.1.2">ℎ</ci><ci id="S4a.p2.2.m2.2.2.2.2.2.2.1.3.cmml" xref="S4a.p2.2.m2.2.2.2.2.2.2.1.3">𝑖</ci></apply></apply><apply id="S4a.p2.2.m2.4.4.4.4.3.cmml" xref="S4a.p2.2.m2.4.4.4.4.2"><apply id="S4a.p2.2.m2.3.3.3.3.1.1.cmml" xref="S4a.p2.2.m2.3.3.3.3.1.1"><csymbol cd="ambiguous" id="S4a.p2.2.m2.3.3.3.3.1.1.1.cmml" xref="S4a.p2.2.m2.3.3.3.3.1.1">subscript</csymbol><min id="S4a.p2.2.m2.3.3.3.3.1.1.2.cmml" xref="S4a.p2.2.m2.3.3.3.3.1.1.2"></min><ci id="S4a.p2.2.m2.3.3.3.3.1.1.3.cmml" xref="S4a.p2.2.m2.3.3.3.3.1.1.3">𝑖</ci></apply><apply id="S4a.p2.2.m2.4.4.4.4.2.2.1.cmml" xref="S4a.p2.2.m2.4.4.4.4.2.2.1"><csymbol cd="ambiguous" id="S4a.p2.2.m2.4.4.4.4.2.2.1.1.cmml" xref="S4a.p2.2.m2.4.4.4.4.2.2.1">subscript</csymbol><ci id="S4a.p2.2.m2.4.4.4.4.2.2.1.2.cmml" xref="S4a.p2.2.m2.4.4.4.4.2.2.1.2">ℎ</ci><ci id="S4a.p2.2.m2.4.4.4.4.2.2.1.3.cmml" xref="S4a.p2.2.m2.4.4.4.4.2.2.1.3">𝑖</ci></apply></apply></apply><ci id="S4a.p2.2.m2.4.4.6.cmml" xref="S4a.p2.2.m2.4.4.6">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4a.p2.2.m2.4c">\max_{i}(h_{i})-\min_{i}(h_{i})\leq S</annotation><annotation encoding="application/x-llamapun" id="S4a.p2.2.m2.4d">roman_max start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) - roman_min start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ≤ italic_S</annotation></semantics></math>, will always tile with themselves. If the difference is larger, one might need to be careful which plaquettes can touch, to prevent neighboring height variables from differing by more than <math alttext="S" class="ltx_Math" display="inline" id="S4a.p2.3.m3.1"><semantics id="S4a.p2.3.m3.1a"><mi id="S4a.p2.3.m3.1.1" xref="S4a.p2.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S4a.p2.3.m3.1b"><ci id="S4a.p2.3.m3.1.1.cmml" xref="S4a.p2.3.m3.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4a.p2.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S4a.p2.3.m3.1d">italic_S</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4a.p3"> <p class="ltx_p" id="S4a.p3.2">This tiling product gives non-null states (specifically the scars) for the tiles constructed in the main text since <math alttext="\max_{i}(h_{i})-\min_{i}(h_{i})=S" class="ltx_Math" display="inline" id="S4a.p3.1.m1.4"><semantics id="S4a.p3.1.m1.4a"><mrow id="S4a.p3.1.m1.4.4" xref="S4a.p3.1.m1.4.4.cmml"><mrow id="S4a.p3.1.m1.4.4.4" xref="S4a.p3.1.m1.4.4.4.cmml"><mrow id="S4a.p3.1.m1.2.2.2.2.2" xref="S4a.p3.1.m1.2.2.2.2.3.cmml"><msub id="S4a.p3.1.m1.1.1.1.1.1.1" xref="S4a.p3.1.m1.1.1.1.1.1.1.cmml"><mi id="S4a.p3.1.m1.1.1.1.1.1.1.2" xref="S4a.p3.1.m1.1.1.1.1.1.1.2.cmml">max</mi><mi id="S4a.p3.1.m1.1.1.1.1.1.1.3" xref="S4a.p3.1.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4a.p3.1.m1.2.2.2.2.2a" xref="S4a.p3.1.m1.2.2.2.2.3.cmml">⁡</mo><mrow id="S4a.p3.1.m1.2.2.2.2.2.2" xref="S4a.p3.1.m1.2.2.2.2.3.cmml"><mo id="S4a.p3.1.m1.2.2.2.2.2.2.2" stretchy="false" xref="S4a.p3.1.m1.2.2.2.2.3.cmml">(</mo><msub id="S4a.p3.1.m1.2.2.2.2.2.2.1" xref="S4a.p3.1.m1.2.2.2.2.2.2.1.cmml"><mi id="S4a.p3.1.m1.2.2.2.2.2.2.1.2" xref="S4a.p3.1.m1.2.2.2.2.2.2.1.2.cmml">h</mi><mi id="S4a.p3.1.m1.2.2.2.2.2.2.1.3" xref="S4a.p3.1.m1.2.2.2.2.2.2.1.3.cmml">i</mi></msub><mo id="S4a.p3.1.m1.2.2.2.2.2.2.3" stretchy="false" xref="S4a.p3.1.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4a.p3.1.m1.4.4.4.5" xref="S4a.p3.1.m1.4.4.4.5.cmml">−</mo><mrow id="S4a.p3.1.m1.4.4.4.4.2" xref="S4a.p3.1.m1.4.4.4.4.3.cmml"><msub id="S4a.p3.1.m1.3.3.3.3.1.1" xref="S4a.p3.1.m1.3.3.3.3.1.1.cmml"><mi id="S4a.p3.1.m1.3.3.3.3.1.1.2" xref="S4a.p3.1.m1.3.3.3.3.1.1.2.cmml">min</mi><mi id="S4a.p3.1.m1.3.3.3.3.1.1.3" xref="S4a.p3.1.m1.3.3.3.3.1.1.3.cmml">i</mi></msub><mo id="S4a.p3.1.m1.4.4.4.4.2a" xref="S4a.p3.1.m1.4.4.4.4.3.cmml">⁡</mo><mrow id="S4a.p3.1.m1.4.4.4.4.2.2" xref="S4a.p3.1.m1.4.4.4.4.3.cmml"><mo id="S4a.p3.1.m1.4.4.4.4.2.2.2" stretchy="false" xref="S4a.p3.1.m1.4.4.4.4.3.cmml">(</mo><msub id="S4a.p3.1.m1.4.4.4.4.2.2.1" xref="S4a.p3.1.m1.4.4.4.4.2.2.1.cmml"><mi id="S4a.p3.1.m1.4.4.4.4.2.2.1.2" xref="S4a.p3.1.m1.4.4.4.4.2.2.1.2.cmml">h</mi><mi id="S4a.p3.1.m1.4.4.4.4.2.2.1.3" xref="S4a.p3.1.m1.4.4.4.4.2.2.1.3.cmml">i</mi></msub><mo id="S4a.p3.1.m1.4.4.4.4.2.2.3" stretchy="false" xref="S4a.p3.1.m1.4.4.4.4.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4a.p3.1.m1.4.4.5" xref="S4a.p3.1.m1.4.4.5.cmml">=</mo><mi id="S4a.p3.1.m1.4.4.6" xref="S4a.p3.1.m1.4.4.6.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S4a.p3.1.m1.4b"><apply id="S4a.p3.1.m1.4.4.cmml" xref="S4a.p3.1.m1.4.4"><eq id="S4a.p3.1.m1.4.4.5.cmml" xref="S4a.p3.1.m1.4.4.5"></eq><apply id="S4a.p3.1.m1.4.4.4.cmml" xref="S4a.p3.1.m1.4.4.4"><minus id="S4a.p3.1.m1.4.4.4.5.cmml" xref="S4a.p3.1.m1.4.4.4.5"></minus><apply id="S4a.p3.1.m1.2.2.2.2.3.cmml" xref="S4a.p3.1.m1.2.2.2.2.2"><apply id="S4a.p3.1.m1.1.1.1.1.1.1.cmml" xref="S4a.p3.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4a.p3.1.m1.1.1.1.1.1.1.1.cmml" xref="S4a.p3.1.m1.1.1.1.1.1.1">subscript</csymbol><max id="S4a.p3.1.m1.1.1.1.1.1.1.2.cmml" xref="S4a.p3.1.m1.1.1.1.1.1.1.2"></max><ci id="S4a.p3.1.m1.1.1.1.1.1.1.3.cmml" xref="S4a.p3.1.m1.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S4a.p3.1.m1.2.2.2.2.2.2.1.cmml" xref="S4a.p3.1.m1.2.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S4a.p3.1.m1.2.2.2.2.2.2.1.1.cmml" xref="S4a.p3.1.m1.2.2.2.2.2.2.1">subscript</csymbol><ci id="S4a.p3.1.m1.2.2.2.2.2.2.1.2.cmml" xref="S4a.p3.1.m1.2.2.2.2.2.2.1.2">ℎ</ci><ci id="S4a.p3.1.m1.2.2.2.2.2.2.1.3.cmml" xref="S4a.p3.1.m1.2.2.2.2.2.2.1.3">𝑖</ci></apply></apply><apply id="S4a.p3.1.m1.4.4.4.4.3.cmml" xref="S4a.p3.1.m1.4.4.4.4.2"><apply id="S4a.p3.1.m1.3.3.3.3.1.1.cmml" xref="S4a.p3.1.m1.3.3.3.3.1.1"><csymbol cd="ambiguous" id="S4a.p3.1.m1.3.3.3.3.1.1.1.cmml" xref="S4a.p3.1.m1.3.3.3.3.1.1">subscript</csymbol><min id="S4a.p3.1.m1.3.3.3.3.1.1.2.cmml" xref="S4a.p3.1.m1.3.3.3.3.1.1.2"></min><ci id="S4a.p3.1.m1.3.3.3.3.1.1.3.cmml" xref="S4a.p3.1.m1.3.3.3.3.1.1.3">𝑖</ci></apply><apply id="S4a.p3.1.m1.4.4.4.4.2.2.1.cmml" xref="S4a.p3.1.m1.4.4.4.4.2.2.1"><csymbol cd="ambiguous" id="S4a.p3.1.m1.4.4.4.4.2.2.1.1.cmml" xref="S4a.p3.1.m1.4.4.4.4.2.2.1">subscript</csymbol><ci id="S4a.p3.1.m1.4.4.4.4.2.2.1.2.cmml" xref="S4a.p3.1.m1.4.4.4.4.2.2.1.2">ℎ</ci><ci id="S4a.p3.1.m1.4.4.4.4.2.2.1.3.cmml" xref="S4a.p3.1.m1.4.4.4.4.2.2.1.3">𝑖</ci></apply></apply></apply><ci id="S4a.p3.1.m1.4.4.6.cmml" xref="S4a.p3.1.m1.4.4.6">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4a.p3.1.m1.4c">\max_{i}(h_{i})-\min_{i}(h_{i})=S</annotation><annotation encoding="application/x-llamapun" id="S4a.p3.1.m1.4d">roman_max start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) - roman_min start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = italic_S</annotation></semantics></math>. In general, scars do not need to have a tiling structure. One eigenstate of <math alttext="E^{2}" class="ltx_Math" display="inline" id="S4a.p3.2.m2.1"><semantics id="S4a.p3.2.m2.1a"><msup id="S4a.p3.2.m2.1.1" xref="S4a.p3.2.m2.1.1.cmml"><mi id="S4a.p3.2.m2.1.1.2" xref="S4a.p3.2.m2.1.1.2.cmml">E</mi><mn id="S4a.p3.2.m2.1.1.3" xref="S4a.p3.2.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4a.p3.2.m2.1b"><apply id="S4a.p3.2.m2.1.1.cmml" xref="S4a.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S4a.p3.2.m2.1.1.1.cmml" xref="S4a.p3.2.m2.1.1">superscript</csymbol><ci id="S4a.p3.2.m2.1.1.2.cmml" xref="S4a.p3.2.m2.1.1.2">𝐸</ci><cn id="S4a.p3.2.m2.1.1.3.cmml" type="integer" xref="S4a.p3.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4a.p3.2.m2.1c">E^{2}</annotation><annotation encoding="application/x-llamapun" id="S4a.p3.2.m2.1d">italic_E start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> is not composed of tilings. It is described in the following section.</p> </div> </section> <section class="ltx_section" id="S5a"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">V </span>A <math alttext="4\times 4" class="ltx_Math" display="inline" id="S5a.1.m1.1"><semantics id="S5a.1.m1.1b"><mrow id="S5a.1.m1.1.1" xref="S5a.1.m1.1.1.cmml"><mn id="S5a.1.m1.1.1.2" xref="S5a.1.m1.1.1.2.cmml">4</mn><mo id="S5a.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5a.1.m1.1.1.1.cmml">×</mo><mn id="S5a.1.m1.1.1.3" xref="S5a.1.m1.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S5a.1.m1.1c"><apply id="S5a.1.m1.1.1.cmml" xref="S5a.1.m1.1.1"><times id="S5a.1.m1.1.1.1.cmml" xref="S5a.1.m1.1.1.1"></times><cn id="S5a.1.m1.1.1.2.cmml" type="integer" xref="S5a.1.m1.1.1.2">4</cn><cn id="S5a.1.m1.1.1.3.cmml" type="integer" xref="S5a.1.m1.