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braid group statistics in nLab
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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="topological_physics">Topological physics</h4> <div class="hide"><div> <p><strong>Topological Physics</strong> – Phenomena in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> controlled by the <a class="existingWikiWord" href="/nlab/show/topology">topology</a> (often: the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>) of the <a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a>.</p> <p>General theory:</p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/extended+topological+field+theory">extended</a>) <a class="existingWikiWord" href="/nlab/show/topological+field+theory">topological field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a>:</p> <ul> <li> <p><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/quantum+material">quantum material</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gapped+Hamiltonian">gapped Hamiltonian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a>, <a class="existingWikiWord" href="/nlab/show/symmetry+protected+trivial+order">symmetry protected trivial order</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+insulator">topological insulator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+Hall+effect">quantum Hall effect</a>, <a class="existingWikiWord" href="/nlab/show/quantum+spin+Hall+effect">quantum spin Hall effect</a></p> <p><a class="existingWikiWord" href="/nlab/show/anyons">anyons</a>, <a class="existingWikiWord" href="/nlab/show/braid+group+statistics">braid group statistics</a></p> </li> </ul> </li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/metamaterials">metamaterials</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+photonics">topological photonics</a> (<a class="existingWikiWord" href="/nlab/show/light+waves">light waves</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+phononics">topological phononics</a> (<a class="existingWikiWord" href="/nlab/show/sound+waves">sound waves</a>)</p> </li> </ul> <p>For <a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a></li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a></li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/quantum+chromodynamics">quantum chromodynamics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></li> </ul> </div></div> <h4 id="quantum_systems">Quantum systems</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>, <a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">dependent</a> <a class="existingWikiWord" href="/nlab/show/linear+type+theory">linear type theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a> in <a class="existingWikiWord" href="/nlab/show/quantum+information+theory+via+dagger-compact+categories">†-compact categories</a></p> <p><a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a></p> </li> </ul> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+systems">quantum systems</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/parameterized+quantum+systems">parameterized</a>, <a class="existingWikiWord" href="/nlab/show/open+quantum+system">open</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+state+collapse">quantum state collapse</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+decoherence">quantum decoherence</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">quantum adiabatic theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a></p> <p><a class="existingWikiWord" href="/nlab/show/Dyson+formula">Dyson formula</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+many-body+physics">quantum many-body physics</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/functorial+quantum+field+theory">functorial quantum field theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-</a>)<a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+material">quantum material</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/topological+phases+of+matter">topological</a>) <a class="existingWikiWord" href="/nlab/show/phases+of+matter">phases of matter</a></p> </li> </ul> <p><br /></p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> – <a class="existingWikiWord" href="/nlab/show/observables">observables</a> and <a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/quasi-state">quasi-state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> <p><a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a></p> <p><a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a></p> <p><a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> <p><a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+amplitude">vacuum amplitude</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+fluctuation">vacuum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/false+vacuum">false vacuum</a>, <a class="existingWikiWord" href="/nlab/show/tachyon">tachyon</a>, <a class="existingWikiWord" href="/nlab/show/Coleman-De+Luccia+instanton">Coleman-De Luccia instanton</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a></p> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/observables">observables</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+effect">quantum effect</a>, <a class="existingWikiWord" href="/nlab/show/effect+algebra">effect algebra</a></p> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>, <a class="existingWikiWord" href="/nlab/show/retarded+product">retarded product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra+of+observables">interacting field algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorems">theorems</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nuiten%27s+lemma">Nuiten's lemma</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> </ul> </li> </ul> </div> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information+via+dagger-compact+categories">quantum information via dagger-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+channel">quantum channel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+teleportation">quantum teleportation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+entanglement+entropy">topological entanglement entropy</a></p> </li> </ul> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+technology">quantum technology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+sensing">quantum sensing</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+communication">quantum communication</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+cryptography">quantum cryptography</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+machine+learning">quantum machine learning</a></p> </li> </ul> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+computing">quantum computing</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/qdit">qdit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+gate">quantum gate</a>, <a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adiabatic+quantum+computation">adiabatic quantum computation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measurement-based+quantum+computation">measurement-based quantum computation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+programming+language">quantum programming language</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+error+correction">quantum error correction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/HaPPY+code">HaPPY code</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Majorana+dimer+code">Majorana dimer code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+resonance+qbit">spin resonance qbit</a></p> </li> <li> <p>quantum algorithms:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grover%27s+algorithm">Grover's algorithm</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Shor%27s+algorithm">Shor's algorithm</a></p> </li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Anyon braiding</a></li> <ul> <li><a href='#AsGeneralizedBosonFermionStatistics'>As generalized boson/fermion statistics?</a></li> <li><a href='#AsAFictitiousAharonovBohmEffect'>Braiding of anyonic quanta – via “fictitious” AB-phases</a></li> <li><a href='#AsBraidingOfDefects'>Braiding of anyonic defects – via adiabatic Berry phases</a></li> <ul> <li><a href='#VortexAnyons'>Vortex anyons</a></li> <li><a href='#AnyonicBandNodes'>Anyonic band nodes?