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Berry connection in nLab
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<div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 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content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="solid_state_physics">Solid state physics</h4> <div class="hide"><div> <ul> <li> <p>basics</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Slater+determinant">Slater determinant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/degeneracy+pressure">degeneracy pressure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystal">crystal</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/crystallographic+group">crystallographic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bravais+lattice">Bravais lattice</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bloch+theory">Bloch theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electronic+band+structure">electronic band structure</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/valence+bundle">valence bundle</a>, <a class="existingWikiWord" href="/nlab/show/conduction+bundle">conduction bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/valence+band">valence band</a>, <a class="existingWikiWord" href="/nlab/show/conduction+band">conduction band</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Berry+connection">Berry connection</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Berry+phase">Berry phase</a>, <a class="existingWikiWord" href="/nlab/show/Zak+phase">Zak phase</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+of+matter">phase of matter</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/insulator">insulator</a>, <a class="existingWikiWord" href="/nlab/show/topological+insulator">topological insulator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metal">metal</a>/<a class="existingWikiWord" href="/nlab/show/conductor">conductor</a>, <a class="existingWikiWord" href="/nlab/show/semi-conductor">semi-conductor</a>, <a class="existingWikiWord" href="/nlab/show/semi-metal">semi-metal</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superconductor">superconductor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strange+metal">strange metal</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+phase+of+matter">topological phase of matter</a>, <a class="existingWikiWord" href="/nlab/show/topological+state+of+matter">topological state of matter</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetry+protected+topological+phase">symmetry protected topological phase</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory+classification+of+topological+phases+of+matter">K-theory classification of topological phases of matter</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a></p> <p><a class="existingWikiWord" href="/nlab/show/anyons">anyons</a></p> </li> <li> <p>Examples</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/graphene">graphene</a>, <a class="existingWikiWord" href="/nlab/show/Haldane+model">Haldane model</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+Hall+effect">quantum Hall effect</a>)</p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS-CFT+in+condensed+matter+physics">AdS-CFT in condensed matter physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Levin-Wen+model">Levin-Wen model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Josephson+effect">Josephson effect</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#for_anyon_statistics'>For anyon statistics</a></li> <li><a href='#experimental_observation'>Experimental observation</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>Generally, a <em>Berry phase</em> is a non-trivial phase picked up by a <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> of definite <a class="existingWikiWord" href="/nlab/show/energy">energy</a> under <a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">adiabatic</a> changes of the <a class="existingWikiWord" href="/nlab/show/quantum+system">quantum system</a>‘s <a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a> around a <a class="existingWikiWord" href="/nlab/show/loop">loop</a> in its parameter space.</p> <p>Specifically, in <a class="existingWikiWord" href="/nlab/show/condensed+matter+theory">condensed matter theory</a> and for a <a class="existingWikiWord" href="/nlab/show/crystal">crystalline</a> material with a set of isolated (gapped) <a class="existingWikiWord" href="/nlab/show/electronic+band+structure">electronic bands</a>, the <a class="existingWikiWord" href="/nlab/show/partial+derivatives">partial derivatives</a> along <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>-<a class="existingWikiWord" href="/nlab/show/vectors">vectors</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/Bloch+states">Bloch states</a>, projected back onto these states, turn out to canonically define a <a class="existingWikiWord" href="/nlab/show/connection+on+a+vector+bundle">connection</a> <a class="existingWikiWord" href="/nlab/show/differential+1-form">1-form</a> on the <a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a> (over the <a class="existingWikiWord" href="/nlab/show/Brillouin+torus">Brillouin