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5a.1.m1.1d">4\times 4</annotation><annotation encoding="application/x-llamapun" id="S5a.1.m1.1e">4 × 4</annotation></semantics></math> Scar for the <math alttext="E^{2}" class="ltx_Math" display="inline" id="S5a.2.m2.1"><semantics id="S5a.2.m2.1b"><msup id="S5a.2.m2.1.1" xref="S5a.2.m2.1.1.cmml"><mi id="S5a.2.m2.1.1.2" xref="S5a.2.m2.1.1.2.cmml">E</mi><mn id="S5a.2.m2.1.1.3" xref="S5a.2.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S5a.2.m2.1c"><apply id="S5a.2.m2.1.1.cmml" xref="S5a.2.m2.1.1"><csymbol cd="ambiguous" id="S5a.2.m2.1.1.1.cmml" xref="S5a.2.m2.1.1">superscript</csymbol><ci id="S5a.2.m2.1.1.2.cmml" xref="S5a.2.m2.1.1.2">𝐸</ci><cn id="S5a.2.m2.1.1.3.cmml" type="integer" xref="S5a.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5a.2.m2.1d">E^{2}</annotation><annotation encoding="application/x-llamapun" id="S5a.2.m2.1e">italic_E start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> potential</h2> <div class="ltx_para" id="S5a.p1"> <p class="ltx_p" id="S5a.p1.1">The remaining observed scar referred to in the main text seems exclusive of the <math alttext="4\times 4" class="ltx_Math" display="inline" id="S5a.p1.1.m1.1"><semantics id="S5a.p1.1.m1.1a"><mrow id="S5a.p1.1.m1.1.1" xref="S5a.p1.1.m1.1.1.cmml"><mn id="S5a.p1.1.m1.1.1.2" xref="S5a.p1.1.m1.1.1.2.cmml">4</mn><mo id="S5a.p1.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5a.p1.1.m1.1.1.1.cmml">×</mo><mn id="S5a.p1.1.m1.1.1.3" xref="S5a.p1.1.m1.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S5a.p1.1.m1.1b"><apply id="S5a.p1.1.m1.1.1.cmml" xref="S5a.p1.1.m1.1.1"><times id="S5a.p1.1.m1.1.1.1.cmml" xref="S5a.p1.1.m1.1.1.1"></times><cn id="S5a.p1.1.m1.1.1.2.cmml" type="integer" xref="S5a.p1.1.m1.1.1.2">4</cn><cn id="S5a.p1.1.m1.1.1.3.cmml" type="integer" xref="S5a.p1.1.m1.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5a.p1.1.m1.1c">4\times 4</annotation><annotation encoding="application/x-llamapun" id="S5a.p1.1.m1.1d">4 × 4</annotation></semantics></math> system. While the fundamental tiling structure still applies, it comes from a different tiling of the states described in the main text. The two plaquettes in a tile lie in the same row or column but have a distance of 2. Explicitly</p> </div> <div class="ltx_para" id="S5a.p2"> <table class="ltx_equation ltx_eqn_table" id="S5.E24"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\ket{\psi_{s}}=\frac{1}{16}\prod_{(n,n^{\prime})\in T}\left(H^{+}_{n}-H^{+}_{n% ^{\prime}}\right)\ket{\mathbf{0}}," class="ltx_Math" display="block" id="S5.E24.m1.5"><semantics id="S5.E24.m1.5a"><mrow id="S5.E24.m1.5.5.1" xref="S5.E24.m1.5.5.1.1.cmml"><mrow id="S5.E24.m1.5.5.1.1" xref="S5.E24.m1.5.5.1.1.cmml"><mrow id="S5.E24.m1.1.1.3" xref="S5.E24.m1.1.1.2.cmml"><mo id="S5.E24.m1.1.1.3.1" stretchy="false" xref="S5.E24.m1.1.1.2.1.cmml">|</mo><msub id="S5.E24.m1.1.1.1.1" xref="S5.E24.m1.1.1.1.1.cmml"><mi id="S5.E24.m1.1.1.1.1.2" xref="S5.E24.m1.1.1.1.1.2.cmml">ψ</mi><mi id="S5.E24.m1.1.1.1.1.3" xref="S5.E24.m1.1.1.1.1.3.cmml">s</mi></msub><mo id="S5.E24.m1.1.1.3.2" stretchy="false" 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,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(24)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5a.p2.1">where the set of tiles <math alttext="T=\left\{T_{1},\dots,T_{8}\right\}" class="ltx_Math" display="inline" id="S5a.p2.1.m1.3"><semantics id="S5a.p2.1.m1.3a"><mrow id="S5a.p2.1.m1.3.3" xref="S5a.p2.1.m1.3.3.cmml"><mi id="S5a.p2.1.m1.3.3.4" xref="S5a.p2.1.m1.3.3.4.cmml">T</mi><mo id="S5a.p2.1.m1.3.3.3" xref="S5a.p2.1.m1.3.3.3.cmml">=</mo><mrow id="S5a.p2.1.m1.3.3.2.2" xref="S5a.p2.1.m1.3.3.2.3.cmml"><mo id="S5a.p2.1.m1.3.3.2.2.3" xref="S5a.p2.1.m1.3.3.2.3.cmml">{</mo><msub id="S5a.p2.1.m1.2.2.1.1.1" xref="S5a.p2.1.m1.2.2.1.1.1.cmml"><mi id="S5a.p2.1.m1.2.2.1.1.1.2" xref="S5a.p2.1.m1.2.2.1.1.1.2.cmml">T</mi><mn id="S5a.p2.1.m1.2.2.1.1.1.3" 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}</annotation></semantics></math> is presented in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S5.F8" title="Figure 8 ‣ V A 4×4 Scar for the 𝐸² potential ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">8</span></a>.</p> </div> <figure class="ltx_figure" id="S5.F8"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="269" id="S5.F8.g1" src="extracted/5828746/images/scar6.png" width="269"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 8: </span>Schematic representation of a tiling that produces a scar for the <math alttext="E^{2}" class="ltx_Math" display="inline" id="S5.F8.3.m1.1"><semantics id="S5.F8.3.m1.1b"><msup id="S5.F8.3.m1.1.1" xref="S5.F8.3.m1.1.1.cmml"><mi id="S5.F8.3.m1.1.1.2" xref="S5.F8.3.m1.1.1.2.cmml">E</mi><mn id="S5.F8.3.m1.1.1.3" xref="S5.F8.3.m1.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S5.F8.3.m1.1c"><apply id="S5.F8.3.m1.1.1.cmml" xref="S5.F8.3.m1.1.1"><csymbol cd="ambiguous" id="S5.F8.3.m1.1.1.1.cmml" xref="S5.F8.3.m1.1.1">superscript</csymbol><ci id="S5.F8.3.m1.1.1.2.cmml" xref="S5.F8.3.m1.1.1.2">𝐸</ci><cn id="S5.F8.3.m1.1.1.3.cmml" type="integer" xref="S5.F8.3.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F8.3.m1.1d">E^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.F8.3.m1.1e">italic_E start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> potential in a <math alttext="4\times 4" class="ltx_Math" display="inline" id="S5.F8.4.m2.1"><semantics id="S5.F8.4.m2.1b"><mrow id="S5.F8.4.m2.1.1" xref="S5.F8.4.m2.1.1.cmml"><mn id="S5.F8.4.m2.1.1.2" xref="S5.F8.4.m2.1.1.2.cmml">4</mn><mo id="S5.F8.4.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.F8.4.m2.1.1.1.cmml">×</mo><mn id="S5.F8.4.m2.1.1.3" xref="S5.F8.4.m2.1.1.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.F8.4.m2.1c"><apply id="S5.F8.4.m2.1.1.cmml" xref="S5.F8.4.m2.1.1"><times id="S5.F8.4.m2.1.1.1.cmml" xref="S5.F8.4.m2.1.1.1"></times><cn id="S5.F8.4.m2.1.1.2.cmml" type="integer" xref="S5.F8.4.m2.1.1.2">4</cn><cn id="S5.F8.4.m2.1.1.3.cmml" type="integer" xref="S5.F8.4.m2.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.F8.4.m2.1d">4\times 4</annotation><annotation encoding="application/x-llamapun" id="S5.F8.4.m2.1e">4 × 4</annotation></semantics></math> lattice.</figcaption> </figure> <div class="ltx_para" id="S5a.p3"> <p class="ltx_p" id="S5a.p3.2">This tiling guarantees that each <math alttext="T_{i}" class="ltx_Math" display="inline" id="S5a.p3.1.m1.1"><semantics id="S5a.p3.1.m1.1a"><msub id="S5a.p3.1.m1.1.1" xref="S5a.p3.1.m1.1.1.cmml"><mi id="S5a.p3.1.m1.1.1.2" xref="S5a.p3.1.m1.1.1.2.cmml">T</mi><mi id="S5a.p3.1.m1.1.1.3" xref="S5a.p3.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5a.p3.1.m1.1b"><apply id="S5a.p3.1.m1.1.1.cmml" xref="S5a.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S5a.p3.1.m1.1.1.1.cmml" xref="S5a.p3.1.m1.1.1">subscript</csymbol><ci id="S5a.p3.1.m1.1.1.2.cmml" xref="S5a.p3.1.m1.1.1.2">𝑇</ci><ci id="S5a.p3.1.m1.1.1.3.cmml" xref="S5a.p3.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5a.p3.1.m1.1c">T_{i}</annotation><annotation encoding="application/x-llamapun" id="S5a.p3.1.m1.1d">italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> shares the same number of neighboring height variables of value 0 and 1 and therefore will be an eigenstate of the potential with eigenvalue <math alttext="16g" class="ltx_Math" display="inline" id="S5a.p3.2.m2.1"><semantics id="S5a.p3.2.m2.1a"><mrow id="S5a.p3.2.m2.1.1" xref="S5a.p3.2.m2.1.1.cmml"><mn id="S5a.p3.2.m2.1.1.2" xref="S5a.p3.2.m2.1.1.2.cmml">16</mn><mo id="S5a.p3.2.m2.1.1.1" xref="S5a.p3.2.m2.1.1.1.cmml">⁢</mo><mi id="S5a.p3.2.m2.1.1.3" xref="S5a.p3.2.m2.1.1.3.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S5a.p3.2.m2.1b"><apply id="S5a.p3.2.m2.1.1.cmml" xref="S5a.p3.2.m2.1.1"><times id="S5a.p3.2.m2.1.1.1.cmml" xref="S5a.p3.2.m2.1.1.1"></times><cn id="S5a.p3.2.m2.1.1.2.cmml" type="integer" xref="S5a.p3.2.m2.1.1.2">16</cn><ci id="S5a.p3.2.m2.1.1.3.cmml" xref="S5a.p3.2.m2.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5a.p3.2.m2.1c">16g</annotation><annotation encoding="application/x-llamapun" id="S5a.p3.2.m2.1d">16 italic_g</annotation></semantics></math>.