</a></li> <li><a href='#DefectBranes'>Defect branes</a></li> </ul> </ul> <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='#AnyonicTopologicalOrderInTermsOfBraidedFusionCategoriesReferences'>Anyonic topological order in terms of braided fusion categories</a></li> <ul> <li><a href='#claim_and_status'>Claim and status</a></li> <li><a href='#AnyonicOrderInMTheoryReferences'>In string/M-theory</a></li> <li><a href='#further_discussion'>Further discussion</a></li> </ul> <li><a href='#AnyonsInTheQuantumHallEffectReferences'>Anyons in fractional quantum Hall systems</a></li> <li><a href='#ObservationOfAnyonsInFQH'>Observation of anyons in fractional quantum Hall systems</a></li> <li><a href='#AnyonsInTopologicalSuperconductorsReferences'>Anyons in topological superconductors</a></li> <li><a href='#ReferencesVortexAnyons'>Defect anyons</a></li> <li><a href='#ReferencesAnyonicBraidingInMomentumSpace'>Anyons in momentum-space</a></li> <li><a href='#TopologicalQuantumComputationWithAnyons'>Topological quantum computation with anyons</a></li> <li><a href='#braid_group_representations_as_topological_quantum_gates'>Braid group representations (as topological quantum gates)</a></li> <li><a href='#CompilationToBraidGates'>Compilation to braid gate circuits</a></li> </ul> </ul> </div> <h2 id="Idea">Anyon braiding</h2> <p>In <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a>, <em>braid group statistics</em> or <em>anyon statistics</em> (sometimes: <em>plektons</em>) refers to an exotic kind of <a class="existingWikiWord" href="/nlab/show/particle+statistics">particle statistics</a> where the <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> of the <a class="existingWikiWord" href="/nlab/show/worldlines">worldlines</a> of certain effective <a class="existingWikiWord" href="/nlab/show/particles">particles</a> (“<a class="existingWikiWord" href="/nlab/show/anyons">anyons</a>”) in an effectively 1+2-dimensional <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> has the effect of transforming the <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> of the total <a class="existingWikiWord" href="/nlab/show/quantum+system">quantum system</a> by <a class="existingWikiWord" href="/nlab/show/unitary+operators">unitary operators</a> which constitute a <a class="existingWikiWord" href="/nlab/show/linear+representation">linear representation</a> of the <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a> – a <em><a class="existingWikiWord" href="/nlab/show/braid+representation">braid representation</a></em>.</p> <p>Often this is motivated as a generalization of the <a class="existingWikiWord" href="/nlab/show/boson">boson</a>- or <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a>-“statistics” which enters the <em><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></em>, see <a href="#AsGeneralizedBosonFermionStatistics">below</a>.</p> <p>But the actual mathematical nature of anyons must be different from that of elements of <a class="existingWikiWord" href="/nlab/show/boson">boson</a>/<a class="existingWikiWord" href="/nlab/show/fermion">fermion</a>-<a class="existingWikiWord" href="/nlab/show/Fock+spaces">Fock spaces</a> and often remains somewhat vague in existing discussions. One may recognize two different more concrete conceptualizations of <em>anyons</em> in the literature:</p> <ol> <li id="AnyonicQuanta"> <p><strong>anyonic quanta</strong> much like <a class="existingWikiWord" href="/nlab/show/boson">boson</a>/<a class="existingWikiWord" href="/nlab/show/fermion">fermion</a> <a class="existingWikiWord" href="/nlab/show/quanta">quanta</a> but subject to an additional global <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> by <a class="existingWikiWord" href="/nlab/show/Aharonov-Bohm+effect">Aharonov-Bohm phases</a> due to a <a class="existingWikiWord" href="/nlab/show/flat+connection">flat</a> <em><a class="existingWikiWord" href="/nlab/show/fictitious+gauge+field">fictitious gauge field</a></em> which is sourced by and coupled to each of the quanta;</p> <blockquote> <p>(this goes back to <a href="#ArovasSchriefferWilczekZee85">Arovas, Schrieffer, Wilczek & Zee 1985</a>, further developed in <a href="#ChenWilczekWittenHalperin89">Chen, Wilczek, Witten & Halperin 1989</a>, see <a href="#AsAFictitiousAharonovBohmEffect">below</a>)</p> </blockquote> </li> <li id="AnyonicDefects"> <p><strong>anyonic defects</strong> like <a class="existingWikiWord" href="/nlab/show/vortices">vortices</a> or other <a class="existingWikiWord" href="/nlab/show/solitons">solitons</a>, whose position is a <a class="existingWikiWord" href="/nlab/show/parameterized+quantum+system">classical parameter</a> to the <a class="existingWikiWord" href="/nlab/show/quantum+system">quantum system</a>, the <em><a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">adiabatic movement</a></em> of which acts by <a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a> on the quantum <a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>.</p> <blockquote> <p>(e.g. <a href="#AvrosSchriefferWilczek84">Avros, Schrieffer & Wilczek 1984</a>, see <a href="#AsBraidingOfDefects">further below</a>)</p> </blockquote> </li> </ol> <p>The concept of anyons is particularly well motivated in <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a>, where effectively 2-dimensional <a class="existingWikiWord" href="/nlab/show/quantum+materials">quantum materials</a> are common place (eg. <a class="existingWikiWord" href="/nlab/show/graphene">graphene</a>) or where particles may otherwise be constrained to move in a <a class="existingWikiWord" href="/nlab/show/plane">plane</a>, such as in the <a class="existingWikiWord" href="/nlab/show/quantum+Hall+effect">quantum Hall effect</a>. There is a multitude of <em><a class="existingWikiWord" href="/nlab/show/model+%28in+theoretical+physics%29">models</a></em> in <a class="existingWikiWord" href="/nlab/show/condensed+matter+theory">condensed matter theory</a> (mostly <a class="existingWikiWord" href="/nlab/show/lattice+models">lattice models</a>, such as <a class="existingWikiWord" href="/nlab/show/string-net+models">string-net models</a>) which <em>theoretically</em> realize anyon braid group statistics, and there are some first <a class="existingWikiWord" href="/nlab/show/experiment">experimental</a> indications of anyonic phenomena in actual materials (see the references under <em><a href="#ObservationOfAnyonsInFQH">Experimental Realization</a></em>) below.</p> <p>Specifically, in the context of <a class="existingWikiWord" href="/nlab/show/topological+phases+of+matter">topological phases of matter</a>, the (potential) presence of anyons has come to be known as the case of <em><a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a></em>, see there for more.</p> <p>Besides general curiosity, much of the interest in anyonic braid group statistics lies in the fact that these <a class="existingWikiWord" href="/nlab/show/braid+representations">braid representations</a> are imagined to potentially serve as <a class="existingWikiWord" href="/nlab/show/quantum+gates">quantum gates</a> in <a class="existingWikiWord" href="/nlab/show/topological+quantum+computers">topological quantum computers</a>. See there for more.</p> <h3 id="AsGeneralizedBosonFermionStatistics">As generalized boson/fermion statistics?</h3> <p>In <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>, one speaks of the “statistics” of a <a class="existingWikiWord" href="/nlab/show/particle">particle</a> species when referring to the <a class="existingWikiWord" href="/nlab/show/linear+representation">linear representation</a> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-particle <a class="existingWikiWord" href="/nlab/show/wavefunctions">wavefunctions</a> form under the the <a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sym</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sym(n)</annotation></semantics></math> which <a class="existingWikiWord" href="/nlab/show/permutation">permutes</a> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> particles.</p> <p>While there is a rich <a class="existingWikiWord" href="/nlab/show/representation+theory+of+the+symmetric+group">representation theory of the symmetric group</a>, the <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> says, when it applies, that for <em><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> quanta</em> only the simplest two possibilities may occur:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bosons">bosons</a> (such as <a class="existingWikiWord" href="/nlab/show/photons">photons</a>) transform under the <a class="existingWikiWord" href="/nlab/show/trivial+representation">trivial representation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sym</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sym(n)</annotation></semantics></math>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> (such as <a class="existingWikiWord" href="/nlab/show/electrons">electrons</a>) transform under the <a class="existingWikiWord" href="/nlab/show/sign+representation">sign representation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sym</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sym(n)</annotation></semantics></math>.