torus</a>) which is spanned by these states. This is called a <em>Berry connection</em>, due to (<a href="#Berry84">Berry 84</a>, <a href="#Simon83">Simon 83</a>). The <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a> along this connection describes the change of <a class="existingWikiWord" href="/nlab/show/Bloch+states">Bloch states</a> under the <a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">adiabatic</a> change of their <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>/<a class="existingWikiWord" href="/nlab/show/wave+vector">wave vector</a>. The <a class="existingWikiWord" href="/nlab/show/curvature+2-form">curvature 2-form</a> of a Berry connection is accordingly called the <em>Berry curvature</em>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a> of such a Berry connection is called a <em><a class="existingWikiWord" href="/nlab/show/Berry+phase">Berry phase</a></em>, in general, and a <em><a class="existingWikiWord" href="/nlab/show/Zak+phase">Zak phase</a></em> (<a href="#Zak89">Zak 89</a>) when evaluated along one of the non-trivial 1-<a class="existingWikiWord" href="/nlab/show/cycles">cycles</a> of the <a class="existingWikiWord" href="/nlab/show/Brillouin+torus">Brillouin torus</a>.</p> <p>By default, this is understood to apply to the <a class="existingWikiWord" href="/nlab/show/valence+bundle">valence bundle</a> of a <a class="existingWikiWord" href="/nlab/show/crystal">crystalline</a> material, but the construction works more generally.</p> <p id="TopologicalOrder"> If in the case of a gapped <a class="existingWikiWord" href="/nlab/show/valence+bundle">valence bundle</a>, hence a <a class="existingWikiWord" href="/nlab/show/topological+insulator">topological insulator</a>-phase, the <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a> of the Berry connection is <a class="existingWikiWord" href="/nlab/show/non-abelian+group">non-abelian</a> (which may happen, as originally highlighted in <a href="#WilczekZee1984">Wilczek & Zee 84</a>) then one also says that the topological phase exhibits <em><a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a></em>.</p> <p>For <a class="existingWikiWord" href="/nlab/show/semimetals">semimetals</a> the Berry phases of the <a class="existingWikiWord" href="/nlab/show/valence+bundle">valence bundle</a> around their nodal loci of <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> 2 are a measure for the <a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> to adiabatically deforming the semimetal such as to open its gap closures, hence to become a (<a class="existingWikiWord" href="/nlab/show/topological+insulator">topological</a>) <a class="existingWikiWord" href="/nlab/show/insulator">insulator</a> (eg. <a href="#Vanderbilt18">Vanderbilt 18, 5.5.2</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Berry%27s+phase">Berry's phase</a> (needs to be merged)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">quantum adiabatic theorem</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The original article:</p> <ul> <li id="Berry84"><a class="existingWikiWord" href="/nlab/show/Michael+V.+Berry">Michael V. Berry</a>, <em>Quantal phase factors accompanying adiabatic changes</em>, Proc. R. Soc. Lond. A <strong>392</strong> (1984) 45–57 (<a href="https://doi.org/10.1098/rspa.1984.0023">doi:10.1098/rspa.1984.0023</a>, <a href="https://www.jstor.org/stable/2397741">jstor:2397741</a>)</li> </ul> <p>The formulation in terms of <a class="existingWikiWord" href="/nlab/show/connections">connections</a> on <a class="existingWikiWord" href="/nlab/show/fiber+bundles">fiber bundles</a> and their <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a>:</p> <ul> <li id="Simon83"><a class="existingWikiWord" href="/nlab/show/Barry+Simon">Barry Simon</a>, <em>Holonomy, the Quantum Adiabatic Theorem, and Berry’s Phase</em>, Phys. Rev. Lett. <strong>51</strong> (1983) 2167 (<a href="https://doi.org/10.1103/PhysRevLett.51.2167">doi:10.1103/PhysRevLett.51.2167</a>)</li> </ul> <p>Generalization to (connections with) <a class="existingWikiWord" href="/nlab/show/non-abelian+group">non-abelian</a> <a class="existingWikiWord" href="/nlab/show/holonomies">holonomies</a>:</p> <ul> <li id="WilczekZee1984"><a class="existingWikiWord" href="/nlab/show/Frank+Wilczek">Frank Wilczek</a>, <a class="existingWikiWord" href="/nlab/show/Anthony+Zee">Anthony Zee</a>, <em>Appearance of gauge structure in simple dynamical systems</em>, Physical Review Letters <strong>52</strong> 24 (1984) 2111 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1103/PhysRevLett.52.2111">doi:10.1103/PhysRevLett.52.2111</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>The special case of Berry phases around the 1-cycles of a <a class="existingWikiWord" href="/nlab/show/Brillouin+torus">Brillouin torus</a> – <a class="existingWikiWord" href="/nlab/show/Zak+phases">Zak phases</a>:</p> <ul> <li id="Zak89"><a class="existingWikiWord" href="/nlab/show/Joshua+Zak">Joshua Zak</a>, <em>Berry’s phase for energy bands in solids</em>, Phys. Rev. Lett. <strong>62</strong> (1989) 2747 (<a href="https://doi.org/10.1103/PhysRevLett.62.2747">doi:10.1103/PhysRevLett.62.2747</a>)</li> </ul> <p>Review and