</p> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">VI </span>Amplitudes of Scars</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">The amplitudes of the isolated low-entropy states match the predictions of the constructed scars. We demonstrate this in the case of the negative energy scars in <math alttext="S=1" class="ltx_Math" display="inline" id="S6.p1.1.m1.1"><semantics id="S6.p1.1.m1.1a"><mrow id="S6.p1.1.m1.1.1" xref="S6.p1.1.m1.1.1.cmml"><mi id="S6.p1.1.m1.1.1.2" xref="S6.p1.1.m1.1.1.2.cmml">S</mi><mo id="S6.p1.1.m1.1.1.1" xref="S6.p1.1.m1.1.1.1.cmml">=</mo><mn id="S6.p1.1.m1.1.1.3" xref="S6.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.1.m1.1b"><apply id="S6.p1.1.m1.1.1.cmml" xref="S6.p1.1.m1.1.1"><eq id="S6.p1.1.m1.1.1.1.cmml" xref="S6.p1.1.m1.1.1.1"></eq><ci id="S6.p1.1.m1.1.1.2.cmml" xref="S6.p1.1.m1.1.1.2">𝑆</ci><cn id="S6.p1.1.m1.1.1.3.cmml" type="integer" xref="S6.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.1.m1.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="S6.p1.1.m1.1d">italic_S = 1</annotation></semantics></math> single-leg ladders.</p> </div> <div class="ltx_para" id="S6.p2"> <p class="ltx_p" id="S6.p2.6">The predicted scars with negative energy are generated by tilings of <math alttext="\ket{\psi_{-}}" class="ltx_Math" display="inline" id="S6.p2.1.m1.1"><semantics id="S6.p2.1.m1.1a"><mrow id="S6.p2.1.m1.1.1.3" xref="S6.p2.1.m1.1.1.2.cmml"><mo id="S6.p2.1.m1.1.1.3.1" stretchy="false" xref="S6.p2.1.m1.1.1.2.1.cmml">|</mo><msub id="S6.p2.1.m1.1.1.1.1" xref="S6.p2.1.m1.1.1.1.1.cmml"><mi id="S6.p2.1.m1.1.1.1.1.2" xref="S6.p2.1.m1.1.1.1.1.2.cmml">ψ</mi><mo id="S6.p2.1.m1.1.1.1.1.3" xref="S6.p2.1.m1.1.1.1.1.3.cmml">−</mo></msub><mo id="S6.p2.1.m1.1.1.3.2" stretchy="false" xref="S6.p2.1.m1.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.1.m1.1b"><apply id="S6.p2.1.m1.1.1.2.cmml" xref="S6.p2.1.m1.1.1.3"><csymbol cd="latexml" id="S6.p2.1.m1.1.1.2.1.cmml" xref="S6.p2.1.m1.1.1.3.1">ket</csymbol><apply id="S6.p2.1.m1.1.1.1.1.cmml" xref="S6.p2.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S6.p2.1.m1.1.1.1.1.1.cmml" xref="S6.p2.1.m1.1.1.1.1">subscript</csymbol><ci id="S6.p2.1.m1.1.1.1.1.2.cmml" xref="S6.p2.1.m1.1.1.1.1.2">𝜓</ci><minus id="S6.p2.1.m1.1.1.1.1.3.cmml" xref="S6.p2.1.m1.1.1.1.1.3"></minus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.1.m1.1c">\ket{\psi_{-}}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.1.m1.1d">| start_ARG italic_ψ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> defined in the main text. In the single-leg ladder, only two tilings exist. This generates two states of equal energy, which we call <math alttext="\ket{\psi_{1}}" class="ltx_Math" display="inline" id="S6.p2.2.m2.1"><semantics id="S6.p2.2.m2.1a"><mrow id="S6.p2.2.m2.1.1.3" xref="S6.p2.2.m2.1.1.2.cmml"><mo id="S6.p2.2.m2.1.1.3.1" stretchy="false" xref="S6.p2.2.m2.1.1.2.1.cmml">|</mo><msub id="S6.p2.2.m2.1.1.1.1" xref="S6.p2.2.m2.1.1.1.1.cmml"><mi id="S6.p2.2.m2.1.1.1.1.2" xref="S6.p2.2.m2.1.1.1.1.2.cmml">ψ</mi><mn id="S6.p2.2.m2.1.1.1.1.3" xref="S6.p2.2.m2.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.p2.2.m2.1.1.3.2" stretchy="false" xref="S6.p2.2.m2.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.2.m2.1b"><apply id="S6.p2.2.m2.1.1.2.cmml" xref="S6.p2.2.m2.1.1.3"><csymbol cd="latexml" id="S6.p2.2.m2.1.1.2.1.cmml" xref="S6.p2.2.m2.1.1.3.1">ket</csymbol><apply id="S6.p2.2.m2.1.1.1.1.cmml" xref="S6.p2.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S6.p2.2.m2.1.1.1.1.1.cmml" xref="S6.p2.2.m2.1.1.1.1">subscript</csymbol><ci id="S6.p2.2.m2.1.1.1.1.2.cmml" xref="S6.p2.2.m2.1.1.1.1.2">𝜓</ci><cn id="S6.p2.2.m2.1.1.1.1.3.cmml" type="integer" xref="S6.p2.2.m2.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.2.m2.1c">\ket{\psi_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.2.m2.1d">| start_ARG italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> and <math alttext="\ket{\psi_{2}}" class="ltx_Math" display="inline" id="S6.p2.3.m3.1"><semantics id="S6.p2.3.m3.1a"><mrow id="S6.p2.3.m3.1.1.3" xref="S6.p2.3.m3.1.1.2.cmml"><mo id="S6.p2.3.m3.1.1.3.1" stretchy="false" xref="S6.p2.3.m3.1.1.2.1.cmml">|</mo><msub id="S6.p2.3.m3.1.1.1.1" xref="S6.p2.3.m3.1.1.1.1.cmml"><mi id="S6.p2.3.m3.1.1.1.1.2" xref="S6.p2.3.m3.1.1.1.1.2.cmml">ψ</mi><mn id="S6.p2.3.m3.1.1.1.1.3" xref="S6.p2.3.m3.1.1.1.1.3.cmml">2</mn></msub><mo id="S6.p2.3.m3.1.1.3.2" stretchy="false" xref="S6.p2.3.m3.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.3.m3.1b"><apply id="S6.p2.3.m3.1.1.2.cmml" xref="S6.p2.3.m3.1.1.3"><csymbol cd="latexml" id="S6.p2.3.m3.1.1.2.1.cmml" xref="S6.p2.3.m3.1.1.3.1">ket</csymbol><apply id="S6.p2.3.m3.1.1.1.1.cmml" xref="S6.p2.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S6.p2.3.m3.1.1.1.1.1.cmml" xref="S6.p2.3.m3.1.1.1.1">subscript</csymbol><ci id="S6.p2.3.m3.1.1.1.1.2.cmml" xref="S6.p2.3.m3.1.1.1.1.2">𝜓</ci><cn id="S6.p2.3.m3.1.1.1.1.3.cmml" type="integer" xref="S6.p2.3.m3.1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.3.m3.1c">\ket{\psi_{2}}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.3.m3.1d">| start_ARG italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math>. They are linearly independent and will be found in a superposition <math alttext="\ket{\psi_{S}}=a\ket{\psi_{1}}+b\ket{\psi_{2}}" class="ltx_Math" display="inline" id="S6.p2.4.m4.3"><semantics id="S6.p2.4.m4.3a"><mrow id="S6.p2.4.m4.3.4" xref="S6.p2.4.m4.3.4.cmml"><mrow id="S6.p2.4.m4.1.1.3" xref="S6.p2.4.m4.1.1.2.cmml"><mo id="S6.p2.4.m4.1.1.3.1" stretchy="false" xref="S6.p2.4.m4.1.1.2.1.cmml">|</mo><msub id="S6.p2.4.m4.1.1.1.1" xref="S6.p2.4.m4.1.1.1.1.cmml"><mi id="S6.p2.4.m4.1.1.1.1.2" xref="S6.p2.4.m4.1.1.1.1.2.cmml">ψ</mi><mi id="S6.p2.4.m4.1.1.1.1.3" xref="S6.p2.4.m4.1.1.1.1.3.cmml">S</mi></msub><mo id="S6.p2.4.m4.1.1.3.2" stretchy="false" xref="S6.p2.4.m4.1.1.2.1.cmml">⟩</mo></mrow><mo id="S6.p2.4.m4.3.4.1" xref="S6.p2.4.m4.3.4.1.cmml">=</mo><mrow id="S6.p2.4.m4.3.4.2" xref="S6.p2.4.m4.3.4.2.cmml"><mrow id="S6.p2.4.m4.3.4.2.2" xref="S6.p2.4.m4.3.4.2.2.cmml"><mi id="S6.p2.4.m4.3.4.2.2.2" xref="S6.p2.4.m4.3.4.2.2.2.cmml">a</mi><mo id="S6.p2.4.m4.3.4.2.2.1" xref="S6.p2.4.m4.3.4.2.2.1.cmml">⁢</mo><mrow id="S6.p2.4.m4.2.2.3" xref="S6.p2.4.m4.2.2.2.cmml"><mo id="S6.p2.4.m4.2.2.3.1" stretchy="false" xref="S6.p2.4.m4.2.2.2.1.cmml">|</mo><msub id="S6.p2.4.m4.2.2.1.1" xref="S6.p2.4.m4.2.2.1.1.cmml"><mi id="S6.p2.4.m4.2.2.1.1.2" xref="S6.p2.4.m4.2.2.1.1.2.cmml">ψ</mi><mn id="S6.p2.4.m4.2.2.1.1.3" xref="S6.p2.4.m4.2.2.1.1.3.cmml">1</mn></msub><mo id="S6.p2.4.m4.2.2.3.2" stretchy="false" xref="S6.p2.4.m4.2.2.2.1.cmml">⟩</mo></mrow></mrow><mo id="S6.p2.4.m4.3.4.2.1" xref="S6.p2.4.m4.3.4.2.1.cmml">+</mo><mrow id="S6.p2.4.m4.3.4.2.3" xref="S6.p2.4.m4.3.4.2.3.cmml"><mi id="S6.p2.4.m4.3.4.2.3.2" xref="S6.p2.4.m4.3.4.2.3.2.cmml">b</mi><mo id="S6.p2.4.m4.3.4.2.3.1" xref="S6.p2.4.m4.3.4.2.3.1.cmml">⁢</mo><mrow id="S6.p2.4.m4.3.3.3" xref="S6.p2.4.m4.3.3.2.cmml"><mo id="S6.p2.4.m4.3.3.3.1" stretchy="false" xref="S6.p2.4.m4.3.3.2.1.cmml">|</mo><msub id="S6.p2.4.m4.3.3.1.1" xref="S6.p2.4.m4.3.3.1.1.cmml"><mi id="S6.p2.4.m4.3.3.1.1.2" xref="S6.p2.4.m4.3.3.1.1.2.cmml">ψ</mi><mn id="S6.p2.4.m4.3.3.1.1.3" xref="S6.p2.4.m4.3.3.1.1.3.cmml">2</mn></msub><mo id="S6.p2.4.m4.3.3.3.2" stretchy="false" xref="S6.p2.4.m4.3.3.2.1.cmml">⟩</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.4.m4.3b"><apply id="S6.p2.4.m4.3.4.cmml" xref="S6.p2.4.m4.3.4"><eq id="S6.p2.4.m4.3.4.1.cmml" xref="S6.p2.4.m4.3.4.1"></eq><apply id="S6.p2.4.m4.1.1.2.cmml" xref="S6.p2.4.m4.1.1.3"><csymbol cd="latexml" id="S6.p2.4.m4.1.1.2.1.cmml" xref="S6.p2.4.m4.1.1.3.1">ket</csymbol><apply id="S6.p2.4.m4.1.1.1.1.cmml" xref="S6.p2.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S6.p2.4.m4.1.1.1.1.1.cmml" xref="S6.p2.4.m4.1.1.1.1">subscript</csymbol><ci id="S6.p2.4.m4.1.1.1.1.2.cmml" xref="S6.p2.4.m4.1.1.1.1.2">𝜓</ci><ci id="S6.p2.4.m4.1.1.1.1.3.cmml" xref="S6.p2.4.m4.1.1.1.1.3">𝑆</ci></apply></apply><apply id="S6.p2.4.m4.3.4.2.cmml" xref="S6.p2.4.m4.3.4.2"><plus id="S6.p2.4.m4.3.4.2.1.cmml" xref="S6.p2.4.m4.3.4.2.1"></plus><apply id="S6.p2.4.m4.3.4.2.2.cmml" xref="S6.p2.4.m4.3.4.2.2"><times id="S6.p2.4.m4.3.4.2.2.1.cmml" xref="S6.p2.4.m4.3.4.2.2.1"></times><ci id="S6.p2.4.m4.3.4.2.2.2.cmml" xref="S6.p2.4.m4.3.4.2.2.2">𝑎</ci><apply id="S6.p2.4.m4.2.2.2.cmml" xref="S6.p2.4.m4.2.2.3"><csymbol cd="latexml" id="S6.p2.4.m4.2.2.2.1.cmml" xref="S6.p2.4.m4.2.2.3.1">ket</csymbol><apply id="S6.p2.4.m4.2.2.1.1.cmml" xref="S6.p2.