</p> <p>(see also at <em><a class="existingWikiWord" href="/nlab/show/Slater+determinant">Slater determinant</a></em>)</p> </li> </ul> <p>Now, the <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> strands covers the symmetric group</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Br</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>↠</mo><mi>Sym</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Br(n) \twoheadrightarrow Sym(n) </annotation></semantics></math></div> <p>which allows one to regard any <a class="existingWikiWord" href="/nlab/show/linear+representation">linear representation</a> of the symmetric group also as a particular <a class="existingWikiWord" href="/nlab/show/braid+group+representation">braid group representation</a>. But by its definition, the <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a> may be understood as the group of <a class="existingWikiWord" href="/nlab/show/isotopy">isotopy</a>-classes of <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint</a> <a class="existingWikiWord" href="/nlab/show/timelike">timelike</a> <a class="existingWikiWord" href="/nlab/show/worldlines">worldlines</a> in an effectively 2+1 dimensional <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, with the group operation being <a class="existingWikiWord" href="/nlab/show/concatenation">concatenation</a> of worldlines.</p> <p>In this sense, one may imagine that <em>any</em> <a class="existingWikiWord" href="/nlab/show/braid+group+representations">braid group representations</a> may generalize the boson/fermion statistics in 2+1 dimensions. Texts typically suggest that this applies to <em><a class="existingWikiWord" href="/nlab/show/quasiparticle">quasiparticles</a></em>.</p> <p>The term <em>anyon</em> (due to <a href="#Wilczek82b">Wilczek 1982b</a>) is a pun on this state of affairs that <em>any</em> statistics “in between” boson- and fermion-statistics may be allowed.</p> <p>On the other hand, anyonic braiding is conceptually different from boson/fermion statistics – if it were on the same footing then the <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> would <em>rule out</em> anyonic braiding. This is acknowledged by <a href="#ChenWilczekWittenHalperin89">Chen, Wilczek, Witten & Halperin 1989, p. 352</a> (cf. also <a href="#Wilczek90">Wilczek 1990, §I.2, pp. 11</a>):</p> <blockquote> <p>Once the <a class="existingWikiWord" href="/nlab/show/permutation+group">permutation group</a> is replaced by the <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a>, the simple construction of passing from the solution to the one-particle problems to the solution of many-particle problems, familiar from the free bosons and free fermions, does not work anymore.</p> </blockquote> <p><br /></p> <h3 id="AsAFictitiousAharonovBohmEffect">Braiding of anyonic quanta – via “fictitious” AB-phases</h3> <p>A concrete model for <em><a href="#AnyonicQuanta">anyonic quanta</a></em> via otherwise <a class="existingWikiWord" href="/nlab/show/free+field">free</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> in 2d <a class="existingWikiWord" href="/nlab/show/interaction">interacting</a> through a <a class="existingWikiWord" href="/nlab/show/flat+connection">flat</a> “<a class="existingWikiWord" href="/nlab/show/fictitious+gauge+field">fictitious gauge field</a>” was proposed in <a href="#ArovasSchriefferWilczekZee85">Arovas, Schrieffer, Wilczek & Zee 1985</a> and developed in <a href="#ChenWilczekWittenHalperin89">Chen, Wilczek, Witten & Halperin 1989</a> and <a href="#IengoLechner92">Iengo & Lechner 1992</a> (the model has been advertized in early reviews, e.g. <a href="#Wilczek90">Wilczek 1990, §I.3</a> and <a href="#Wilczek91">Wilczek 1991</a>, but seems not to have been developed much since):</p> <p>This model regards anyons as <em>a priori</em> <a class="existingWikiWord" href="/nlab/show/free+field">free</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a>, but equipped now with a non-local mutual <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> via a “fictitious gauge field” (<a href="#ChenWilczekWittenHalperin89">CWWH89, §2</a>), in that each of the particles is modeled as the singular source of a <a class="existingWikiWord" href="/nlab/show/flat+connection">flat</a> <a class="existingWikiWord" href="/nlab/show/circle+bundle+with+connection">circle connection</a> (a <a class="existingWikiWord" href="/nlab/show/vector+potential">vector potential</a> with vanishing <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a>), which hence exerts no <a class="existingWikiWord" href="/nlab/show/Lorentz+force">Lorentz force</a> but has the effect that globally each other particle is subject to the same <a class="existingWikiWord" href="/nlab/show/Aharonov-Bohm+effect">Aharonov-Bohm effect</a> as would be caused by a tuple of infinite <a class="existingWikiWord" href="/nlab/show/solenoids">solenoids</a> piercing through each of the other particle’s positions.</p> <p>For emphasis, from <a href="#ChenWilczekWittenHalperin89">CWWH89, p. 359</a>:</p> <blockquote> <p>Here the particles are to be regarded (in the absence of interactions) as fermions; the interaction then makes them anyons with statistical parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>=</mo><mi>π</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>1</mn><mo stretchy="false">/</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\theta = \pi(1 - 1/n)</annotation></semantics></math>.</p> </blockquote> <p>It follows (<a href="#Wu84">Wu 1984</a>, <a href="#ImboImboSudarshan90">Imbo, Imbo & Sudarshan 1990</a>) that (quoting from <a href="#FroehlichGabbianiMarchetti90">Fröhlich, Gabbiani & Marchetti 1990, p. 20</a>):</p> <blockquote> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>∈</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\theta \in\!\!\!\!\!/ \frac{1}{2}\mathbb{Z}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">Hilbert space</a> of <a class="existingWikiWord" href="/nlab/show/anyon">anyon</a> <a class="existingWikiWord" href="/nlab/show/wave+functions">wave functions</a> must be chosen to be a space of <a class="existingWikiWord" href="/nlab/show/multi-valued+functions">multi-valued functions</a> with half-<a class="existingWikiWord" href="/nlab/show/monodromies">monodromies</a> given by the <a class="existingWikiWord" href="/nlab/show/complex+phase">phase</a> factors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi mathvariant="normal">i</mi><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">exp(2 \pi \mathrm{i} \theta)</annotation></semantics></math>. Such wave functions can be viewed as single-valued functions on the <a class="existingWikiWord" href="/nlab/show/universal+cover">universal cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>M</mi><mo>˜</mo></mover> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\widetilde M_n</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">M_n</annotation></semantics></math> [the <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">configuration space of points</a>&rbrack.</p> </blockquote> <p>Further discussion of anyon-<a class="existingWikiWord" href="/nlab/show/wavefunctions">wavefunctions</a> as <a class="existingWikiWord" href="/nlab/show/multi-valued+functions">multi-valued functions</a> on a <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">configuration space of points</a>, hence <a class="existingWikiWord" href="/nlab/show/equivariant+functions">equivariant functions</a> on its <a class="existingWikiWord" href="/nlab/show/universal+cover">universal cover</a>: <a href="#BCMS93">BCMS93, §1</a>, <a href="#MundSchrader93">Mund & Schrader 1995</a>, <a href="#DFT97">DFT97, §1</a> <a href="#Myrheim99">Myrheim 1999</a>, <a href="#DMV03">DMV03</a>, <a href="#MurthyShankar09">Murthy & Shankar 2009</a></p> <p>Incidentally, the <a class="existingWikiWord" href="/nlab/show/quasi-particle">quasi-particle</a>-excitations <em>of</em> (or <em>in</em>) a gas of such Aharonov-Bohm phased anyons are argued to be <a class="existingWikiWord" href="/nlab/show/vortices">vortices</a> (<a href="#ChenWilczekWittenHalperin89">CWWH89, p. 457</a>):</p> <blockquote> <p>we are led to conclude that <em>in anyon superconductivity, charged quasi-particles and vortices do not constitute two separate sorts of elementary excitations - they are one and the same.</em></p> </blockquote> <p>This seamlessly leads over to:</p> <p><br /></p> <h3 id="AsBraidingOfDefects">Braiding of anyonic defects – via adiabatic Berry phases</h3> <p>In practice, many (most?) incarnations of the concept of anyons are <em><a href="#AnyonicDefects">anyonic defects</a></em> – <a class="existingWikiWord" href="/nlab/show/non-perturbative+effect">non-perturbative</a> <a class="existingWikiWord" href="/nlab/show/soliton">solitonic</a> <a class="existingWikiWord" href="/nlab/show/defects">defects</a> (of <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a>=2), akin to <em><a class="existingWikiWord" href="/nlab/show/vortices">vortices</a></em> in fluids:</p> <blockquote> <p><em>Anyonic particles are best viewed as a kind of topological defects that reveal non-trivial properties of the ground state.</em> [<a href="#Kitaev06">Kitaev 2006, p. 4</a>]</p> <p><em>Anyons can arise in two ways: as localised excitations of an interacting quantum Hamiltonian or as defects in an ordered system.