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t-J model - Wikipedia
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href="https://bg.wikipedia.org/wiki/T-J_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB" title="T-J модел – Bulgarian" lang="bg" hreflang="bg" data-title="T-J модел" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Model_t-J" title="Model t-J – Catalan" lang="ca" hreflang="ca" data-title="Model t-J" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Mod%C3%A8le_t-J" title="Modèle t-J – French" lang="fr" hreflang="fr" data-title="Modèle t-J" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Modello_t-J" title="Modello t-J – Italian" lang="it" hreflang="it" data-title="Modello t-J" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Model_t-J" title="Model t-J – Polish" lang="pl" hreflang="pl" data-title="Model t-J" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/T-J%E6%A8%A1%E5%9E%8B" title="T-J模型 – Chinese" lang="zh" hreflang="zh" data-title="T-J模型" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/2D-Hubbard-model.png/330px-2D-Hubbard-model.png 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a8/2D-Hubbard-model.png 2x" data-file-width="355" data-file-height="258" /></a><figcaption>2D Hubbard model. The t-J model is the Hubbard model for <i>U</i> >> <i>t</i></figcaption></figure> <p>In <a href="/wiki/Solid-state_physics" title="Solid-state physics">solid-state physics</a>, the <b><i>t</i>-<i>J</i> model</b> is a model first derived by <a href="/w/index.php?title=J%C3%B3zef_Spa%C5%82ek&action=edit&redlink=1" class="new" title="Józef Spałek (page does not exist)">Józef Spałek</a><sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> to explain <a href="/wiki/Antiferromagnetism" title="Antiferromagnetism">antiferromagnetic</a> properties of <a href="/wiki/Mott_insulator" title="Mott insulator">Mott insulators</a>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> taking into account experimental results about the strength of electron-electron repulsion in these materials.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Material" title="Material">material</a> is modelled as a <a href="/wiki/Lattice_model_(physics)" title="Lattice model (physics)">lattice</a> with atoms in the sites with conduction <a href="/wiki/Electron" title="Electron">electrons</a> (or holes) moving between them, like in the <a href="/wiki/Hubbard_model" title="Hubbard model">Hubbard model</a>. Unlike the Hubbard model, the electrons are <a href="/wiki/Strongly_correlated_material" title="Strongly correlated material">strongly-correlated</a>, meaning the electrons are sensitive to reciprocal <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">coulombic repulsion</a>, and so are less likely to occupy lattice sites already occupied by another electron. In the basic Hubbard model, the repulsion, indicated by <i>U</i>, can be small or even zero, and electrons are more free to jump (<i>hopping</i>, parametrized by <i>t</i> as <i>transfer</i> or <a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling"><i>tunnel</i></a>) from one site to another. In the <i>t</i>-<i>J</i> model, instead of <i>U</i>, there is the parameter <i>J</i>, <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of the ratio <i>t</i>/<i>U</i>. </p><p>Like the Hubbard model, it is a prospective microscopic theory of <a href="/wiki/High_temperature_superconductivity" class="mw-redirect" title="High temperature superconductivity">high temperature superconductivity</a> in <a href="/wiki/Cuprate_superconductor" title="Cuprate superconductor">cuprate superconductors</a> which arise from <a href="/wiki/Dopant" title="Dopant">doped</a> antiferromagnets, particularly in the case where the lattice considered is the two-dimensional lattice.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Cuprate superconductors are currently (as of 2024) the superconductors with the highest known superconducting transition temperature at ambient pressure, but there is no consensus on the microscopic theory responsible for their superconducting transition. </p> <div class="mw-heading mw-heading2"><h2 id="The_Hamiltonian">The Hamiltonian</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-J_model&action=edit&section=1" title="Edit section: The Hamiltonian"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In quantum physics, system's models are usually based on the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> <a href="/wiki/Operator_(physics)" title="Operator (physics)">operator</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^<!-- ^ --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb06de5217295d7fbdbf68fb9c5309a513fc99e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\hat {H}}}"></span>, corresponding to the total energy of that system, including both <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> and <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a>. </p><p>The <i>t</i>-<i>J</i> Hamiltonian can be derived from the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^<!