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S6.p2.4.m4.2.2.1.1.1.cmml" xref="S6.p2.4.m4.2.2.1.1">subscript</csymbol><ci id="S6.p2.4.m4.2.2.1.1.2.cmml" xref="S6.p2.4.m4.2.2.1.1.2">𝜓</ci><cn id="S6.p2.4.m4.2.2.1.1.3.cmml" type="integer" xref="S6.p2.4.m4.2.2.1.1.3">1</cn></apply></apply></apply><apply id="S6.p2.4.m4.3.4.2.3.cmml" xref="S6.p2.4.m4.3.4.2.3"><times id="S6.p2.4.m4.3.4.2.3.1.cmml" xref="S6.p2.4.m4.3.4.2.3.1"></times><ci id="S6.p2.4.m4.3.4.2.3.2.cmml" xref="S6.p2.4.m4.3.4.2.3.2">𝑏</ci><apply id="S6.p2.4.m4.3.3.2.cmml" xref="S6.p2.4.m4.3.3.3"><csymbol cd="latexml" id="S6.p2.4.m4.3.3.2.1.cmml" xref="S6.p2.4.m4.3.3.3.1">ket</csymbol><apply id="S6.p2.4.m4.3.3.1.1.cmml" xref="S6.p2.4.m4.3.3.1.1"><csymbol cd="ambiguous" id="S6.p2.4.m4.3.3.1.1.1.cmml" xref="S6.p2.4.m4.3.3.1.1">subscript</csymbol><ci id="S6.p2.4.m4.3.3.1.1.2.cmml" xref="S6.p2.4.m4.3.3.1.1.2">𝜓</ci><cn id="S6.p2.4.m4.3.3.1.1.3.cmml" type="integer" xref="S6.p2.4.m4.3.3.1.1.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.4.m4.3c">\ket{\psi_{S}}=a\ket{\psi_{1}}+b\ket{\psi_{2}}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.4.m4.3d">| start_ARG italic_ψ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT end_ARG ⟩ = italic_a | start_ARG italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ⟩ + italic_b | start_ARG italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math>. All non-zero amplitudes <math alttext="\braket{\phi_{i}}{\psi_{1,2}}" class="ltx_Math" display="inline" id="S6.p2.5.m5.2"><semantics id="S6.p2.5.m5.2a"><mrow id="S6.p2.5.m5.2.2.4" xref="S6.p2.5.m5.2.2.3.cmml"><mo id="S6.p2.5.m5.2.2.4.1" stretchy="false" xref="S6.p2.5.m5.2.2.3.1.cmml">⟨</mo><msub id="S6.p2.5.m5.1.1.1.1" xref="S6.p2.5.m5.1.1.1.1.cmml"><mi id="S6.p2.5.m5.1.1.1.1.2" xref="S6.p2.5.m5.1.1.1.1.2.cmml">ϕ</mi><mi id="S6.p2.5.m5.1.1.1.1.3" xref="S6.p2.5.m5.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.p2.5.m5.2.2.4.2" lspace="0em" rspace="0.170em" xref="S6.p2.5.m5.2.2.3.1.cmml">|</mo><msub id="S6.p2.5.m5.2.2.2.2" xref="S6.p2.5.m5.2.2.2.2.cmml"><mi id="S6.p2.5.m5.2.2.2.2.4" xref="S6.p2.5.m5.2.2.2.2.4.cmml">ψ</mi><mrow id="S6.p2.5.m5.2.2.2.2.2.2.4" xref="S6.p2.5.m5.2.2.2.2.2.2.3.cmml"><mn id="S6.p2.5.m5.2.2.2.2.1.1.1" xref="S6.p2.5.m5.2.2.2.2.1.1.1.cmml">1</mn><mo id="S6.p2.5.m5.2.2.2.2.2.2.4.1" xref="S6.p2.5.m5.2.2.2.2.2.2.3.cmml">,</mo><mn id="S6.p2.5.m5.2.2.2.2.2.2.2" xref="S6.p2.5.m5.2.2.2.2.2.2.2.cmml">2</mn></mrow></msub><mo id="S6.p2.5.m5.2.2.4.3" stretchy="false" xref="S6.p2.5.m5.2.2.3.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.5.m5.2b"><apply id="S6.p2.5.m5.2.2.3.cmml" xref="S6.p2.5.m5.2.2.4"><csymbol cd="latexml" id="S6.p2.5.m5.2.2.3.1.cmml" xref="S6.p2.5.m5.2.2.4.1">inner-product</csymbol><apply id="S6.p2.5.m5.1.1.1.1.cmml" xref="S6.p2.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S6.p2.5.m5.1.1.1.1.1.cmml" xref="S6.p2.5.m5.1.1.1.1">subscript</csymbol><ci id="S6.p2.5.m5.1.1.1.1.2.cmml" xref="S6.p2.5.m5.1.1.1.1.2">italic-ϕ</ci><ci id="S6.p2.5.m5.1.1.1.1.3.cmml" xref="S6.p2.5.m5.1.1.1.1.3">𝑖</ci></apply><apply id="S6.p2.5.m5.2.2.2.2.cmml" xref="S6.p2.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="S6.p2.5.m5.2.2.2.2.3.cmml" xref="S6.p2.5.m5.2.2.2.2">subscript</csymbol><ci id="S6.p2.5.m5.2.2.2.2.4.cmml" xref="S6.p2.5.m5.2.2.2.2.4">𝜓</ci><list id="S6.p2.5.m5.2.2.2.2.2.2.3.cmml" xref="S6.p2.5.m5.2.2.2.2.2.2.4"><cn id="S6.p2.5.m5.2.2.2.2.1.1.1.cmml" type="integer" xref="S6.p2.5.m5.2.2.2.2.1.1.1">1</cn><cn id="S6.p2.5.m5.2.2.2.2.2.2.2.cmml" type="integer" xref="S6.p2.5.m5.2.2.2.2.2.2.2">2</cn></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.5.m5.2c">\braket{\phi_{i}}{\psi_{1,2}}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.5.m5.2d">⟨ start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG | start_ARG italic_ψ start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math>, in the electric field basis, are <math alttext="\pm\frac{1}{\sqrt{2^{L/2}}}" class="ltx_Math" display="inline" id="S6.p2.6.m6.1"><semantics id="S6.p2.6.m6.1a"><mrow id="S6.p2.6.m6.1.1" xref="S6.p2.6.m6.1.1.cmml"><mo id="S6.p2.6.m6.1.1a" xref="S6.p2.6.m6.1.1.cmml">±</mo><mfrac id="S6.p2.6.m6.1.1.2" xref="S6.p2.6.m6.1.1.2.cmml"><mn id="S6.p2.6.m6.1.1.2.2" xref="S6.p2.6.m6.1.1.2.2.cmml">1</mn><msqrt id="S6.p2.6.m6.1.1.2.3" xref="S6.p2.6.m6.1.1.2.3.cmml"><msup id="S6.p2.6.m6.1.1.2.3.2" xref="S6.p2.6.m6.1.1.2.3.2.cmml"><mn id="S6.p2.6.m6.1.1.2.3.2.2" xref="S6.p2.6.m6.1.1.2.3.2.2.cmml">2</mn><mrow id="S6.p2.6.m6.1.1.2.3.2.3" xref="S6.p2.6.m6.1.1.2.3.2.3.cmml"><mi id="S6.p2.6.m6.1.1.2.3.2.3.2" xref="S6.p2.6.m6.1.1.2.3.2.3.2.cmml">L</mi><mo id="S6.p2.6.m6.1.1.2.3.2.3.1" xref="S6.p2.6.m6.1.1.2.3.2.3.1.cmml">/</mo><mn id="S6.p2.6.m6.1.1.2.3.2.3.3" xref="S6.p2.6.m6.1.1.2.3.2.3.3.cmml">2</mn></mrow></msup></msqrt></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.6.m6.1b"><apply id="S6.p2.6.m6.1.1.cmml" xref="S6.p2.6.m6.1.1"><csymbol cd="latexml" id="S6.p2.6.m6.1.1.1.cmml" xref="S6.p2.6.m6.1.1">plus-or-minus</csymbol><apply id="S6.p2.6.m6.1.1.2.cmml" xref="S6.p2.6.m6.1.1.2"><divide id="S6.p2.6.m6.1.1.2.1.cmml" xref="S6.p2.6.m6.1.1.2"></divide><cn id="S6.p2.6.m6.1.1.2.2.cmml" type="integer" xref="S6.p2.6.m6.1.1.2.2">1</cn><apply id="S6.p2.6.m6.1.1.2.3.cmml" xref="S6.p2.6.m6.1.1.2.3"><root id="S6.p2.6.m6.1.1.2.3a.cmml" xref="S6.p2.6.m6.1.1.2.3"></root><apply id="S6.p2.6.m6.1.1.2.3.2.cmml" xref="S6.p2.6.m6.1.1.2.3.2"><csymbol cd="ambiguous" id="S6.p2.6.m6.1.1.2.3.2.1.cmml" xref="S6.p2.6.m6.1.1.2.3.2">superscript</csymbol><cn id="S6.p2.6.m6.1.1.2.3.2.2.cmml" type="integer" xref="S6.p2.6.m6.1.1.2.3.2.2">2</cn><apply id="S6.p2.6.m6.1.1.2.3.2.3.cmml" xref="S6.p2.6.m6.1.1.2.3.2.3"><divide id="S6.p2.6.m6.1.1.2.3.2.3.1.cmml" xref="S6.p2.6.m6.1.1.2.3.2.3.1"></divide><ci id="S6.p2.6.m6.1.1.2.3.2.3.2.cmml" xref="S6.p2.6.m6.1.1.2.3.2.3.2">𝐿</ci><cn id="S6.p2.6.m6.1.1.2.3.2.3.3.cmml" type="integer" xref="S6.p2.6.m6.1.1.2.3.2.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.6.m6.1c">\pm\frac{1}{\sqrt{2^{L/2}}}</annotation><annotation encoding="application/x-llamapun" id="S6.p2.6.m6.1d">± divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 start_POSTSUPERSCRIPT italic_L / 2 end_POSTSUPERSCRIPT end_ARG end_ARG</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S6.F9"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="314" id="S6.F9.g1" src="extracted/5828746/images/overlapS=1.jpeg" width="419"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 9: </span>Overlap of electric field basis states <math alttext="\ket{\phi_{i}}" class="ltx_Math" display="inline" id="S6.F9.5.m1.1"><semantics id="S6.F9.5.m1.1b"><mrow id="S6.F9.5.m1.1.1.3" xref="S6.F9.5.m1.1.1.2.cmml"><mo id="S6.F9.5.m1.1.1.3.1" stretchy="false" xref="S6.F9.5.m1.1.1.2.1.cmml">|</mo><msub id="S6.F9.5.m1.1.1.1.1" xref="S6.F9.5.m1.1.1.1.1.cmml"><mi id="S6.F9.5.m1.1.1.1.1.2" xref="S6.F9.5.m1.1.1.1.1.2.cmml">ϕ</mi><mi id="S6.F9.5.m1.1.1.1.1.3" xref="S6.F9.5.m1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.F9.5.m1.1.1.3.2" stretchy="false" xref="S6.F9.5.m1.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.F9.5.m1.1c"><apply id="S6.F9.5.m1.1.1.2.cmml" xref="S6.F9.5.m1.1.1.3"><csymbol cd="latexml" id="S6.F9.5.m1.1.1.2.1.cmml" xref="S6.F9.5.m1.1.1.3.1">ket</csymbol><apply id="S6.F9.5.m1.1.1.1.1.cmml" xref="S6.F9.5.m1.1.1.1.1"><csymbol cd="ambiguous" id="S6.F9.5.m1.1.1.1.1.1.cmml" xref="S6.F9.5.m1.1.1.1.1">subscript</csymbol><ci id="S6.F9.5.m1.1.1.1.1.2.cmml" xref="S6.F9.5.m1.1.1.1.1.2">italic-ϕ</ci><ci id="S6.F9.5.m1.1.1.1.1.3.cmml" xref="S6.F9.5.m1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F9.5.m1.1d">\ket{\phi_{i}}</annotation><annotation encoding="application/x-llamapun" id="S6.F9.5.m1.1e">| start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> with the lowest entropy scar in Fig. 3 of the main text. It depicts the entanglement entropy of <math alttext="10\times 1" class="ltx_Math" display="inline" id="S6.F9.6.m2.1"><semantics id="S6.F9.6.m2.1b"><mrow id="S6.F9.6.m2.1.1" xref="S6.F9.6.m2.1.1.cmml"><mn id="S6.F9.6.m2.1.1.2" xref="S6.F9.6.m2.1.1.2.cmml">10</mn><mo id="S6.F9.6.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S6.F9.6.m2.1.1.1.cmml">×</mo><mn id="S6.F9.6.m2.1.1.3" xref="S6.F9.6.