</em> [<a href="topological+quantum+computation#DasSarmaFreedmanNayak15">Das Sarma, Freedman & Nayak 2015, p. 1</a>]</p> </blockquote> <p>(Compare also the original discussions in <a href="#GoldinMenikoffSharp81">Goldin, Menikoff & Sharp 1981, §III</a>, <a href="#Wilczek82a">Wilczek 1982a</a> & <a href="#Wilczek90">Wilczek 1990, p. 5</a>, which offer a quantum particle “<a class="existingWikiWord" href="/nlab/show/bound+state">bound</a>” to a classical & infinite <a class="existingWikiWord" href="/nlab/show/solenoid">solenoid</a> – hence a 2d <a class="existingWikiWord" href="/nlab/show/magnetic+monopole">magnetic monopole</a> <a class="existingWikiWord" href="/nlab/show/defect">defect</a> – as a decent model for an anyon.)</p> <p>But <a class="existingWikiWord" href="/nlab/show/defects">defects</a> are a kind of <a class="existingWikiWord" href="/nlab/show/boundary+conditions">boundary conditions</a>, hence <em>external parameters</em> or <em><a class="existingWikiWord" href="/nlab/show/background+fields">background fields</a></em> for the actual <a class="existingWikiWord" href="/nlab/show/quantum+field">quantum field</a>.</p> <p>Concretely, a widely appreciated proposal (<a href="#MooreRead91">Moore & Read 1991</a>, <a href="#ReadRezayi99">Read & Rezayi 1999</a>) identifies anyonic ground state <a class="existingWikiWord" href="/nlab/show/wavefunctions">wavefunctions</a> with <a class="existingWikiWord" href="/nlab/show/conformal+blocks">conformal blocks</a> of a <a class="existingWikiWord" href="/nlab/show/2d+CFT">2d CFT</a> – <a class="existingWikiWord" href="/nlab/show/AdS3-CFT2+and+CS-WZW+correspondence">hence</a> with <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> <a class="existingWikiWord" href="/nlab/show/quantum+states">states</a> – with prescribed poles at the location of the anyons.</p> <p>Now the <a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">quantum adiabatic theorem</a> says that the sufficiently slow motion of such external parameters transforms the quantum <a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a> by <a class="existingWikiWord" href="/nlab/show/unitary+operators">unitary operators</a> (“<a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a>”, see also at <em><a class="existingWikiWord" href="/nlab/show/adiabatic+quantum+computation">adiabatic quantum computation</a></em>). This suggests (<a href="#AvrosSchriefferWilczek84">Avros, Schrieffer & Wilczek 1984, p. 1</a>, <a href="#FreedmanKitaevLarsenWang03">Freedman, Kitaev, Larsen & Wang 2003, pp. 6</a>, <a href="#NayakSimonSternFreedman08">Nayak, Simon, Stern & Freedman 2008, §II.A.2 (p. 6)</a>, <a href="adiabatic+quantum+computation#ChengGalitskiDasSarma11">Cheng, Galitski & Das Sarma 2011, p. 1</a>) that:</p> <p> <div class='num_defn' id='DefectBraiding'> <h6>Definition</h6> <p><strong>(adiabatic defect braiding)</strong> <br /> <em>Anyon braiding statistics is the <a class="existingWikiWord" href="/nlab/show/braid+group+representation">braid group representation</a> on a quantum <a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a> induced by <a class="existingWikiWord" href="/nlab/show/adiabatic+theorem">adiabatic</a> <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> of <a class="existingWikiWord" href="/nlab/show/TQFT">topological</a> <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a>=2 <a class="existingWikiWord" href="/nlab/show/defects">defects</a> in their <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">configuration space</a>.</em></p> <p>Here</p> <ol> <li> <p>for point-defects the “configuration space” is the <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">configuration space of points</a> in a <a class="existingWikiWord" href="/nlab/show/surface">surface</a> (as briefly touched upon already in <a href="#LeinaasMyrheim77">Leinaas & Myrheim 1977, pp. 22</a>, <a href="#Wilczek82b">Wilczek 1982b, p. 959</a>), such as in the <a class="existingWikiWord" href="/nlab/show/plane">plane</a>, in which case its fundamental group is the <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a>;</p> </li> <li> <p>“topological” is meant as in <a class="existingWikiWord" href="/nlab/show/topological+quantum+field+theory">topological quantum field theory</a>: The induced <a class="existingWikiWord" href="/nlab/show/adiabatic+theorem">adiabatic</a> <a class="existingWikiWord" href="/nlab/show/unitary+transformation">unitary transformation</a> is demanded/assumed to depend only on the <a class="existingWikiWord" href="/nlab/show/isotopy">isotopy</a>-class of the defect-<a class="existingWikiWord" href="/nlab/show/worldlines">worldlines</a>, hence only on the underlying <a class="existingWikiWord" href="/nlab/show/braid+group">braid</a>-pattern.</p> </li> </ol> <p></p> </div> </p> <p>This notion of anyon “statistics” is at least tacitly implicit in much of the literature on anyons in <a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a>, such as in the popular graphics depicting anyon <a class="existingWikiWord" href="/nlab/show/worldlines">worldlines</a> as the <a class="existingWikiWord" href="/nlab/show/Wilson+lines">Wilson lines</a> in <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>. (see the graphics <a href="#BraidingOfNodalPointsInMomentumSpace">below</a>).</p> <p>Indeed, the effects of <a class="existingWikiWord" href="/nlab/show/adiabatic+theorem">adiabatic</a> braiding of <a class="existingWikiWord" href="/nlab/show/defects">defects</a> in <a class="existingWikiWord" href="/nlab/show/quantum+materials">quantum materials</a> has been understood and discussed before and in parallel to the term “anyon” becoming established: <a href="#Mermin79">Mermin 1979</a>, <a href="#LoPreskill93">Lo & Preskill 1993</a>.</p> <p>A concrete realistic example of defect anyons are <a class="existingWikiWord" href="/nlab/show/vortex">vortex</a> anyons see <a href="#VortexAnyons">below</a>. But the notion of codimension=2 defects subsumes situations that are quite different from the <a class="existingWikiWord" href="/nlab/show/quasiparticle">quasiparticle</a>-excitations imagined in traditional texts on anyons, such as:</p> <ul> <li> <p><a href="#AnyonicBandNodes">anyonic band nodes</a></p> </li> <li> <p><a href="#DefectBranes">defect branes</a>.</p> </li> </ul> <h4 id="VortexAnyons">Vortex anyons</h4> <p>Specifically, <em><a class="existingWikiWord" href="/nlab/show/vortex">vortex</a> anyons</em> are realized in <a class="existingWikiWord" href="/nlab/show/Bose-Einstein+condensates">Bose-Einstein condensates</a> (<a href="#MPSS19">MPSS19</a>, following <a href="#PFCZ01">PFCZ01</a>) and in (other) <a class="existingWikiWord" href="/nlab/show/superfluids">superfluids</a> (<a href="#MMN21">MMN21</a>).</p> <p id="VorticesInAnyonCondensate"> In fact, defect-type <em>vortex anyons</em> generically appear in <a class="existingWikiWord" href="/nlab/show/condensates">condensates</a> <em>of non-defect anyons</em> (<a href="#CDLR19">CDLR19</a>):</p> <div style="margin: -20px 0px 20px 10px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/VorticesInAnyonCondensate.jpg" width="430px" /> <figcaption style="text-align: center">From <a href="#CDLR19">CDLR19</a></figcaption> </figure> </div> <p>A theoretical model of vortex anyons in a <a class="existingWikiWord" href="/nlab/show/Higgs+field">Higgs field</a> <a class="existingWikiWord" href="/nlab/show/interaction">coupled to</a> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> is discussed in <a href="#FroehlichMarchetti88">Fröhlich & Marchetti 1988</a>. An instructive <a class="existingWikiWord" href="/nlab/show/lattice+model">lattice model</a> of vortex anyons is analyzed in detail in <a href="#Kitaev06">Kitaev 2006</a>.</p> <p>Much attention in current efforts towards realizing <a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a> is being paid to anyons realized as <a class="existingWikiWord" href="/nlab/show/Majorana+zero+modes">Majorana zero modes</a> <a class="existingWikiWord" href="/nlab/show/bound+state">bound</a> to <a class="existingWikiWord" href="/nlab/show/vortices">vortices</a> (<a href="topological+quantum+computation#DasSarmaFreedmanNayak15">Das Sarma, Freedman & Nayak 2015</a>, cf. <a href="su2-anyon#MMBDRSC19">MMBDRSC19</a>).</p> <p id="MayGeneralize"> This situation may generalize to <a class="existingWikiWord" href="/nlab/show/parafermion">parafermion</a>-<a class="existingWikiWord" href="/nlab/show/su%282%29-anyons">su(2)-anyons</a>, where</p> <blockquote> <p>each (anti)<a class="existingWikiWord" href="/nlab/show/soliton">soliton</a> carries <a class="existingWikiWord" href="/nlab/show/parafermion">parafermion</a> zero mode which supplies it with the non-Abelian statistics [<a href="parafermion#Tsvelik2014a">Tsvelik 2014a</a>, <a href="https://arxiv.org/pdf/1404.2840.pdf#page=2">p. 2</a>, cf. <a href="parafermion#Borcherding18">Borcherding 2018, pp. 3</a>].