-- ^ --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb06de5217295d7fbdbf68fb9c5309a513fc99e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\hat {H}}}"></span> of the Hubbard model using the <a href="/wiki/Schrieffer%E2%80%93Wolff_transformation" title="Schrieffer–Wolff transformation">Schrieffer–Wolff transformation</a>, with the transformation generator depending on <i>t</i>/<i>U</i> and excluding the possibility for electrons to doubly occupy a lattice's site,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> which results in:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}=-t\sum _{\langle ij\rangle ,\sigma }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)+J\sum _{\langle ij\rangle }\left(\mathbf {S} _{i}\cdot \mathbf {S} _{j}-{\frac {n_{i}n_{j}}{4}}\right)+O(t^{3}/U^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^<!-- ^ --></mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−<!-- − --></mo> <mi>t</mi> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo> <mi>i</mi> <mi>j</mi> <mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo> <mo>,</mo> <mi>σ<!-- σ --></mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>σ<!-- σ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†<!-- † --></mo> </mrow> </msubsup> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>σ<!-- σ --></mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> <mo>.</mo> <mi mathvariant="normal">c</mi> <mo>.</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>J</mi> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo> <mi>i</mi> <mi>j</mi> <mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅<!-- ⋅ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=-t\sum _{\langle ij\rangle ,\sigma }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)+J\sum _{\langle ij\rangle }\left(\mathbf {S} _{i}\cdot \mathbf {S} _{j}-{\frac {n_{i}n_{j}}{4}}\right)+O(t^{3}/U^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41547eac904e3d222a9d5be511900db5e8f77cd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:66.502ex; height:7.176ex;" alt="{\displaystyle {\hat {H}}=-t\sum _{\langle ij\rangle ,\sigma }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)+J\sum _{\langle ij\rangle }\left(\mathbf {S} _{i}\cdot \mathbf {S} _{j}-{\frac {n_{i}n_{j}}{4}}\right)+O(t^{3}/U^{2})}"></span></dd></dl> <p>where the term in <i>t</i> corresponds to the kinetic energy and is equal to the one in the Hubbard model. The second one is the potential energy approximated at the second order, because this is an approximation of the Hubbard model in the limit <i>U</i> >> <i>t</i> developed in power of <i>t</i>. Terms at higher order can be added.<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The parameters are: </p> <ul><li><span class="sfrac nowrap"><span style="display:none;display:inline-block; vertical-align:top; text-align:center;"><span style="display:block;padding:0 0.1em;">Σ</span><span style="display:block; font-size:70%; line-height:1em;padding:0 0.1em;"><span style="position:relative; line-height:1em; margin-top:-0.5em; top:-0.5em;">⟨<i>ij</i>⟩</span></span></span></span> is the sum over nearest-neighbor sites <i>i</i> and <i>j</i>, for all sites, typically on a <a href="/wiki/Two-dimensional" class="mw-redirect" title="Two-dimensional">two-dimensional</a> <a href="/wiki/Square_lattice" title="Square lattice">square lattice</a>,</li> <li><i>c</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">†</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>iσ</i></sub></span></span>, <i>c</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>iσ</i></sub></span></span> are the <a href="/wiki/Fermion" title="Fermion">fermionic</a> <a href="/wiki/Creation_and_annihilation_operator" class="mw-redirect" title="Creation and annihilation operator">creation and annihilation operators</a> at site <i>i</i>,</li> <li><i>σ</i> is the <a href="/wiki/Spin_polarization" title="Spin polarization">spin polarization</a>,</li> <li><i>t</i> is the <a href="/wiki/Tight_binding#Second_quantization" title="Tight binding">hopping integral</a>,</li> <li><i>J</i> is the antiferromagnetic <a href="/wiki/Exchange_interaction" title="Exchange interaction">exchange coupling</a>, <i>J</i> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">4<i>t</i><sup>2</sup></span><span class="sr-only">/</span><span class="den"><i>U</i></span></span>⁠</span>,</li> <li><i>U</i> is the on-site <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">coulombic repulsion</a>, that must satisfy the condition for <i>U</i> >> <i>t</i>,</li> <li><i>n<sub>i</sub></i> = <span class="sfrac nowrap"><span style="display:none;display:inline-block; vertical-align:top; text-align:center;"><span style="display:block;padding:0 0.1em;">Σ</span><span style="display:block; font-size:70%; line-height:1em;padding:0 0.1em;"><span