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.F9.6.m2.1c"><apply id="S6.F9.6.m2.1.1.cmml" xref="S6.F9.6.m2.1.1"><times id="S6.F9.6.m2.1.1.1.cmml" xref="S6.F9.6.m2.1.1.1"></times><cn id="S6.F9.6.m2.1.1.2.cmml" type="integer" xref="S6.F9.6.m2.1.1.2">10</cn><cn id="S6.F9.6.m2.1.1.3.cmml" type="integer" xref="S6.F9.6.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F9.6.m2.1d">10\times 1</annotation><annotation encoding="application/x-llamapun" id="S6.F9.6.m2.1e">10 × 1</annotation></semantics></math> <math alttext="S=1" class="ltx_Math" display="inline" id="S6.F9.7.m3.1"><semantics id="S6.F9.7.m3.1b"><mrow id="S6.F9.7.m3.1.1" xref="S6.F9.7.m3.1.1.cmml"><mi id="S6.F9.7.m3.1.1.2" xref="S6.F9.7.m3.1.1.2.cmml">S</mi><mo id="S6.F9.7.m3.1.1.1" xref="S6.F9.7.m3.1.1.1.cmml">=</mo><mn id="S6.F9.7.m3.1.1.3" xref="S6.F9.7.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.F9.7.m3.1c"><apply id="S6.F9.7.m3.1.1.cmml" xref="S6.F9.7.m3.1.1"><eq id="S6.F9.7.m3.1.1.1.cmml" xref="S6.F9.7.m3.1.1.1"></eq><ci id="S6.F9.7.m3.1.1.2.cmml" xref="S6.F9.7.m3.1.1.2">𝑆</ci><cn id="S6.F9.7.m3.1.1.3.cmml" type="integer" xref="S6.F9.7.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.F9.7.m3.1d">S=1</annotation><annotation encoding="application/x-llamapun" id="S6.F9.7.m3.1e">italic_S = 1</annotation></semantics></math> plaquettes with periodic boundary conditions in the <math alttext="x" class="ltx_Math" display="inline" id="S6.F9.8.m4.1"><semantics id="S6.F9.8.m4.1b"><mi id="S6.F9.8.m4.1.1" xref="S6.F9.8.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S6.F9.8.m4.1c"><ci id="S6.F9.8.m4.1.1.cmml" xref="S6.F9.8.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.F9.8.m4.1d">x</annotation><annotation encoding="application/x-llamapun" id="S6.F9.8.m4.1e">italic_x</annotation></semantics></math> direction. Most states have zero overlap. </figcaption> </figure> <div class="ltx_para" id="S6.p3"> <p class="ltx_p" id="S6.p3.1">There are therefore four different possible magnitudes of the amplitudes <math alttext="A=\braket{\phi_{i}}{\psi_{S}}" class="ltx_Math" display="inline" id="S6.p3.1.m1.2"><semantics id="S6.p3.1.m1.2a"><mrow id="S6.p3.1.m1.2.3" xref="S6.p3.1.m1.2.3.cmml"><mi id="S6.p3.1.m1.2.3.2" xref="S6.p3.1.m1.2.3.2.cmml">A</mi><mo id="S6.p3.1.m1.2.3.1" xref="S6.p3.1.m1.2.3.1.cmml">=</mo><mrow id="S6.p3.1.m1.2.2.4" xref="S6.p3.1.m1.2.2.3.cmml"><mo id="S6.p3.1.m1.2.2.4.1" stretchy="false" xref="S6.p3.1.m1.2.2.3.1.cmml">⟨</mo><msub id="S6.p3.1.m1.1.1.1.1" xref="S6.p3.1.m1.1.1.1.1.cmml"><mi id="S6.p3.1.m1.1.1.1.1.2" xref="S6.p3.1.m1.1.1.1.1.2.cmml">ϕ</mi><mi id="S6.p3.1.m1.1.1.1.1.3" xref="S6.p3.1.m1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.p3.1.m1.2.2.4.2" lspace="0em" rspace="0.170em" xref="S6.p3.1.m1.2.2.3.1.cmml">|</mo><msub id="S6.p3.1.m1.2.2.2.2" xref="S6.p3.1.m1.2.2.2.2.cmml"><mi id="S6.p3.1.m1.2.2.2.2.2" xref="S6.p3.1.m1.2.2.2.2.2.cmml">ψ</mi><mi id="S6.p3.1.m1.2.2.2.2.3" xref="S6.p3.1.m1.2.2.2.2.3.cmml">S</mi></msub><mo id="S6.p3.1.m1.2.2.4.3" stretchy="false" xref="S6.p3.1.m1.2.2.3.1.cmml">⟩</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p3.1.m1.2b"><apply id="S6.p3.1.m1.2.3.cmml" xref="S6.p3.1.m1.2.3"><eq id="S6.p3.1.m1.2.3.1.cmml" xref="S6.p3.1.m1.2.3.1"></eq><ci id="S6.p3.1.m1.2.3.2.cmml" xref="S6.p3.1.m1.2.3.2">𝐴</ci><apply id="S6.p3.1.m1.2.2.3.cmml" xref="S6.p3.1.m1.2.2.4"><csymbol cd="latexml" id="S6.p3.1.m1.2.2.3.1.cmml" xref="S6.p3.1.m1.2.2.4.1">inner-product</csymbol><apply id="S6.p3.1.m1.1.1.1.1.cmml" xref="S6.p3.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S6.p3.1.m1.1.1.1.1.1.cmml" xref="S6.p3.1.m1.1.1.1.1">subscript</csymbol><ci id="S6.p3.1.m1.1.1.1.1.2.cmml" xref="S6.p3.1.m1.1.1.1.1.2">italic-ϕ</ci><ci id="S6.p3.1.m1.1.1.1.1.3.cmml" xref="S6.p3.1.m1.1.1.1.1.3">𝑖</ci></apply><apply id="S6.p3.1.m1.2.2.2.2.cmml" xref="S6.p3.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S6.p3.1.m1.2.2.2.2.1.cmml" xref="S6.p3.1.m1.2.2.2.2">subscript</csymbol><ci id="S6.p3.1.m1.2.2.2.2.2.cmml" xref="S6.p3.1.m1.2.2.2.2.2">𝜓</ci><ci id="S6.p3.1.m1.2.2.2.2.3.cmml" xref="S6.p3.1.m1.2.2.2.2.3">𝑆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.1.m1.2c">A=\braket{\phi_{i}}{\psi_{S}}</annotation><annotation encoding="application/x-llamapun" id="S6.p3.1.m1.2d">italic_A = ⟨ start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG | start_ARG italic_ψ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math>:</p> <ol class="ltx_enumerate" id="S6.I1"> <li class="ltx_item" id="S6.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S6.I1.i1.p1"> <p class="ltx_p" id="S6.I1.i1.p1.3">The basis state <math alttext="\ket{\phi_{i}}" class="ltx_Math" display="inline" id="S6.I1.i1.p1.1.m1.1"><semantics id="S6.I1.i1.p1.1.m1.1a"><mrow id="S6.I1.i1.p1.1.m1.1.1.3" xref="S6.I1.i1.p1.1.m1.1.1.2.cmml"><mo id="S6.I1.i1.p1.1.m1.1.1.3.1" stretchy="false" xref="S6.I1.i1.p1.1.m1.1.1.2.1.cmml">|</mo><msub id="S6.I1.i1.p1.1.m1.1.1.1.1" xref="S6.I1.i1.p1.1.m1.1.1.1.1.cmml"><mi id="S6.I1.i1.p1.1.m1.1.1.1.1.2" xref="S6.I1.i1.p1.1.m1.1.1.1.1.2.cmml">ϕ</mi><mi id="S6.I1.i1.p1.1.m1.1.1.1.1.3" xref="S6.I1.i1.p1.1.m1.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.I1.i1.p1.1.m1.1.1.3.2" stretchy="false" xref="S6.I1.i1.p1.1.m1.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.p1.1.m1.1b"><apply id="S6.I1.i1.p1.1.m1.1.1.2.cmml" xref="S6.I1.i1.p1.1.m1.1.1.3"><csymbol cd="latexml" id="S6.I1.i1.p1.1.m1.1.1.2.1.cmml" xref="S6.I1.i1.p1.1.m1.1.1.3.1">ket</csymbol><apply id="S6.I1.i1.p1.1.m1.1.1.1.1.cmml" xref="S6.I1.i1.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.p1.1.m1.1.1.1.1.1.cmml" xref="S6.I1.i1.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S6.I1.i1.p1.1.m1.1.1.1.1.2.cmml" xref="S6.I1.i1.p1.1.m1.1.1.1.1.2">italic-ϕ</ci><ci id="S6.I1.i1.p1.1.m1.1.1.1.1.3.cmml" xref="S6.I1.i1.p1.1.m1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.p1.1.m1.1c">\ket{\phi_{i}}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.p1.1.m1.1d">| start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> has zero overlap with both <math alttext="\ket{\psi_{1}}" class="ltx_Math" display="inline" id="S6.I1.i1.p1.2.m2.1"><semantics id="S6.I1.i1.p1.2.m2.1a"><mrow id="S6.I1.i1.p1.2.m2.1.1.3" xref="S6.I1.i1.p1.2.m2.1.1.2.cmml"><mo id="S6.I1.i1.p1.2.m2.1.1.3.1" stretchy="false" xref="S6.I1.i1.p1.2.m2.1.1.2.1.cmml">|</mo><msub id="S6.I1.i1.p1.2.m2.1.1.1.1" xref="S6.I1.i1.p1.2.m2.1.1.1.1.cmml"><mi id="S6.I1.i1.p1.2.m2.1.1.1.1.2" xref="S6.I1.i1.p1.2.m2.1.1.1.1.2.cmml">ψ</mi><mn id="S6.I1.i1.p1.2.m2.1.1.1.1.3" xref="S6.I1.i1.p1.2.m2.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.I1.i1.p1.2.m2.1.1.3.2" stretchy="false" xref="S6.I1.i1.p1.2.m2.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.p1.2.m2.1b"><apply id="S6.I1.i1.p1.2.m2.1.1.2.cmml" xref="S6.I1.i1.p1.2.m2.1.1.3"><csymbol cd="latexml" id="S6.I1.i1.p1.2.m2.1.1.2.1.cmml" xref="S6.I1.i1.p1.2.m2.1.1.3.1">ket</csymbol><apply id="S6.I1.i1.p1.2.m2.1.1.1.1.cmml" xref="S6.I1.i1.p1.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.p1.2.m2.1.1.1.1.1.cmml" xref="S6.I1.i1.p1.2.m2.1.1.1.1">subscript</csymbol><ci id="S6.I1.i1.p1.2.m2.1.1.1.1.2.cmml" xref="S6.I1.i1.p1.2.m2.1.1.1.1.2">𝜓</ci><cn id="S6.I1.i1.p1.2.m2.1.1.1.1.3.cmml" type="integer" xref="S6.I1.i1.p1.2.m2.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.p1.2.m2.1c">\ket{\psi_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.p1.2.m2.1d">| start_ARG italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> and <math alttext="\ket{\psi_{2}}" class="ltx_Math" display="inline" id="S6.I1.i1.p1.3.m3.1"><semantics id="S6.I1.i1.p1.3.m3.1a"><mrow id="S6.I1.i1.p1.3.m3.1.1.3" xref="S6.I1.i1.p1.3.m3.1.1.2.cmml"><mo id="S6.I1.i1.p1.3.m3.1.1.3.1" stretchy="false" xref="S6.I1.i1.p1.3.m3.1.1.2.1.cmml">|</mo><msub id="S6.I1.i1.p1.3.m3.1.1.1.1" xref="S6.I1.i1.p1.3.m3.1.1.1.1.cmml"><mi id="S6.I1.i1.p1.3.m3.1.1.1.1.2" xref="S6.I1.i1.p1.3.m3.1.1.1.1.2.cmml">ψ</mi><mn id="S6.I1.i1.p1.3.m3.1.1.1.1.3" xref="S6.I1.i1.p1.3.m3.1.1.1.1.3.cmml">2</mn></msub><mo id="S6.I1.i1.p1.3.m3.1.1.3.2" stretchy="false" xref="S6.I1.i1.p1.3.m3.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.p1.3.m3.1b"><apply id="S6.I1.i1.p1.3.m3.1.1.2.cmml" xref="S6.I1.i1.p1.3.m3.1.1.3"><csymbol cd="latexml" id="S6.I1.i1.p1.3.m3.1.1.2.1.cmml" xref="S6.I1.i1.p1.3.m3.1.1.3.1">ket</csymbol><apply id="S6.I1.i1.p1.3.m3.1.1.1.1.cmml" xref="S6.I1.i1.p1.