</p> </blockquote> <p id="BraidingOfNodalPointsInMomentumSpace"> <strong>Braiding of nodal points in momentum space</strong> (graphics from <a href="su2-anyon#MMBDRSC19">MMBDRSC19</a>, <a href="https://www.nature.com/articles/s41467-019-10397-5.pdf#page=2">Fig. 1</a>):</p> <div style="margin: -20px 0px 20px 10px"> <img src="/nlab/files/MZMVortex-220530.jpg" width="440px" /> </div> <h4 id="AnyonicBandNodes">Anyonic band nodes?</h4> <p>Around 2020 the view has been emerging that also defects “in momentum/reciprocal space” may behave as anyonic defects under braiding in momentum space. This applies concretely to nodal points (where <a class="existingWikiWord" href="/nlab/show/electron+bands">electron bands</a> touch or cross) in the momentum space <a class="existingWikiWord" href="/nlab/show/Brillouin+torus">Brillouin torus</a> of <a class="existingWikiWord" href="/nlab/show/topological+semi-metals">topological semi-metals</a>:</p> <blockquote> <p>here are band crossing points, henceforth called <a class="existingWikiWord" href="/nlab/show/vortices">vortices</a> [<a href="#AhnParkYang19">Ahnm Park & Yang 2019</a>]</p> <p>a new type non-Abelian “braiding” of nodal-line rings inside the momentum space [<a href="#TiwariBzdusek20">Tiwari & Bzdušek 2020</a>]</p> </blockquote> <div style="margin: -20px 0px 20px 10px"> <img src="/nlab/files/BraidingOfBandNodes-220529.jpg" width="500px" /> </div> <blockquote> <p>(graphics from <a class="existingWikiWord" href="/schreiber/show/Topological+Quantum+Computation+in+TED-K">SS22</a>)</p> </blockquote> <p>Curiously, these reciprocal/momemtum space anyons lend themselves to tractable laboratory manipulation in a way that has remained notoriously elusive for “position space” anyons:</p> <blockquote> <p>Our work opens up routes to readily manipulate Weyl nodes using only slight external parameter changes, paving the way for the practical realization of reciprocal space braiding [<a href="#CBSM22">CBSM22</a>],</p> </blockquote> <p>specifically</p> <blockquote> <p>it is possible to controllably braid Kagome band nodes in monolayer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">Si</mi> <mn>2</mn></msub><msub><mi mathvariant="normal">O</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{Si}_2 \mathrm{O}_3</annotation></semantics></math> using strain and/or an external electric field [<a href="#PBMS22">PBMS22</a>],</p> </blockquote> <p>leading to:</p> <blockquote> <p>new opportunities for exploring non-Abelian braiding of band crossing points (nodes) in reciprocal space, providing an alternative to the real space braiding exploited by other strategies.</p> <p>Real space braiding is practically constrained to boundary states, which has made experimental observation and manipulation difficult; instead, reciprocal space braiding occurs in the bulk states of the band structures and we demonstrate in this work that this provides a straightforward platform for non-Abelian braiding. [<a href="#PBSM22">PBSM22</a>].</p> </blockquote> <h4 id="DefectBranes">Defect branes</h4> <p>In <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>, <em><a class="existingWikiWord" href="/nlab/show/defect+branes">defect branes</a></em> are <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> or <a class="existingWikiWord" href="/nlab/show/M-branes">M-branes</a> of <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a>=2, such as <a class="existingWikiWord" href="/nlab/show/D7-branes">D7-branes</a> in <a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">type IIB string theory</a> or <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a>-<a class="existingWikiWord" href="/nlab/show/brane+intersection">intersections</a> on “<a class="existingWikiWord" href="/nlab/show/3-brane+in+6d">M3-branes</a>” in <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>.</p> <p>It has been suggest in <a href="defect+brane#deBoerShigemori12">deBoer & Shigemori 2012, p. 65</a> that these could behave like anyons. This is further substantiated in <a href="defect+brane#SS22">SS22</a>. See <a href="defect+brane#RelationToAnyons">there</a> for more.</p> <p><br /></p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parastatistics">parastatistics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/su%282%29-anyon">su(2)-anyon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+fusion+category">unitary fusion category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+Hall+effect">quantum Hall effect</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The concept of <a class="existingWikiWord" href="/nlab/show/anyons">anyons</a> satisfying <a class="existingWikiWord" href="/nlab/show/braid+group+statistics">braid group statistics</a> originated independently in:</p> <ul> <li id="LeinaasMyrheim77"> <p><a class="existingWikiWord" href="/nlab/show/Jon+Magne+Leinaas">Jon Magne Leinaas</a>, <a class="existingWikiWord" href="/nlab/show/Jan+Myrheim">Jan Myrheim</a>, <em>On the theory of identical particles</em>, <em>К теории тождествениых частиц</em>, Nuovo Cim B 37, 1–23 (1977) (<a href="https://doi.org/10.1007/BF02727953">doi:10.1007/BF02727953</a>)</p> </li> <li id="GoldinMenikoffSharp81"> <p>G. A. Goldin, R. Menikoff, D. H. Sharp, <em>Representations of a local current algebra in nonsimply connected space and the Aharonov–Bohm effect</em>, J. Math. Phys. <strong>22</strong> 1664 (1981) [<a href="https://doi.org/10.1063/1.525110">doi:10.1063/1.525110</a>]</p> </li> <li id="Wilczek82a"> <p><a class="existingWikiWord" href="/nlab/show/Frank+Wilczek">Frank Wilczek</a>, <em>Magnetic Flux, Angular Momentum, and Statistics</em>, Phys. Rev. Lett. <strong>48</strong> (1982) 1144 (reprinted in <a href="#Wilczek90">Wilczek 1990, p. 163-165</a>) [<a href="https://doi.org/10.1103/PhysRevLett.48.1144">doi:10.1103/PhysRevLett.48.1144</a>]</p> </li> </ul> <p>The term <em>anyon</em> was introduced in:</p> <ul> <li id="Wilczek82b"><a class="existingWikiWord" href="/nlab/show/Frank+Wilczek">Frank Wilczek</a>, <em>Quantum Mechanics of Fractional-Spin Particles</em>, Phys. Rev. Lett. <strong>49</strong> (1982) 957 (reprinted in <a href="#Wilczek90">Wilczek 1990, p. 166-168</a>) [<a href="https://doi.org/10.1103/PhysRevLett.49.957">doi:10.1103/PhysRevLett.49.957</a>]</li> </ul> <p>Identification of anyon phases (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 an <a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">adiabatic transport</a> 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) [<a href="https://doi.org/10.1103/PhysRevLett.53.722">doi:10.1103/PhysRevLett.53.722</a>]</li> </ul> <p>The “<a class="existingWikiWord" href="/nlab/show/fictitious+gauge+field">fictitious gauge field</a>”-method for modelling anyons:</p> <ul> <li id="ArovasSchriefferWilczekZee85"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+P.+Arovas">Daniel P. Arovas</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Schrieffer">Robert Schrieffer</a>, <a class="existingWikiWord" href="/nlab/show/Frank+Wilczek">Frank Wilczek</a>, <a class="existingWikiWord" href="/nlab/show/Anthony+Zee">Anthony Zee</a>, <em>Statistical mechanics of anyons</em>, Nuclear Physics B <strong>251</strong> (1985) 117-126 (reprinted in <a href="#Wilczek90">Wilczek 1990, p. 173-182</a>) [<a href="https://doi.org/10.1016/0550-3213(85)90252-4">doi:10.1016/0550-3213(85)90252-4</a>]</p> </li> <li id="ChenWilczekWittenHalperin89"> <p><a class="existingWikiWord" href="/nlab/show/Yi-Hong+Chen">Yi-Hong Chen</a>, <a class="existingWikiWord" href="/nlab/show/Frank+Wilczek">Frank Wilczek</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <a class="existingWikiWord" href="/nlab/show/Bertrand+Halperin">Bertrand Halperin</a>, <em>On Anyon Superconductivity</em>, International Journal of Modern Physics B <strong>03</strong> 07 (1989) 1001-1067 (reprinted in <a href="#Wilczek90">Wilczek 1990, p. 342-408</a>) [<a href="https://doi.org/10.1142/S0217979289000725">doi:10.1142/S0217979289000725</a>, <a class="existingWikiWord" href="/nlab/files/CWWH-AnyonSuperfluidity.pdf" title="pdf">pdf</a>]</p> </li> <li id="Wilczek91"> <p><a class="existingWikiWord" href="/nlab/show/Frank+Wilczek">Frank Wilczek</a>, <em>States of Anyon Matter</em>, International Journal of Modern Physics B <strong>05</strong> 09 (1991) 1273-1312 [<a href="https://doi.org/10.1142/S0217979291000626">doi:10.1142/S0217979291000626</a>]</p> </li> </ul> <p>and with specific emphasis on the resulting (abelian!) <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>:</p> <ul> <li id="IengoLechner92"><a class="existingWikiWord" href="/nlab/show/Roberto+Iengo">Roberto Iengo</a>, <a class="existingWikiWord" href="/nlab/show/Kurt+Lechner">Kurt Lechner</a>, <em>Anyon quantum mechanics and Chern-Simons theory</em>, Physics Reports <strong>213</strong> 4 (1992) 179-269 [<a href="https://doi.org/10.1016/0370-1573(92)90039-3">doi:10.1016/0370-1573(92)90039-3</a>]</li> </ul> <p>The suggestion that the anyonic <a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>-<a class="existingWikiWord" href="/nlab/show/wavefunctions">wavefunctions</a> are