style="position:relative; line-height:1em; margin-top:-0.5em; top:-0.5em;"><i>σ</i></span></span></span></span><i>c</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">†</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>iσ</i></sub></span></span><i>c</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>iσ</i></sub></span></span> is the particle number at site <i>i</i> and can be maximum 1, so that double occupancy is forbidden (in the Hubbard model is possible),</li> <li><b>S</b><sub><i>i</i></sub> and <b>S</b><sub><i>j</i></sub> are the <a href="/wiki/Spin_(physics)#Mathematical_formulation" title="Spin (physics)">spins</a> on sites <i>i</i> and <i>j</i>,</li> <li>h. c. stands for <a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">Hermitian conjugate</a>,</li></ul> <p>If <i>n<sub>i</sub></i> = 1, that is when in the <a href="/wiki/Ground_state" title="Ground state">ground state</a>, there is just one electron per lattice's site (half-filling), the model reduces to the <a href="/wiki/Quantum_Heisenberg_model" title="Quantum Heisenberg model">Heisenberg model</a> and the ground state reproduce a <a href="/wiki/Dielectric" title="Dielectric">dielectric</a> antiferromagnets (<a href="/wiki/Mott_insulator" title="Mott insulator">Mott insulator</a>).<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>The model can be further extended considering also the next-nearest-neighbor sites and the <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</a> to set the ground state in function of the total number of particles:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {\hat {H}}}=t_{1}\sum \limits _{\langle i,j\rangle }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)\ +\ t_{2}\sum \limits _{\langle \langle i,j\rangle \rangle }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)\ +\ J\sum \limits _{\langle i,j\rangle }\left(\mathbf {S} _{i}\cdot \mathbf {S} _{j}-{\frac {n_{i}n_{j}}{4}}\right)-\ \mu \sum \limits _{i}n_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> <mo class="MJX-tex-caligraphic" mathvariant="script" stretchy="false">^<!-- ^ --></mo> </mover> </mrow> </mrow> </mrow> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <munder> <mo movablelimits="false">∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>σ<!-- σ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†<!-- † --></mo> </mrow> </msubsup> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>σ<!-- σ --></mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> <mo>.</mo> <mi mathvariant="normal">c</mi> <mo>.</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <munder> <mo movablelimits="false">∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo> <mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo> <mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>σ<!-- σ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†<!-- † --></mo> </mrow> </msubsup> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>σ<!-- σ --></mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> <mo>.</mo> <mi mathvariant="normal">c</mi> <mo>.</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <mi>J</mi> <munder> <mo movablelimits="false">∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅<!-- ⋅ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−<!-- − --></mo> <mtext> </mtext> <mi>μ<!-- μ --></mi> <munder> <mo movablelimits="false">∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {\hat {H}}}=t_{1}\sum \limits _{\langle i,j\rangle }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)\ +\ t_{2}\sum \limits _{\langle \langle i,j\rangle \rangle }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)\ +\ J\sum \limits _{\langle i,j\rangle }\left(\mathbf {S} _{i}\cdot \mathbf {S} _{j}-{\frac {n_{i}n_{j}}{4}}\right)-\ \mu \sum \limits _{i}n_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c792e7dda07d103adbf8006fdf09302bd6b1c59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:92.591ex; height:7.176ex;" alt="{\displaystyle {\mathcal {\hat {H}}}=t_{1}\sum \limits _{\langle i,j\rangle }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)\ +\ t_{2}\sum \limits _{\langle \langle i,j\rangle \rangle }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)\ +\ J\sum \limits _{\langle i,j\rangle }\left(\mathbf {S} _{i}\cdot \mathbf {S} _{j}-{\frac {n_{i}n_{j}}{4}}\right)-\ \mu \sum \limits _{i}n_{i},}"></span></dd></dl> <p>where ⟨...⟩ and ⟨⟨...⟩⟩ denote the nearest and next-nearest neighbors, respectively, with two different values for the hopping integral (<i>t</i><sub>1</sub> and <i>t</i><sub>2</sub>) and <i>μ</i> is the chemical potential. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-J_model&action=edit&section=2" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output 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(2020-01-31). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevResearch.2.013108">"Pairing in the two-dimensional Hubbard model from weak to strong coupling"</a>. <i>Physical Review Research</i>. <b>2</b> (1): 013108. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1909.00627">1909.00627</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020PhRvR...2a3108R">2020PhRvR...2a3108R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevResearch.2.013108">10.1103/PhysRevResearch.2.013108</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:202540002">202540002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Research&rft.atitle=Pairing+in+the+two-dimensional+Hubbard+model+from+weak+to+strong+coupling&rft.volume=2&rft.issue=1&rft.pages=013108&rft.date=2020-01-31&rft_id=info%3Aarxiv%2F1909.00627&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A202540002%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevResearch.2.013108&rft_id=info%3Abibcode%2F2020PhRvR...2a3108R&rft.aulast=R%C3%B8mer&rft.aufirst=Astrid+T.&rft.au=Maier%2C+Thomas+A.&rft.au=Kreisel%2C+Andreas&rft.au=Eremin%2C+Ilya&rft.au=Hirschfeld%2C+P.+J.&rft.au=Andersen%2C+Brian+M.&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevResearch.2.013108&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">This is done using a <a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">projector quantum operator</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo stretchy="false">^<!-- ^ --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a46a8cf7bc789e8c4f8de7ca0d9946a46fb8842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.812ex; height:2.843ex;" alt="{\displaystyle {\hat {P}}}"></span> that project <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb044130844bbac0e4a80636cf4058a4eaddccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.89ex; height:3.176ex;" alt="{\displaystyle {\hat {H}}_{t}}"></span> on the subspace where fermionic operators can not add an electron on a site already occupied (see next note)</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEckle2019" class="citation book cs1">Eckle, Hans-Peter (2019). <a rel="nofollow" class="external text" href="http://physics.gu.se/~tfkhj/HubbardEckle.pdf">"8.8.1 From the Hubbard to the t–J model: non-half filled band case"</a> <span class="cs1-format">(PDF)</span>. <i>Models of Quantum Matter</i>. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780199678839" title="Special:BookSources/9780199678839"><bdi>9780199678839</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=8.8.1+From+the+Hubbard+to+the+t%E2%80%93J+model%3A+non-half+filled+band+case&rft.btitle=Models+of+Quantum+Matter&rft.pub=Oxford+University+Press&rft.date=2019&rft.isbn=9780199678839&rft.aulast=Eckle&rft.aufirst=Hans-Peter&rft_id=http%3A%2F%2Fphysics.gu.se%2F~tfkhj%2FHubbardEckle.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIzyumovChashchin1998" class="citation journal cs1">Izyumov, Yu. 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(1998). <a rel="nofollow" class="external text" href="https://doi.org/10.5488%2FCMP.1.1.41">"tJ -model in terms of equations with variational derivatives"</a>. <i><a href="/wiki/Condensed_Matter_Physics" class="mw-redirect" title="Condensed Matter Physics">Condensed Matter Physics</a></i>. <b>1</b> (1): 41–56. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998CMPh....1...41I">1998CMPh....1...41I</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.5488%2FCMP.1.1.41">10.5488/CMP.1.1.41</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11254082">11254082</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Condensed+Matter+Physics&rft.atitle=tJ+-model+in+terms+of+equations+with+variational+derivatives&rft.volume=1&rft.issue=1&rft.pages=41-56&rft.date=1998&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11254082%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.5488%2FCMP.1.1.41&rft_id=info%3Abibcode%2F1998CMPh....1...41I&rft.aulast=Izyumov&rft.aufirst=Yu.+A.&rft.au=Chashchin%2C+N.+I.&rft_id=https%3A%2F%2Fdoi.org%2F10.5488%252FCMP.1.1.41&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarchev1998" class="citation journal cs1">Karchev, Naoum (1998). "Generalized <i>CP</i><sup>1</sup> model from the <i>t</i><sub>1</sub>-<i>t</i><sub>2</sub>-<i>J</i> model". <i>Phys. Rev. B</i>. <b>57</b> (17): 10913. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/9706105">cond-mat/9706105</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998PhRvB..5710913K">1998PhRvB..5710913K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevB.57.10913">10.1103/PhysRevB.57.10913</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:12865671">12865671</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.+B&rft.atitle=Generalized+CP%3Csup%3E1%3C%2Fsup%3E+model+from+the+t%3Csub%3E1%3C%2Fsub%3E-t%3Csub%3E2%3C%2Fsub%3E-J+model&rft.volume=57&rft.issue=17&rft.pages=10913&rft.date=1998&rft_id=info%3Aarxiv%2Fcond-mat%2F9706105&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A12865671%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevB.57.10913&rft_id=info%3Abibcode%2F1998PhRvB..5710913K&rft.aulast=Karchev&rft.aufirst=Naoum&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYanagisawa2008" class="citation journal cs1">Yanagisawa, Takashi (2008-02-12). <a rel="nofollow" class="external text" href="https://iopscience.iop.org/article/10.1088/1367-2630/10/2/023014">"Phase diagram of the<i>t</i>–<i>U</i><sup>2</sup>Hamiltonian of the weak coupling Hubbard model"</a>. <i>New Journal of Physics</i>. <b>10</b> (2): 023014. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0803.1739">0803.1739</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008NJPh...10b3014Y">2008NJPh...10b3014Y</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1367-2630%2F10%2F2%2F023014">10.1088/1367-2630/10/2/023014</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1367-2630">1367-2630</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:55405863">55405863</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=New+Journal+of+Physics&rft.atitle=Phase+diagram+of+thet%E2%80%93U%3Csup%3E2%3C%2Fsup%3EHamiltonian+of+the+weak+coupling+Hubbard+model&rft.volume=10&rft.issue=2&rft.pages=023014&rft.date=2008-02-12&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A55405863%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2008NJPh...10b3014Y&rft_id=info%3Aarxiv%2F0803.1739&rft.issn=1367-2630&rft_id=info%3Adoi%2F10.1088%2F1367-2630%2F10%2F2%2F023014&rft.aulast=Yanagisawa&rft.aufirst=Takashi&rft_id=https%3A%2F%2Fiopscience.iop.org%2Farticle%2F10.1088%2F1367-2630%2F10%2F2%2F023014&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-J_model&action=edit&section=3" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFazekas1999" class="citation book cs1">Fazekas, Patrik (1999). <i>Lectures on Correlation and Magnetism</i>. Series in Modern Condensed Matter Physics: Volume 5. Vol. 5. <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>. p. 199. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F2945">10.1142/2945</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4499-62-0" title="Special:BookSources/978-981-4499-62-0"><bdi>978-981-4499-62-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Correlation+and+Magnetism&rft.series=Series+in+Modern+Condensed+Matter+Physics%3A+Volume+5&rft.pages=199&rft.pub=World+Scientific&rft.date=1999&rft_id=info%3Adoi%2F10.1142%2F2945&rft.isbn=978-981-4499-62-0&rft.aulast=Fazekas&rft.aufirst=Patrik&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpałek2007" class="citation journal cs1">Spałek, Józef (2007). "<i>t</i>-<i>J</i> model then and now: A personal perspective from the pioneering times". <i>Acta Phys. Pol. A</i>. <b>111</b> (4): 409–424. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0706.4236">0706.4236</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007AcPPA.111..409S">2007AcPPA.111..409S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.12693%2FAPhysPolA.111.409">10.12693/APhysPolA.111.409</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:53117123">53117123</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Phys.+Pol.+A&rft.atitle=t-J+model+then+and+now%3A+A+personal+perspective+from+the+pioneering+times&rft.volume=111&rft.issue=4&rft.pages=409-424&rft.date=2007&rft_id=info%3Aarxiv%2F0706.4236&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A53117123%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.12693%2FAPhysPolA.111.409&rft_id=info%3Abibcode%2F2007AcPPA.111..409S&rft.aulast=Spa%C5%82ek&rft.aufirst=J%C3%B3zef&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDr_Mitchell" class="citation cs2">Dr Mitchell, <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=wtYQlELm3tk"><i>Electron interactions and the Hubbard model</i></a><span class="reference-accessdate">, retrieved <span class="nowrap">2022-08-29</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electron+interactions+and+the+Hubbard+model&rft.au=Dr+Mitchell&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DwtYQlELm3tk&rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></li></ul> <!-- NewPP limit report Parsed by mw‐web.eqiad.main‐6b6c9bdc8b‐r5vlk Cached time: 20241102164359 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1] CPU time usage: 0.213 seconds Real time usage: 0.384 seconds Preprocessor visited node count: 856/1000000 Post‐expand include size: 32592/2097152 bytes Template argument size: 123/2097152 bytes Highest expansion depth: 8/100 Expensive parser function count: 1/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 44996/5000000 bytes Lua time usage: 0.110/10.000 seconds Lua memory usage: 4249265/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 256.380 1 -total 78.74% 201.875 1 Template:Reflist 52.39% 134.322 9 Template:Cite_journal 7.48% 19.168 4 Template:Su 5.18% 13.293 1 Template:Sfrac 4.52% 11.585 2 Template:Cite_book 2.62% 6.713 1 Template:Citation 1.84% 4.721 2 Template:Underset 0.81% 2.084 1 Template:Main_other --> <!-- Saved in parser cache with key enwiki:pcache:idhash:10477190-0!canonical and timestamp 20241102164359 and revision id 1232260190. 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