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i1.p1.3.m3.1.1.1.1.1.cmml" xref="S6.I1.i1.p1.3.m3.1.1.1.1">subscript</csymbol><ci id="S6.I1.i1.p1.3.m3.1.1.1.1.2.cmml" xref="S6.I1.i1.p1.3.m3.1.1.1.1.2">𝜓</ci><cn id="S6.I1.i1.p1.3.m3.1.1.1.1.3.cmml" type="integer" xref="S6.I1.i1.p1.3.m3.1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.p1.3.m3.1c">\ket{\psi_{2}}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.p1.3.m3.1d">| start_ARG italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> and therefore the amplitude is zero.</p> </div> </li> <li class="ltx_item" id="S6.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S6.I1.i2.p1"> <p class="ltx_p" id="S6.I1.i2.p1.2">The basis state has non-zero overlap with only <math alttext="\ket{\psi_{1}}" class="ltx_Math" display="inline" id="S6.I1.i2.p1.1.m1.1"><semantics id="S6.I1.i2.p1.1.m1.1a"><mrow id="S6.I1.i2.p1.1.m1.1.1.3" xref="S6.I1.i2.p1.1.m1.1.1.2.cmml"><mo id="S6.I1.i2.p1.1.m1.1.1.3.1" stretchy="false" xref="S6.I1.i2.p1.1.m1.1.1.2.1.cmml">|</mo><msub id="S6.I1.i2.p1.1.m1.1.1.1.1" xref="S6.I1.i2.p1.1.m1.1.1.1.1.cmml"><mi id="S6.I1.i2.p1.1.m1.1.1.1.1.2" xref="S6.I1.i2.p1.1.m1.1.1.1.1.2.cmml">ψ</mi><mn id="S6.I1.i2.p1.1.m1.1.1.1.1.3" xref="S6.I1.i2.p1.1.m1.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.I1.i2.p1.1.m1.1.1.3.2" stretchy="false" xref="S6.I1.i2.p1.1.m1.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i2.p1.1.m1.1b"><apply id="S6.I1.i2.p1.1.m1.1.1.2.cmml" xref="S6.I1.i2.p1.1.m1.1.1.3"><csymbol cd="latexml" id="S6.I1.i2.p1.1.m1.1.1.2.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1.3.1">ket</csymbol><apply id="S6.I1.i2.p1.1.m1.1.1.1.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i2.p1.1.m1.1.1.1.1.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S6.I1.i2.p1.1.m1.1.1.1.1.2.cmml" xref="S6.I1.i2.p1.1.m1.1.1.1.1.2">𝜓</ci><cn id="S6.I1.i2.p1.1.m1.1.1.1.1.3.cmml" type="integer" xref="S6.I1.i2.p1.1.m1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i2.p1.1.m1.1c">\ket{\psi_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i2.p1.1.m1.1d">| start_ARG italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> and has the amplitude <math alttext="A=\pm\frac{a}{\sqrt{2^{L/2}}}" class="ltx_Math" display="inline" id="S6.I1.i2.p1.2.m2.1"><semantics id="S6.I1.i2.p1.2.m2.1a"><mrow id="S6.I1.i2.p1.2.m2.1.1" xref="S6.I1.i2.p1.2.m2.1.1.cmml"><mi id="S6.I1.i2.p1.2.m2.1.1.2" xref="S6.I1.i2.p1.2.m2.1.1.2.cmml">A</mi><mo id="S6.I1.i2.p1.2.m2.1.1.1" xref="S6.I1.i2.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S6.I1.i2.p1.2.m2.1.1.3" xref="S6.I1.i2.p1.2.m2.1.1.3.cmml"><mo id="S6.I1.i2.p1.2.m2.1.1.3a" xref="S6.I1.i2.p1.2.m2.1.1.3.cmml">±</mo><mfrac id="S6.I1.i2.p1.2.m2.1.1.3.2" xref="S6.I1.i2.p1.2.m2.1.1.3.2.cmml"><mi id="S6.I1.i2.p1.2.m2.1.1.3.2.2" xref="S6.I1.i2.p1.2.m2.1.1.3.2.2.cmml">a</mi><msqrt id="S6.I1.i2.p1.2.m2.1.1.3.2.3" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3.cmml"><msup id="S6.I1.i2.p1.2.m2.1.1.3.2.3.2" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.cmml"><mn id="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.2" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.2.cmml">2</mn><mrow id="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3.cmml"><mi id="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3.2" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3.2.cmml">L</mi><mo id="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3.1" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3.1.cmml">/</mo><mn id="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3.3" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3.3.cmml">2</mn></mrow></msup></msqrt></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i2.p1.2.m2.1b"><apply id="S6.I1.i2.p1.2.m2.1.1.cmml" xref="S6.I1.i2.p1.2.m2.1.1"><eq id="S6.I1.i2.p1.2.m2.1.1.1.cmml" xref="S6.I1.i2.p1.2.m2.1.1.1"></eq><ci id="S6.I1.i2.p1.2.m2.1.1.2.cmml" xref="S6.I1.i2.p1.2.m2.1.1.2">𝐴</ci><apply id="S6.I1.i2.p1.2.m2.1.1.3.cmml" xref="S6.I1.i2.p1.2.m2.1.1.3"><csymbol cd="latexml" id="S6.I1.i2.p1.2.m2.1.1.3.1.cmml" xref="S6.I1.i2.p1.2.m2.1.1.3">plus-or-minus</csymbol><apply id="S6.I1.i2.p1.2.m2.1.1.3.2.cmml" xref="S6.I1.i2.p1.2.m2.1.1.3.2"><divide id="S6.I1.i2.p1.2.m2.1.1.3.2.1.cmml" xref="S6.I1.i2.p1.2.m2.1.1.3.2"></divide><ci id="S6.I1.i2.p1.2.m2.1.1.3.2.2.cmml" xref="S6.I1.i2.p1.2.m2.1.1.3.2.2">𝑎</ci><apply id="S6.I1.i2.p1.2.m2.1.1.3.2.3.cmml" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3"><root id="S6.I1.i2.p1.2.m2.1.1.3.2.3a.cmml" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3"></root><apply id="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.cmml" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.1.cmml" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3.2">superscript</csymbol><cn id="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.2.cmml" type="integer" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.2">2</cn><apply id="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3.cmml" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3"><divide id="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3.1.cmml" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3.1"></divide><ci id="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3.2.cmml" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3.2">𝐿</ci><cn id="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3.3.cmml" type="integer" xref="S6.I1.i2.p1.2.m2.1.1.3.2.3.2.3.3">2</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i2.p1.2.m2.1c">A=\pm\frac{a}{\sqrt{2^{L/2}}}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i2.p1.2.m2.1d">italic_A = ± divide start_ARG italic_a end_ARG start_ARG square-root start_ARG 2 start_POSTSUPERSCRIPT italic_L / 2 end_POSTSUPERSCRIPT end_ARG end_ARG</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S6.I1.i3.p1"> <p class="ltx_p" id="S6.I1.i3.p1.2">The basis state has non-zero overlap with only <math alttext="\ket{\psi_{2}}" class="ltx_Math" display="inline" id="S6.I1.i3.p1.1.m1.1"><semantics id="S6.I1.i3.p1.1.m1.1a"><mrow id="S6.I1.i3.p1.1.m1.1.1.3" xref="S6.I1.i3.p1.1.m1.1.1.2.cmml"><mo id="S6.I1.i3.p1.1.m1.1.1.3.1" stretchy="false" xref="S6.I1.i3.p1.1.m1.1.1.2.1.cmml">|</mo><msub id="S6.I1.i3.p1.1.m1.1.1.1.1" xref="S6.I1.i3.p1.1.m1.1.1.1.1.cmml"><mi id="S6.I1.i3.p1.1.m1.1.1.1.1.2" xref="S6.I1.i3.p1.1.m1.1.1.1.1.2.cmml">ψ</mi><mn id="S6.I1.i3.p1.1.m1.1.1.1.1.3" xref="S6.I1.i3.p1.1.m1.1.1.1.1.3.cmml">2</mn></msub><mo id="S6.I1.i3.p1.1.m1.1.1.3.2" stretchy="false" xref="S6.I1.i3.p1.1.m1.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i3.p1.1.m1.1b"><apply id="S6.I1.i3.p1.1.m1.1.1.2.cmml" xref="S6.I1.i3.p1.1.m1.1.1.3"><csymbol cd="latexml" id="S6.I1.i3.p1.1.m1.1.1.2.1.cmml" xref="S6.I1.i3.p1.1.m1.1.1.3.1">ket</csymbol><apply id="S6.I1.i3.p1.1.m1.1.1.1.1.cmml" xref="S6.I1.i3.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i3.p1.1.m1.1.1.1.1.1.cmml" xref="S6.I1.i3.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S6.I1.i3.p1.1.m1.1.1.1.1.2.cmml" xref="S6.I1.i3.p1.1.m1.1.1.1.1.2">𝜓</ci><cn id="S6.I1.i3.p1.1.m1.1.1.1.1.3.cmml" type="integer" xref="S6.I1.i3.p1.1.m1.1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i3.p1.1.m1.1c">\ket{\psi_{2}}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i3.p1.1.m1.1d">| start_ARG italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> and has the amplitude <math alttext="A=\pm\frac{b}{\sqrt{2^{L/2}}}" class="ltx_Math" display="inline" id="S6.I1.i3.p1.2.m2.1"><semantics id="S6.I1.i3.p1.2.m2.1a"><mrow id="S6.I1.i3.p1.2.m2.1.1" xref="S6.I1.i3.p1.2.m2.1.1.cmml"><mi id="S6.I1.i3.p1.2.m2.1.1.2" xref="S6.I1.i3.p1.2.m2.1.1.2.cmml">A</mi><mo id="S6.I1.i3.p1.2.m2.1.1.1" xref="S6.I1.i3.