essentially <a class="existingWikiWord" href="/nlab/show/conformal+blocks">conformal blocks</a> of <a class="existingWikiWord" href="/nlab/show/2d+CFT">2d CFT</a> (notably for <a class="existingWikiWord" href="/nlab/show/su%282%29-anyons">su(2)-anyons</a>):</p> <ul> <li id="MooreRead91"> <p><a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+Read">Nicholas Read</a>, <em>Nonabelions in the fractional quantum hall effect</em>, Nuclear Physics B <strong>360</strong> 2–3 (1991) 362-396 [<a href="https://doi.org/10.1016/0550-3213(91)90407-O">doi:10.1016/0550-3213(91)90407-O</a>, <a href="https://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf">pdf</a>]</p> </li> <li id="ReadRezayi99"> <p><a class="existingWikiWord" href="/nlab/show/Nicholas+Read">Nicholas Read</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Rezayi">Edward Rezayi</a>, <em>Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level</em>, Phys. Rev. B <strong>59</strong> (1999) 8084 [<a href="https://doi.org/10.1103/PhysRevB.59.8084">doi:10.1103/PhysRevB.59.8084</a>]</p> </li> </ul> <p>More comprehensive accounts of anyons:</p> <ul> <li id="Wilczek90"> <p><a class="existingWikiWord" href="/nlab/show/Frank+Wilczek">Frank Wilczek</a>, <em>Fractional Statistics and Anyon Superconductivity</em>, World Scientific (1990) [<a href="https://doi.org/10.1142/0961">doi:10.1142/0961</a>]</p> </li> <li id="Kitaev06"> <p><a class="existingWikiWord" href="/nlab/show/Alexei+Kitaev">Alexei Kitaev</a>, <em>Anyons in an exactly solved model and beyond</em>, Annals of Physics <strong>321</strong> 1 (2006) 2-111 [<a href="https://doi.org/10.1016/j.aop.2005.10.005">doi:10.1016/j.aop.2005.10.005</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eduardo+Fradkin">Eduardo Fradkin</a>, <em>Anyon superconductivity</em>, Chapter 11 in: <em>Field Theories of Condensed Matter Physics</em>, Cambridge University Press (2013) 414-431 [ISBN: 9781139015509, <a href="https://doi.org/10.1017/CBO9781139015509.013">doi:10.1017/CBO9781139015509.013</a>]</p> </li> </ul> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Martin+Greiter">Martin Greiter</a>, <a class="existingWikiWord" href="/nlab/show/Frank+Wilczek">Frank Wilczek</a>, <em>Fractional Statistics</em> [<a href="https://arxiv.org/abs/2210.02530">arXiv:2210.02530</a>]</li> </ul> <p>On the history of the concept:</p> <ul> <li>Gerald A. Goldin: <em>The Prediction of Anyons: Its History and Wider Implications</em>, SciPost Phys. Proc. <strong>14</strong> 005 (2023) [<a href="https://arxiv.org/abs/2212.12632">arXiv:2212.12632</a>, <a href="https://scipost.org/SciPostPhysProc.14.005">doi: 10.21468/SciPostPhysProc.14.005</a>]</li> </ul> <p>Rigorous discussion in terms of <a class="existingWikiWord" href="/nlab/show/superselection+sectors">superselection sectors</a> in <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a>:</p> <ul> <li id="FroehlichMarchetti88"> <p><a class="existingWikiWord" href="/nlab/show/J%C3%BCrg+Fr%C3%B6hlich">Jürg Fröhlich</a>, <a class="existingWikiWord" href="/nlab/show/Pieralberto+Marchetti">Pieralberto Marchetti</a>, <em>Quantum field theory of anyons</em>, Lett. Math. Phys. <strong>16</strong> (1988) 347–358 (reprinted in <a href="#Wilczek90">Wilczek 1990, p. 202-213</a>) [<a href="https://doi.org/10.1007/BF00402043">doi:10.1007/BF00402043</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/J%C3%BCrg+Fr%C3%B6hlich">Jürg Fröhlich</a>, <a class="existingWikiWord" href="/nlab/show/Fabrizio+Gabbiani">Fabrizio Gabbiani</a>, <em>Braid statistics in local quantum theory</em>, Reviews in Mathematical Physics, <strong>2</strong> 03 (1990) 251-353 [<a href="https://doi.org/10.1142/S0129055X90000107">doi:10.1142/S0129055X90000107</a>]</p> </li> <li id="FroehlichGabbianiMarchetti90"> <p><a class="existingWikiWord" href="/nlab/show/J%C3%BCrg+Fr%C3%B6hlich">Jürg Fröhlich</a>, <a class="existingWikiWord" href="/nlab/show/Fabrizio+Gabbiani">Fabrizio Gabbiani</a>, <a class="existingWikiWord" href="/nlab/show/Pieralberto+Marchetti">Pieralberto Marchetti</a>, <em>Braid statistics in three-dimensional local quantum field theory</em>, in: H.C. Lee (ed.) <em>Physics, Geometry and Topology</em> NATO ASI Series, <strong>238</strong> Springer (1990) [<a href="https://doi.org/10.1007/978-1-4615-3802-8_2">doi:10.1007/978-1-4615-3802-8_2</a>, <a class="existingWikiWord" href="/nlab/files/FroehlichGabbianiMarchetti-BraidStatistics.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, Karl-Henning Rehren, <a class="existingWikiWord" href="/nlab/show/Bert+Schroer">Bert Schroer</a>, <em>Superselection sectors with braid group statistics and exchange algebras – I: General theory</em>, Comm. Math. Phys. Volume 125, Number 2 (1989), 201-226. (<a href="http://projecteuclid.org/euclid.cmp/1104179464">euclid:cmp/1104179464</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, Karl-Henning Rehren, <a class="existingWikiWord" href="/nlab/show/Bert+Schroer">Bert Schroer</a>, <em>Superselection sectors with braid group statistics and exchange algebras – II: Geometric aspects and conformal covariance</em>, Reviews in Mathematical PhysicsVol. 04, No. spec01, pp. 113-157 (1992) (<a href="https://doi.org/10.1142/S0129055X92000170">doi:10.1142/S0129055X92000170</a> <a href="ftp://ftp.theorie.physik.uni-goettingen.de/pub/papers/rehren/92/superselection_sectors_II_RMP.pdf">pdf</a>)</p> </li> </ul> <p>Discussion of anyon-<a class="existingWikiWord" href="/nlab/show/wavefunctions">wavefunctions</a> as <a class="existingWikiWord" href="/nlab/show/multi-valued+functions">multi-valued functions</a> on a <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">configuration space of points</a>:</p> <ul> <li id="Wu84"> <p><a class="existingWikiWord" href="/nlab/show/Yong-Shi+Wu">Yong-Shi Wu</a>, <em>Multiparticle Quantum Mechanics Obeying Fractional Statistics</em>, Phys. Rev. Lett. <strong>53</strong> (1984) 111 [<a href="https://doi.org/10.1103/PhysRevLett.53.111">doi:10.1103/PhysRevLett.53.111</a>, <a href="https://core.ac.uk/download/pdf/276286925.pdf">pdf</a>]</p> </li> <li> <p><a href="#FroehlichGabbianiMarchetti90">Fröhlich, Gabbiani & Marchetti 1990, p. 20</a></p> </li> <li id="ImboImboSudarshan90"> <p>Tom Imbo, Chandni Shah Imbo, E. C. G. Sudarshan, <em>Identical particles, exotic statistics and braid groups</em>, Physics Letters B <strong>234</strong> 1–2, (1990) 103-107 [<a href="https://doi.org/10.1016/0370-2693(90)92010-G">doi:10.1016/0370-2693(90)92010-G</a>, <a href="https://lib-extopc.kek.jp/preprints/PDF/2000/0033/0033552.pdf">pdf</a>]</p> </li> <li id="BCMS93"> <p>Garth A. Baker, Geoff S. Canright, Shashikant B. Mulay, Carl Sundberg, <em>On the spectral problem for anyons</em>, Communications in Mathematical Physics <strong>153</strong> (1993) 277–295 [<a href="https://doi.org/10.1007/BF02096644">doi:10.1007/BF02096644</a>]</p> </li> <li id="MundSchrader93"> <p>J. Mund, <a class="existingWikiWord" href="/nlab/show/Robert+Schrader">Robert Schrader</a>, <em>Hilbert Spaces for Nonrelativistic and Relativistic “Free” Plektons (Particles with Braid Group Statistics)</em>, in <em>Advances in dynamical systems and quantum physics</em> (Capri, 1993), World Sci. (1995) 235–259 [<a href="https://arxiv.org/abs/hep-th/9310054">arXiv:hep-th/9310054</a>]</p> <p>followed up by:</p> <p><a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Matthias+Gaberdiel">Matthias Gaberdiel</a> and Stefan M. Rüger, <em>Scattering states of plektons (particles with braid group statistics) in 2+1 dimensional quantum field theory</em>, Communications in Mathematical Physics <strong>175</strong> (1996) 319–335 [<a href="https://doi.org/10.1007/BF02102411">doi:10.1007/BF02102411</a>]</p> </li> <li id="DFT97"> <p>Gianfausto Dell’Antonio, Rodolfo Figari, Alessandro Teta: <em>Statistics in Space Dimension Two</em>, Letters in Mathematical Physics <strong>40</strong> (1997) 235–256 [<a href="https://doi.org/10.1023/A:1007361832622">doi:10.1023/A:1007361832622</a>]</p> </li> <li id="Myrheim99"> <p><a class="existingWikiWord" href="/nlab/show/Jan+Myrheim">Jan Myrheim</a>, <em>Anyons</em>, p. 265-414 in: <em>Topological aspects of low dimensional systems</em>, Les Houches LXIX, Springer (1999) [<a href="https://doi.org/10.1007/3-540-46637-1">doi:10.1007/3-540-46637-1</a>, <a class="existingWikiWord" href="/nlab/files/Myrheim-Anyons.pdf" title="pdf">pdf</a>]</p> </li> <li id="MurthyShankar09"> <p><a class="existingWikiWord" href="/nlab/show/M.V.N.+Murthy">M.V.N. Murthy</a>, <a class="existingWikiWord" href="/nlab/show/Ramamurti+Shankar">Ramamurti Shankar</a>, <em>Exclusion Statistics: From Pauli to Haldane</em> (1999, 2009) [<a href="https://dspace.imsc.res.in/xmlui/handle/123456789/334">dspace:123456789/334</a>, <a href="https://www.imsc.res.in/xmlui/bitstream/handle/123456789/334/MR120.pdf?sequence=1">pdf</a>, <a class="existingWikiWord" href="/nlab/files/MurthyShankar-ExclusionStatistics.pdf" title="pdf">pdf</a>]</p> </li> <li id="DMV03"> <p>G. Date, <a class="existingWikiWord" href="/nlab/show/M.V.N.