further discussion:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Berry">Michael Berry</a>, <em>The quantum phase, five years after</em>, in: <em>Geometric phases in physics</em>, Adv. Ser. Math. Phys. <strong>5</strong>, World Scientific (1989) 7–28 (<a class="existingWikiWord" href="/nlab/files/Berry-FiveYearsAfter.pdf" title="pdf">pdf</a>, <a href="https://doi.org/10.1142/0613">doi:10.1142/0613</a>)</p> </li> <li> <p>Ming-Che Chang, Qian Niu, <em>Berry curvature, orbital moment, and effective quantum theory of electrons in electromagnetic fields</em>, J. Phys.: Condens. Matter <strong>20</strong> (2008) 193202 (<a href="https://iopscience.iop.org/article/10.1088/0953-8984/20/19/193202">doi:10.1088/0953-8984/20/19/193202</a>)</p> </li> <li> <p>Di Xiao, Ming-Che Chang, Qian Niu, <em>Berry Phase Effects on Electronic Properties</em>, Rev. Mod. Phys. <strong>82</strong> (2010) 1959-2007 (<a href="https://arxiv.org/abs/0907.2021">arXiv:0907.2021</a>, <a href="https://doi.org/10.1103/RevModPhys.82.1959">doi:10.1103/RevModPhys.82.1959</a>)</p> </li> </ul> <p>With focus on <a class="existingWikiWord" href="/nlab/show/topological+phases+of+matter">topological phases of matter</a> (<a class="existingWikiWord" href="/nlab/show/topological+insulators">topological insulators</a>, <a class="existingWikiWord" href="/nlab/show/semimetals">semimetals</a>, etc.):</p> <ul> <li id="Vanderbilt18"> <p><a class="existingWikiWord" href="/nlab/show/David+Vanderbilt">David Vanderbilt</a>, <em>Berry Phases in Electronic Structure Theory – Electric Polarization, Orbital Magnetization and Topological Insulators</em>, Cambridge University Press (2018) (<a href="https://doi.org/10.1017/9781316662205">doi:10.1017/9781316662205</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/J%C3%A9r%C3%B4me+Cayssol">Jérôme Cayssol</a>, <a class="existingWikiWord" href="/nlab/show/Jean-No%C3%ABl+Fuchs">Jean-Noël Fuchs</a>, Section IV.C of: <em>Topological and geometrical aspects of band theory</em>, J. Phys. Mater. <strong>4</strong> (2021) 034007 (<a href="https://arxiv.org/abs/2012.11941">arXiv:2012.11941</a>, <a href="https://doi.org/10.1088/2515-7639/abf0b5">doi:10.1088/2515-7639/abf0b5</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tudor+D.+Stanescu">Tudor D. Stanescu</a>, Chapter 2 of: <em>Introduction to Topological Quantum Matter & Quantum Computation</em>, CRC Press 2020 (<a href="https://www.routledge.com/Introduction-to-Topological-Quantum-Matter--Quantum-Computation/Stanescu/p/book/9780367574116">ISBN:9780367574116</a>)</p> </li> </ul> <p>See also:</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Berry_connection_and_curvature">Berry connection and curvature</a></em></li> </ul> <h3 id="for_anyon_statistics">For anyon statistics</h3> <p>On <a class="existingWikiWord" href="/nlab/show/anyon+statistics">anyon phases</a> (specifically in the <a class="existingWikiWord" href="/nlab/show/quantum+Hall+effect">quantum Hall effect</a>) as <a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a> of a <a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">adiabatic</a> transport of anyon positions:</p> <ul> <li id="AvrosSchriefferWilczek84"><a class="existingWikiWord" href="/nlab/show/Daniel+P.+Arovas">Daniel P. Arovas</a>, <a class="existingWikiWord" href="/nlab/show/John+Robert+Schrieffer">John Robert Schrieffer</a>, <a class="existingWikiWord" href="/nlab/show/Frank+Wilczek">Frank Wilczek</a>, <em>Fractional Statistics and the Quantum Hall Effect</em>, Phys. Rev. Lett. 53, 722 (1984) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1103/PhysRevLett.53.722">doi:10.1103/PhysRevLett.53.722</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <h3 id="experimental_observation">Experimental observation</h3> <p>Experimental observation of <a class="existingWikiWord" href="/nlab/show/Zak+phases">Zak phases</a>:</p> <ul> <li>Marcos Atala, Monika Aidelsburger, Julio T. Barreiro, Dmitry Abanin, Takuya Kitagawa, Eugene Demler, Immanuel Bloch: <em>Direct measurement of the Zak phase in topological Bloch bands</em>, Nature Physics <strong>9</strong> (2013) 795–800 (<a href="https://doi.org/10.1038/nphys2790">doi:10.1038/nphys2790</a>)</li> </ul> <p>Proposal for experimental realization of <a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a> around <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a>=2 nodal loci of (quantum simulations of time+space inversion symmetric) <a class="existingWikiWord" href="/nlab/show/semi-metals">semi-metals</a>:</p> <ul> <li> <p>Dan-Wei Zhang, Y. X. Zhao, Rui-Bin Liu, Zheng-Yuan Xue, Shi-Liang Zhu, Z. D. Wang, <em>Quantum simulation of exotic PT-invariant topological nodal loop bands with ultracold atoms in an optical lattice</em>, Phys. Rev. A <strong>93</strong> (2016) 043617 (<a href="https://arxiv.org/abs/1601.00371">arXiv:1601.00371</a>, <a href="https://doi.org/10.1103/PhysRevA.93.043617">doi:10.1103/PhysRevA.93.043617</a>)</p> <blockquote> <p>(see sec II.A, these authors stand out as mentioning the relevant <a class="existingWikiWord" href="/nlab/show/KO-theory">KO-theory</a>)</p> </blockquote> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 13, 2022 at 21:50:05. 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