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S6.I1.i3.p1.2.m2.1.1.3" xref="S6.I1.i3.p1.2.m2.1.1.3.cmml"><mo id="S6.I1.i3.p1.2.m2.1.1.3a" xref="S6.I1.i3.p1.2.m2.1.1.3.cmml">±</mo><mfrac id="S6.I1.i3.p1.2.m2.1.1.3.2" xref="S6.I1.i3.p1.2.m2.1.1.3.2.cmml"><mi id="S6.I1.i3.p1.2.m2.1.1.3.2.2" xref="S6.I1.i3.p1.2.m2.1.1.3.2.2.cmml">b</mi><msqrt id="S6.I1.i3.p1.2.m2.1.1.3.2.3" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3.cmml"><msup id="S6.I1.i3.p1.2.m2.1.1.3.2.3.2" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.cmml"><mn id="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.2" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.2.cmml">2</mn><mrow id="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3.cmml"><mi id="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3.2" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3.2.cmml">L</mi><mo id="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3.1" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3.1.cmml">/</mo><mn id="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3.3" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3.3.cmml">2</mn></mrow></msup></msqrt></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i3.p1.2.m2.1b"><apply id="S6.I1.i3.p1.2.m2.1.1.cmml" xref="S6.I1.i3.p1.2.m2.1.1"><eq id="S6.I1.i3.p1.2.m2.1.1.1.cmml" xref="S6.I1.i3.p1.2.m2.1.1.1"></eq><ci id="S6.I1.i3.p1.2.m2.1.1.2.cmml" xref="S6.I1.i3.p1.2.m2.1.1.2">𝐴</ci><apply id="S6.I1.i3.p1.2.m2.1.1.3.cmml" xref="S6.I1.i3.p1.2.m2.1.1.3"><csymbol cd="latexml" id="S6.I1.i3.p1.2.m2.1.1.3.1.cmml" xref="S6.I1.i3.p1.2.m2.1.1.3">plus-or-minus</csymbol><apply id="S6.I1.i3.p1.2.m2.1.1.3.2.cmml" xref="S6.I1.i3.p1.2.m2.1.1.3.2"><divide id="S6.I1.i3.p1.2.m2.1.1.3.2.1.cmml" xref="S6.I1.i3.p1.2.m2.1.1.3.2"></divide><ci id="S6.I1.i3.p1.2.m2.1.1.3.2.2.cmml" xref="S6.I1.i3.p1.2.m2.1.1.3.2.2">𝑏</ci><apply id="S6.I1.i3.p1.2.m2.1.1.3.2.3.cmml" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3"><root id="S6.I1.i3.p1.2.m2.1.1.3.2.3a.cmml" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3"></root><apply id="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.cmml" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.1.cmml" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3.2">superscript</csymbol><cn id="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.2.cmml" type="integer" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.2">2</cn><apply id="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3.cmml" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3"><divide id="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3.1.cmml" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3.1"></divide><ci id="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3.2.cmml" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3.2">𝐿</ci><cn id="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3.3.cmml" type="integer" xref="S6.I1.i3.p1.2.m2.1.1.3.2.3.2.3.3">2</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i3.p1.2.m2.1c">A=\pm\frac{b}{\sqrt{2^{L/2}}}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i3.p1.2.m2.1d">italic_A = ± divide start_ARG italic_b end_ARG start_ARG square-root start_ARG 2 start_POSTSUPERSCRIPT italic_L / 2 end_POSTSUPERSCRIPT end_ARG end_ARG</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S6.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">4.</span> <div class="ltx_para" id="S6.I1.i4.p1"> <p class="ltx_p" id="S6.I1.i4.p1.3">The basis state has non-zero overlap with both <math alttext="\ket{\psi_{1}}" class="ltx_Math" display="inline" id="S6.I1.i4.p1.1.m1.1"><semantics id="S6.I1.i4.p1.1.m1.1a"><mrow id="S6.I1.i4.p1.1.m1.1.1.3" xref="S6.I1.i4.p1.1.m1.1.1.2.cmml"><mo id="S6.I1.i4.p1.1.m1.1.1.3.1" stretchy="false" xref="S6.I1.i4.p1.1.m1.1.1.2.1.cmml">|</mo><msub id="S6.I1.i4.p1.1.m1.1.1.1.1" xref="S6.I1.i4.p1.1.m1.1.1.1.1.cmml"><mi id="S6.I1.i4.p1.1.m1.1.1.1.1.2" xref="S6.I1.i4.p1.1.m1.1.1.1.1.2.cmml">ψ</mi><mn id="S6.I1.i4.p1.1.m1.1.1.1.1.3" xref="S6.I1.i4.p1.1.m1.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.I1.i4.p1.1.m1.1.1.3.2" stretchy="false" xref="S6.I1.i4.p1.1.m1.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i4.p1.1.m1.1b"><apply id="S6.I1.i4.p1.1.m1.1.1.2.cmml" xref="S6.I1.i4.p1.1.m1.1.1.3"><csymbol cd="latexml" id="S6.I1.i4.p1.1.m1.1.1.2.1.cmml" xref="S6.I1.i4.p1.1.m1.1.1.3.1">ket</csymbol><apply id="S6.I1.i4.p1.1.m1.1.1.1.1.cmml" xref="S6.I1.i4.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i4.p1.1.m1.1.1.1.1.1.cmml" xref="S6.I1.i4.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S6.I1.i4.p1.1.m1.1.1.1.1.2.cmml" xref="S6.I1.i4.p1.1.m1.1.1.1.1.2">𝜓</ci><cn id="S6.I1.i4.p1.1.m1.1.1.1.1.3.cmml" type="integer" xref="S6.I1.i4.p1.1.m1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i4.p1.1.m1.1c">\ket{\psi_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i4.p1.1.m1.1d">| start_ARG italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> and <math alttext="\ket{\psi_{2}}" class="ltx_Math" display="inline" id="S6.I1.i4.p1.2.m2.1"><semantics id="S6.I1.i4.p1.2.m2.1a"><mrow id="S6.I1.i4.p1.2.m2.1.1.3" xref="S6.I1.i4.p1.2.m2.1.1.2.cmml"><mo id="S6.I1.i4.p1.2.m2.1.1.3.1" stretchy="false" xref="S6.I1.i4.p1.2.m2.1.1.2.1.cmml">|</mo><msub id="S6.I1.i4.p1.2.m2.1.1.1.1" xref="S6.I1.i4.p1.2.m2.1.1.1.1.cmml"><mi id="S6.I1.i4.p1.2.m2.1.1.1.1.2" xref="S6.I1.i4.p1.2.m2.1.1.1.1.2.cmml">ψ</mi><mn id="S6.I1.i4.p1.2.m2.1.1.1.1.3" xref="S6.I1.i4.p1.2.m2.1.1.1.1.3.cmml">2</mn></msub><mo id="S6.I1.i4.p1.2.m2.1.1.3.2" stretchy="false" xref="S6.I1.i4.p1.2.m2.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i4.p1.2.m2.1b"><apply id="S6.I1.i4.p1.2.m2.1.1.2.cmml" xref="S6.I1.i4.p1.2.m2.1.1.3"><csymbol cd="latexml" id="S6.I1.i4.p1.2.m2.1.1.2.1.cmml" xref="S6.I1.i4.p1.2.m2.1.1.3.1">ket</csymbol><apply id="S6.I1.i4.p1.2.m2.1.1.1.1.cmml" xref="S6.I1.i4.p1.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S6.I1.i4.p1.2.m2.1.1.1.1.1.cmml" xref="S6.I1.i4.p1.2.m2.1.1.1.1">subscript</csymbol><ci id="S6.I1.i4.p1.2.m2.1.1.1.1.2.cmml" xref="S6.I1.i4.p1.2.m2.1.1.1.1.2">𝜓</ci><cn id="S6.I1.i4.p1.2.m2.1.1.1.1.3.cmml" type="integer" xref="S6.I1.i4.p1.2.m2.1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i4.p1.2.m2.1c">\ket{\psi_{2}}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i4.p1.2.m2.1d">| start_ARG italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> and has the amplitude <math alttext="A=\pm\frac{a+b}{\sqrt{2^{L/2}}}" class="ltx_Math" display="inline" id="S6.I1.i4.p1.3.m3.1"><semantics id="S6.I1.i4.p1.3.m3.1a"><mrow id="S6.I1.i4.p1.3.m3.1.1" xref="S6.I1.i4.p1.3.m3.1.1.cmml"><mi id="S6.I1.i4.p1.3.m3.1.1.2" xref="S6.I1.i4.p1.3.m3.1.1.2.cmml">A</mi><mo id="S6.I1.i4.p1.3.m3.1.1.1" xref="S6.I1.i4.p1.3.m3.1.1.1.cmml">=</mo><mrow id="S6.I1.i4.p1.3.m3.1.1.3" xref="S6.I1.i4.p1.3.m3.1.1.3.cmml"><mo id="S6.I1.i4.p1.3.m3.1.1.3a" xref="S6.I1.i4.p1.3.m3.1.1.3.cmml">±</mo><mfrac id="S6.I1.i4.p1.3.m3.1.1.3.2" xref="S6.I1.i4.p1.3.m3.1.1.3.2.cmml"><mrow id="S6.I1.i4.p1.3.m3.1.1.3.2.2" xref="S6.I1.i4.p1.3.m3.1.1.3.2.2.cmml"><mi id="S6.I1.i4.p1.3.m3.1.1.3.2.2.2" xref="S6.I1.i4.p1.3.m3.1.1.3.2.2.2.cmml">a</mi><mo id="S6.I1.i4.p1.3.m3.1.1.3.2.2.1" xref="S6.I1.i4.p1.3.m3.1.1.3.2.2.1.cmml">+</mo><mi id="S6.I1.i4.p1.3.m3.1.1.3.2.2.3" xref="S6.I1.i4.p1.3.m3.1.1.3.2.2.3.cmml">b</mi></mrow><msqrt id="S6.I1.i4.p1.3.m3.1.1.3.2.3" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3.cmml"><msup id="S6.I1.i4.p1.3.m3.1.1.3.2.3.2" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.cmml"><mn id="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.2" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.2.cmml">2</mn><mrow id="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3.cmml"><mi id="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3.2" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3.2.cmml">L</mi><mo id="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3.1" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3.1.cmml">/</mo><mn id="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3.3" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3.3.cmml">2</mn></mrow></msup></msqrt></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i4.p1.3.m3.1b"><apply id="S6.I1.i4.p1.3.m3.1.1.cmml" xref="S6.I1.i4.p1.3.m3.1.1"><eq id="S6.I1.i4.p1.3.m3.1.1.1.cmml" xref="S6.I1.i4.p1.3.m3.1.1.1"></eq><ci id="S6.I1.i4.p1.3.m3.1.1.2.cmml" xref="S6.I1.i4.p1.3.m3.1.1.2">𝐴</ci><apply id="S6.I1.i4.p1.3.m3.1.1.3.cmml" xref="S6.I1.i4.p1.3.m3.1.1.3"><csymbol cd="latexml" id="S6.I1.i4.p1.3.m3.1.1.3.1.cmml" xref="S6.I1.i4.p1.3.m3.1.1.3">plus-or-minus</csymbol><apply id="S6.I1.i4.p1.3.m3.1.1.3.2.cmml" xref="S6.I1.i4.p1.3.m3.1.1.3.2"><divide id="S6.I1.i4.p1.3.m3.1.1.3.2.1.cmml" xref="S6.I1.i4.p1.3.m3.1.1.3.2"></divide><apply id="S6.I1.i4.p1.3.m3.1.1.3.2.2.cmml" xref="S6.I1.i4.p1.3.m3.1.1.3.2.2"><plus id="S6.I1.i4.p1.3.m3.1.1.3.2.2.1.cmml" xref="S6.I1.i4.p1.3.m3.1.1.3.2.2.1"></plus><ci id="S6.I1.i4.p1.3.m3.1.1.3.2.2.2.cmml" xref="S6.I1.i4.p1.3.m3.1.1.3.2.2.2">𝑎</ci><ci