+Murthy">M.V.N. Murthy</a>, <a class="existingWikiWord" href="/nlab/show/Radhika+Vathsan">Radhika Vathsan</a>, <em>Classical and Quantum Mechanics of Anyons</em> [<a href="https://arxiv.org/abs/cond-mat/0302019">arXiv:cond-mat/0302019</a>]</p> </li> </ul> <p>The topic of <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> of non-abelian anyons is crucial to their identification in <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a> but has received little attenion, exceptions being:</p> <ul> <li id="BondersonShtengelSlingerland07"> <p><a class="existingWikiWord" href="/nlab/show/Parsa+Bonderson">Parsa Bonderson</a>, <a class="existingWikiWord" href="/nlab/show/Kirill+Shtengel">Kirill Shtengel</a>, <a class="existingWikiWord" href="/nlab/show/Joost+Slingerland">Joost Slingerland</a>, <em>Decoherence of Anyonic Charge in Interferometry Measurements</em>, Phys. Rev. Lett. <strong>98</strong> (2007) 070401 [<a href="https://doi.org/10.1103/PhysRevLett.98.070401">doi:10.1103/PhysRevLett.98.070401</a>, <a href="https://arxiv.org/abs/quant-ph/0608119">arXiv:quant-ph/0608119</a>]</p> </li> <li id="BondersonShtengelSlingerland08"> <p><a class="existingWikiWord" href="/nlab/show/Parsa+Bonderson">Parsa Bonderson</a>, <a class="existingWikiWord" href="/nlab/show/Kirill+Shtengel">Kirill Shtengel</a>, <a class="existingWikiWord" href="/nlab/show/Joost+Slingerland">Joost Slingerland</a>, <em>Interferometry of non-Abelian Anyons</em>, Annals Phys. <strong>323</strong> (2008) 2709-2755 [<a href="https://doi.org/10.1016/j.aop.2008.01.012">doi:10.1016/j.aop.2008.01.012</a>, <a href="https://arxiv.org/abs/0707.4206">arXiv:0707.4206</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Andrey+Morozov">Andrey Morozov</a>, <em>On measuring the topological charge of anyons</em> [<a href="https://arxiv.org/abs/2403.07847">arXiv:2403.07847</a>]</p> </li> </ul> <div> <h3 id="AnyonicTopologicalOrderInTermsOfBraidedFusionCategoriesReferences">Anyonic topological order in terms of braided fusion categories</h3> <h4 id="claim_and_status">Claim and status</h4> <p>In <a class="existingWikiWord" href="/nlab/show/condensed+matter+theory">condensed matter theory</a> it is <a class="existingWikiWord" href="/nlab/show/folklore">folklore</a> that species of <a class="existingWikiWord" href="/nlab/show/anyon">anyonic</a> <a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a> correspond to <a class="existingWikiWord" href="/nlab/show/braided+monoidal+categories">braided</a> <a class="existingWikiWord" href="/nlab/show/unitary+fusion+category">unitary</a> <a class="existingWikiWord" href="/nlab/show/fusion+categories">fusion categories</a>/<a class="existingWikiWord" href="/nlab/show/modular+tensor+categories">modular tensor categories</a>.</p> <p>The origin of the claim is:</p> <ul> <li id="Kitaev06"><a class="existingWikiWord" href="/nlab/show/Alexei+Kitaev">Alexei Kitaev</a>, Section 8 and Appendix E of: <em>Anyons in an exactly solved model and beyond</em>, Annals of Physics <strong>321</strong> 1 (2006) 2-111 [<a href="https://doi.org/10.1016/j.aop.2005.10.005">doi:10.1016/j.aop.2005.10.005</a>]</li> </ul> <p>Early accounts re-stating this claim (without attribution):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chetan+Nayak">Chetan Nayak</a>, <a class="existingWikiWord" href="/nlab/show/Steven+H.+Simon">Steven H. Simon</a>, <a class="existingWikiWord" href="/nlab/show/Ady+Stern">Ady Stern</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Freedman">Michael Freedman</a>, <a class="existingWikiWord" href="/nlab/show/Sankar+Das+Sarma">Sankar Das Sarma</a>, pp. 28 of: <em>Non-Abelian Anyons and Topological Quantum Computation</em>, Rev. Mod. Phys. <strong>80</strong> 1083 (2008) [<a href="https://doi.org/10.1103/RevModPhys.80.1083">doi:10.1103/RevModPhys.80.1083</a>, <a href="http://arxiv.org/abs/0707.1889">arXiv:0707.1888</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zhenghan+Wang">Zhenghan Wang</a>, Section 6.3 of: <em>Topological Quantum Computation</em>, CBMS Regional Conference Series in Mathematics <strong>112</strong>, AMS (2010) [<a href="http://www.ams.org/publications/authors/books/postpub/cbms-112">ISBN-13: 978-0-8218-4930-9]</a></p> </li> </ul> <p>Further discussion (mostly review and mostly without attribution):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Simon+Burton">Simon Burton</a>, <em>A Short Guide to Anyons and Modular Functors</em> [<a href="https://arxiv.org/abs/1610.05384">arXiv:1610.05384</a>]</p> <blockquote> <p>(this one stands out as still attributing the claim to <a href="#Kitaev06">Kitaev (2006), Appendix E</a>)</p> </blockquote> </li> <li id="RowellWang18"> <p><a class="existingWikiWord" href="/nlab/show/Eric+C.+Rowell">Eric C. Rowell</a>, <a class="existingWikiWord" href="/nlab/show/Zhenghan+Wang">Zhenghan Wang</a>, <em>Mathematics of Topological Quantum Computing</em>, Bull. Amer. Math. Soc. 55 (2018), 183-238 (<a href="https://arxiv.org/abs/1705.06206">arXiv:1705.06206</a>, <a href="https://doi.org/10.1090/bull/1605">doi:10.1090/bull/1605</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tian+Lan">Tian Lan</a>, <em>A Classification of (2+1)D Topological Phases with Symmetries</em> [<a href="https://arxiv.org/abs/1801.01210">arXiv:1801.01210</a>]</p> </li> <li> <p><em>From categories to anyons: a travelogue</em> [<a href="https://arxiv.org/abs/1811.06670">arXiv:1811.06670</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Colleen+Delaney">Colleen Delaney</a>, <em>A categorical perspective on symmetry, topological order, and quantum information</em> (2019) [<a href="https://escholarship.org/uc/item/5z384290">uc:5z384290>, </a><a href="https://crdelane.pages.iu.edu/dissertationch1-5.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Colleen+Delaney">Colleen Delaney</a>, <em>Lecture notes on modular tensor categories and braid group representations</em> (2019) [<a href="http://web.math.ucsb.edu/~cdelaney/MTC_Notes.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/DelaneyModularTensorCategories.pdf" title="pdf">pdf</a>]</p> </li> <li> <p>Liang Wang, <a class="existingWikiWord" href="/nlab/show/Zhenghan+Wang">Zhenghan Wang</a>, <em>In and around Abelian anyon models</em>, J. Phys. A: Math. Theor. <strong>53</strong> 505203 (2020) [<a href="https://iopscience.iop.org/article/10.1088/1751-8121/abc6c0">doi:10.1088/1751-8121/abc6c0</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Parsa+Bonderson">Parsa Bonderson</a>, <em>Measuring Topological Order</em>, Phys. Rev. Research <strong>3</strong>, 033110 (2021) [<a href="https://doi.org/10.1103/PhysRevResearch.3.033110">doi:10.1103/PhysRevResearch.3.033110</a>, <a href="https://arxiv.org/abs/2102.05677">arXiv:2102.05677</a>]</p> </li> <li> <p>Zhuan Li, Roger S.K. Mong, <em>Detecting topological order from modular transformations of ground states on the torus</em> [<a href="https://arxiv.org/abs/2203.04329">arXiv:2203.04329</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+C.+Rowell">Eric C. Rowell</a>, <em>Braids, Motions and Topological Quantum Computing</em> [<a href="https://arxiv.org/abs/2208.11762">arXiv:2208.11762</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sachin+Valera">Sachin Valera</a>, <em>A Quick Introduction to the Algebraic Theory of Anyons</em>, talk at <em><a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a> Initial Researcher Meeting</em> (Sep 2022) <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 class="existingWikiWord" href="/nlab/files/CQTS-InitialResearcherMeeting-Valera-220914.pdf" title="pdf">pdf</a>]</p> </li> <li> <p>Willie Aboumrad, <em>Quantum computing with anyons: an F-matrix and braid calculator</em> [<a href="https://arxiv.org/abs/2212.00831">arXiv:2212.00831</a>]</p> </li> </ul> <p>Emphasis that the expected description of <a class="existingWikiWord" href="/nlab/show/anyons">anyons</a> by <a class="existingWikiWord" href="/nlab/show/braided+fusion+categories">braided fusion categories</a> had remained <a class="existingWikiWord" href="/nlab/show/folklore">folklore</a>, together with a list of minimal assumptions that would need to be shown:</p> <ul> <li id="Valera21"><a class="existingWikiWord" href="/nlab/show/Sachin+J.