id="S6.I1.i4.p1.3.m3.1.1.3.2.2.3.cmml" xref="S6.I1.i4.p1.3.m3.1.1.3.2.2.3">𝑏</ci></apply><apply id="S6.I1.i4.p1.3.m3.1.1.3.2.3.cmml" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3"><root id="S6.I1.i4.p1.3.m3.1.1.3.2.3a.cmml" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3"></root><apply id="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.cmml" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.1.cmml" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3.2">superscript</csymbol><cn id="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.2.cmml" type="integer" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.2">2</cn><apply id="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3.cmml" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3"><divide id="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3.1.cmml" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3.1"></divide><ci id="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3.2.cmml" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3.2">𝐿</ci><cn id="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3.3.cmml" type="integer" xref="S6.I1.i4.p1.3.m3.1.1.3.2.3.2.3.3">2</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i4.p1.3.m3.1c">A=\pm\frac{a+b}{\sqrt{2^{L/2}}}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i4.p1.3.m3.1d">italic_A = ± divide start_ARG italic_a + italic_b end_ARG start_ARG square-root start_ARG 2 start_POSTSUPERSCRIPT italic_L / 2 end_POSTSUPERSCRIPT end_ARG end_ARG</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S6.p3.7">The fourth case only occurs for the states <math alttext="\ket{10101010...}" class="ltx_Math" display="inline" id="S6.p3.2.m1.1"><semantics id="S6.p3.2.m1.1a"><mrow id="S6.p3.2.m1.1.1.3" xref="S6.p3.2.m1.1.1.2.cmml"><mo id="S6.p3.2.m1.1.1.3.1" stretchy="false" xref="S6.p3.2.m1.1.1.2.1.cmml">|</mo><mrow id="S6.p3.2.m1.1.1.1.1" xref="S6.p3.2.m1.1.1.1.1.cmml"><mn id="S6.p3.2.m1.1.1.1.1.2" xref="S6.p3.2.m1.1.1.1.1.2.cmml">10101010</mn><mo id="S6.p3.2.m1.1.1.1.1.1" xref="S6.p3.2.m1.1.1.1.1.1.cmml">⁢</mo><mi id="S6.p3.2.m1.1.1.1.1.3" mathvariant="normal" xref="S6.p3.2.m1.1.1.1.1.3.cmml">…</mi></mrow><mo id="S6.p3.2.m1.1.1.3.2" stretchy="false" xref="S6.p3.2.m1.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.p3.2.m1.1b"><apply id="S6.p3.2.m1.1.1.2.cmml" xref="S6.p3.2.m1.1.1.3"><csymbol cd="latexml" id="S6.p3.2.m1.1.1.2.1.cmml" xref="S6.p3.2.m1.1.1.3.1">ket</csymbol><apply id="S6.p3.2.m1.1.1.1.1.cmml" xref="S6.p3.2.m1.1.1.1.1"><times id="S6.p3.2.m1.1.1.1.1.1.cmml" xref="S6.p3.2.m1.1.1.1.1.1"></times><cn id="S6.p3.2.m1.1.1.1.1.2.cmml" type="integer" xref="S6.p3.2.m1.1.1.1.1.2">10101010</cn><ci id="S6.p3.2.m1.1.1.1.1.3.cmml" xref="S6.p3.2.m1.1.1.1.1.3">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.2.m1.1c">\ket{10101010...}</annotation><annotation encoding="application/x-llamapun" id="S6.p3.2.m1.1d">| start_ARG 10101010 … end_ARG ⟩</annotation></semantics></math> and <math alttext="\ket{01010101...}" class="ltx_Math" display="inline" id="S6.p3.3.m2.1"><semantics id="S6.p3.3.m2.1a"><mrow id="S6.p3.3.m2.1.1.3" xref="S6.p3.3.m2.1.1.2.cmml"><mo id="S6.p3.3.m2.1.1.3.1" stretchy="false" xref="S6.p3.3.m2.1.1.2.1.cmml">|</mo><mrow id="S6.p3.3.m2.1.1.1.1" xref="S6.p3.3.m2.1.1.1.1.cmml"><mn id="S6.p3.3.m2.1.1.1.1.2" xref="S6.p3.3.m2.1.1.1.1.2.cmml">01010101</mn><mo id="S6.p3.3.m2.1.1.1.1.1" xref="S6.p3.3.m2.1.1.1.1.1.cmml">⁢</mo><mi id="S6.p3.3.m2.1.1.1.1.3" mathvariant="normal" xref="S6.p3.3.m2.1.1.1.1.3.cmml">…</mi></mrow><mo id="S6.p3.3.m2.1.1.3.2" stretchy="false" xref="S6.p3.3.m2.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.p3.3.m2.1b"><apply id="S6.p3.3.m2.1.1.2.cmml" xref="S6.p3.3.m2.1.1.3"><csymbol cd="latexml" id="S6.p3.3.m2.1.1.2.1.cmml" xref="S6.p3.3.m2.1.1.3.1">ket</csymbol><apply id="S6.p3.3.m2.1.1.1.1.cmml" xref="S6.p3.3.m2.1.1.1.1"><times id="S6.p3.3.m2.1.1.1.1.1.cmml" xref="S6.p3.3.m2.1.1.1.1.1"></times><cn id="S6.p3.3.m2.1.1.1.1.2.cmml" type="integer" xref="S6.p3.3.m2.1.1.1.1.2">01010101</cn><ci id="S6.p3.3.m2.1.1.1.1.3.cmml" xref="S6.p3.3.m2.1.1.1.1.3">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.3.m2.1c">\ket{01010101...}</annotation><annotation encoding="application/x-llamapun" id="S6.p3.3.m2.1d">| start_ARG 01010101 … end_ARG ⟩</annotation></semantics></math>, which exist in both tilings. The eight different amplitudes are visible in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2403.08892v3#S6.F9" title="Figure 9 ‣ VI Amplitudes of Scars ‣ Quantum Many-Body Scars for Arbitrary Integer Spin in 2+1D Abelian Gauge Theories"><span class="ltx_text ltx_ref_tag">9</span></a>. It shows the overlap of the electric field basis states <math alttext="\ket{\phi_{i}}" class="ltx_Math" display="inline" id="S6.p3.4.m3.1"><semantics id="S6.p3.4.m3.1a"><mrow id="S6.p3.4.m3.1.1.3" xref="S6.p3.4.m3.1.1.2.cmml"><mo id="S6.p3.4.m3.1.1.3.1" stretchy="false" xref="S6.p3.4.m3.1.1.2.1.cmml">|</mo><msub id="S6.p3.4.m3.1.1.1.1" xref="S6.p3.4.m3.1.1.1.1.cmml"><mi id="S6.p3.4.m3.1.1.1.1.2" xref="S6.p3.4.m3.1.1.1.1.2.cmml">ϕ</mi><mi id="S6.p3.4.m3.1.1.1.1.3" xref="S6.p3.4.m3.1.1.1.1.3.cmml">i</mi></msub><mo id="S6.p3.4.m3.1.1.3.2" stretchy="false" xref="S6.p3.4.m3.1.1.2.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.p3.4.m3.1b"><apply id="S6.p3.4.m3.1.1.2.cmml" xref="S6.p3.4.m3.1.1.3"><csymbol cd="latexml" id="S6.p3.4.m3.1.1.2.1.cmml" xref="S6.p3.4.m3.1.1.3.1">ket</csymbol><apply id="S6.p3.4.m3.1.1.1.1.cmml" xref="S6.p3.4.m3.1.1.1.1"><csymbol cd="ambiguous" id="S6.p3.4.m3.1.1.1.1.1.cmml" xref="S6.p3.4.m3.1.1.1.1">subscript</csymbol><ci id="S6.p3.4.m3.1.1.1.1.2.cmml" xref="S6.p3.4.m3.1.1.1.1.2">italic-ϕ</ci><ci id="S6.p3.4.m3.1.1.1.1.3.cmml" xref="S6.p3.4.m3.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.4.m3.1c">\ket{\phi_{i}}</annotation><annotation encoding="application/x-llamapun" id="S6.p3.4.m3.1d">| start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩</annotation></semantics></math> with a scar of a <math alttext="10\times 1" class="ltx_Math" display="inline" id="S6.p3.5.m4.1"><semantics id="S6.p3.5.m4.1a"><mrow id="S6.p3.5.m4.1.1" xref="S6.p3.5.m4.1.1.cmml"><mn id="S6.p3.5.m4.1.1.2" xref="S6.p3.5.m4.1.1.2.cmml">10</mn><mo id="S6.p3.5.m4.1.1.1" lspace="0.222em" rspace="0.222em" xref="S6.p3.5.m4.1.1.1.cmml">×</mo><mn id="S6.p3.5.m4.1.1.3" xref="S6.p3.5.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p3.5.m4.1b"><apply id="S6.p3.5.m4.1.1.cmml" xref="S6.p3.5.m4.1.1"><times id="S6.p3.5.m4.1.1.1.cmml" xref="S6.p3.5.m4.1.1.1"></times><cn id="S6.p3.5.m4.1.1.2.cmml" type="integer" xref="S6.p3.5.m4.1.1.2">10</cn><cn id="S6.p3.5.m4.1.1.3.cmml" type="integer" xref="S6.p3.5.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.5.m4.1c">10\times 1</annotation><annotation encoding="application/x-llamapun" id="S6.p3.5.m4.1d">10 × 1</annotation></semantics></math> ladder, with spin <math alttext="S=1" class="ltx_Math" display="inline" id="S6.p3.6.m5.1"><semantics id="S6.p3.6.m5.1a"><mrow id="S6.p3.6.m5.1.1" xref="S6.p3.6.m5.1.1.cmml"><mi id="S6.p3.6.m5.1.1.2" xref="S6.p3.6.m5.1.1.2.cmml">S</mi><mo id="S6.p3.6.m5.1.1.1" xref="S6.p3.6.m5.1.1.1.cmml">=</mo><mn id="S6.p3.6.m5.1.1.3" xref="S6.p3.6.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p3.6.m5.1b"><apply id="S6.p3.6.m5.1.1.cmml" xref="S6.p3.6.m5.1.1"><eq id="S6.p3.6.m5.1.1.1.cmml" xref="S6.p3.6.m5.1.1.1"></eq><ci id="S6.p3.6.m5.1.1.2.cmml" xref="S6.p3.6.m5.1.1.2">𝑆</ci><cn id="S6.p3.6.m5.1.1.3.cmml" type="integer" xref="S6.p3.6.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.6.m5.1c">S=1</annotation><annotation encoding="application/x-llamapun" id="S6.p3.6.m5.1d">italic_S = 1</annotation></semantics></math>, under periodic boundary conditions in the <math alttext="x" class="ltx_Math" display="inline" id="S6.p3.7.m6.1"><semantics id="S6.p3.7.m6.1a"><mi id="S6.p3.7.m6.1.1" xref="S6.p3.7.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S6.p3.7.m6.1b"><ci id="S6.p3.7.m6.1.1.cmml" xref="S6.p3.7.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.7.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S6.p3.7.m6.1d">italic_x</annotation></semantics></math> direction.</p> </div> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Mon Sep 2 22:02:52 2024 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; 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