+Valera">Sachin J. Valera</a>, <em>Fusion Structure from Exchange Symmetry in (2+1)-Dimensions</em>, Annals of Physics <strong>429</strong> (2021) [<a href="https://doi.org/10.1016/j.aop.2021.168471">doi:10.1016/j.aop.2021.168471</a>, <a href="https://arxiv.org/abs/2004.06282">arXiv:2004.06282</a>]</li> </ul> <p>An argument that the statement at least for <a class="existingWikiWord" href="/nlab/show/SU%282%29-anyons">SU(2)-anyons</a> does follow from an enhancement of the <a class="existingWikiWord" href="/nlab/show/K-theory+classification+of+topological+phases+of+matter">K-theory classification of topological phases of matter</a> to interacting <a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Anyonic+topological+order+in+TED+K-theory">Anyonic topological order in TED K-theory</a></em>, Rev. Math. Phys. <strong>35</strong> 03 (2023) 2350001 [<a href="https://doi.org/10.1142/S0129055X23500010">doi:10.1142/S0129055X23500010</a>,<a href="https://arxiv.org/abs/2206.13563">arXiv:2206.13563</a>]</li> </ul> <h4 id="AnyonicOrderInMTheoryReferences">In string/M-theory</h4> <p>Arguments realizing such anyonic topological order in the <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>-<a class="existingWikiWord" href="/nlab/show/quantum+field+theory">field theory</a> on <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a>:</p> <p>Via <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> on <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed</a> <a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a> (<a class="existingWikiWord" href="/nlab/show/Seifert+manifolds">Seifert manifolds</a>) analogous to the <a class="existingWikiWord" href="/nlab/show/3d-3d+correspondence">3d-3d correspondence</a> (which instead uses <a class="existingWikiWord" href="/nlab/show/hyperbolic+3-manifolds">hyperbolic 3-manifolds</a>):</p> <ul> <li id="CGK20"> <p>Gil Young Cho, <a class="existingWikiWord" href="/nlab/show/Dongmin+Gang">Dongmin Gang</a>, Hee-Cheol Kim: <em>M-theoretic Genesis of Topological Phases</em>, J. High Energ. Phys. <strong>2020</strong> 115 (2020) [<a href="https://arxiv.org/abs/2007.01532">arXiv:2007.01532</a>, <a href=" https://doi.org/10.1007/JHEP11(2020)115">doi:10.1007/JHEP11(2020)115</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Shawn+X.+Cui">Shawn X. Cui</a>, Paul Gustafson, <a class="existingWikiWord" href="/nlab/show/Yang+Qiu">Yang Qiu</a>, Qing Zhang, <em>From Torus Bundles to Particle-Hole Equivariantization</em>, Lett Math Phys <strong>112</strong> 15 (2022) [<a href="https://doi.org/10.1007/s11005-022-01508-3">doi:10.1007/s11005-022-01508-3</a>, <a href="https://arxiv.org/abs/2106.01959">arXiv:2106.01959</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Shawn+X.+Cui">Shawn X. Cui</a>, <a class="existingWikiWord" href="/nlab/show/Yang+Qiu">Yang Qiu</a>, <a class="existingWikiWord" href="/nlab/show/Zhenghan+Wang">Zhenghan Wang</a>, <em>From Three Dimensional Manifolds to Modular Tensor Categories</em>, Commun. Math. Phys. <strong>397</strong> (2023) 1191–1235 [<a href="https://doi.org/10.1007/s00220-022-04517-4">doi:10.1007/s00220-022-04517-4</a>, <a href="https://arxiv.org/abs/2101.01674">arXiv:2101.01674</a>]</p> </li> <li> <p>Sunjin Choi, <a class="existingWikiWord" href="/nlab/show/Dongmin+Gang">Dongmin Gang</a>, Hee-Cheol Kim: <em>Infrared phases of 3D Class R theories</em>, J. High Energ. Phys. <strong>2022</strong> 151 (2022) [<a href="https://doi.org/10.1007/JHEP11(2022)151">doi:10.1007/JHEP11(2022)151</a>, <a href="https://arxiv.org/abs/2206.11982">arXiv:2206.11982</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Federico+Bonetti">Federico Bonetti</a>, <a class="existingWikiWord" href="/nlab/show/Sakura+Sch%C3%A4fer-Nameki">Sakura Schäfer-Nameki</a>, Jingxiang Wu, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MTC</mi><mo stretchy="false">[</mo><msub><mi>M</mi> <mn>3</mn></msub><mo>,</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">MTC[M_3,G]</annotation></semantics></math>: 3d Topological Order Labeled by Seifert Manifolds</em> [<a href="https://arxiv.org/abs/2403.03973">arXiv:2403.03973</a>]</p> </li> <li id="SS24"> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <em><a class="existingWikiWord" href="/schreiber/show/Anyons+on+M5-Probes+of+Seifert+3-Orbifolds">Anyons on M5-Probes of Seifert 3-Orbifolds</a></em> [<a href="https://arxiv.org/abs/2411.16852">arXiv:2411.16852</a>]</p> </li> <li id="SS25"> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <em><a class="existingWikiWord" href="/schreiber/show/Engineering+of+Anyons+on+M5-Probes+via+Flux-Quantization">Engineering of Anyons on M5-Probes via Flux-Quantization</a></em> &lbrack;<a href="https://arxiv.org/abs/2501.17927">arXiv:2501.17927</a>&rbrack;</p> </li> </ul> <p>Via <a class="existingWikiWord" href="/nlab/show/3-brane+in+6d">3-brane</a> <a class="existingWikiWord" href="/nlab/show/defect+branes">defects</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <em><a class="existingWikiWord" href="/schreiber/show/Anyonic+defect+branes+in+TED+K-theory">Anyonic Defect Branes and Conformal Blocks in Twisted Equivariant Differential K-Theory</a></em>, Reviews in Mathematical Physics <strong>35</strong> 06 (2023) 2350009 [<a href="https://arxiv.org/abs/2203.11838">arXiv:2203.11838</a>, <a href="https://doi.org/10.1142/S0129055X23500095">doi:10.1142/S0129055X23500095</a>]</li> </ul> <h4 id="further_discussion">Further discussion</h4> <p>Relation to <a class="existingWikiWord" href="/nlab/show/ZX-calculus">ZX-calculus</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Fatimah+Rita+Ahmadi">Fatimah Rita Ahmadi</a>, <a class="existingWikiWord" href="/nlab/show/Aleks+Kissinger">Aleks Kissinger</a>, <em>Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics</em> [<a href="https://arxiv.org/abs/2211.03855">arXiv:2211.03855</a>]</li> </ul> <p>On detection of <a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a> by observing <a class="existingWikiWord" href="/nlab/show/modular+transformations">modular transformations</a> on the <a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Zhuan+Li">Zhuan Li</a>, <a class="existingWikiWord" href="/nlab/show/Roger+S.+K.+Mong">Roger S. K. Mong</a>, <em>Detecting topological order from modular transformations of ground states on the torus</em>, Phys. Rev. B <strong>106</strong> (2022) 235115 [<a href="https://doi.org/10.1103/PhysRevB.106.235115">doi:10.1103/PhysRevB.106.235115</a> <a href="https://arxiv.org/abs/2203.04329">arXiv:2203.04329</a>]</li> </ul> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Liang+Kong">Liang Kong</a>, <em>Topological Wick Rotation and Holographic Dualities</em>, <a href="CQTS#KongOct2022">talk at</a> <a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a> (Oct 2022) <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 class="existingWikiWord" href="/nlab/files/Kong-TalkAtCQTS-20221019.pdf" title="pdf">pdf</a>]</li> </ul> </div><div> <h3 id="AnyonsInTheQuantumHallEffectReferences">Anyons in fractional quantum Hall systems</h3> <p>References on <a class="existingWikiWord" href="/nlab/show/anyon">anyon</a>-excitations (satisfying <a class="existingWikiWord" href="/nlab/show/braid+group+statistics">braid group statistics</a>) in the <a class="existingWikiWord" href="/nlab/show/quantum+Hall+effect">quantum Hall effect</a> (for more on the application to <a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a> see the references <a href="topological+quantum}computation#TopologicalQuantumComputationWithAnyons">there</a>):</p> <p>The prediction of <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> <a class="existingWikiWord" href="/nlab/show/anyon">anyon</a>-excitations in the <a class="existingWikiWord" href="/nlab/show/quantum+Hall+effect">quantum Hall effect</a> (i.e. satisfying <a class="existingWikiWord" href="/nlab/show/braid+group+statistics">braid group statistics</a> in 1-dimensional <a class="existingWikiWord" href="/nlab/show/linear+representations">linear representations</a> of the <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bertrand+I.+Halperin">Bertrand I. Halperin</a>: <em>Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States</em>, Phys. Rev. Lett. <strong>52</strong> (1984) 1583 [<a href="https://doi.org/10.1103/PhysRevLett.52.1583">doi:10.1103/PhysRevLett.52.1583</a>]</p> <p>Erratum, Phys. Rev. Lett. <strong>52</strong> (1984) 2390 [<a href="https://doi.org/10.1103/PhysRevLett.52.2390.4">doi:10.1103/PhysRevLett.52.2390.4</a>]</p> </li> <li id="ArovasSchriefferWilczek84"> <p><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. <strong>53</strong> (1984) 722 [<a href="https://doi.org/10.1103/PhysRevLett.53.722">doi:10.1103/PhysRevLett.53.722</a>]</p> </li> </ul> <p>The original discussion of <a class="existingWikiWord" href="/nlab/show/non-abelian+group">non-abelian</a> <a class="existingWikiWord" href="/nlab/show/anyon">anyon</a>-excitations in the <a class="existingWikiWord" href="/nlab/show/quantum+Hall+effect">quantum Hall effect</a> (i.e. satisfying <a class="existingWikiWord" href="/nlab/show/braid+group+statistics">braid group statistics</a> in higher dimensional <a class="existingWikiWord" href="/nlab/show/linear+representations">linear representations</a> of the <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a>, related to <a class="existingWikiWord" href="/nlab/show/modular+tensor+categories">modular tensor categories</a>):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+Read">Nicholas Read</a>, <em>Nonabelions in the fractional quantum Hall effect</em>, Nucl. Phys. 360B(1991)362 (<a href="http://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf">pdf</a>, <a href="https://doi.org/10.1016/0550-3213(91)90407-O">doi:10.1016/0550-3213(91)90407-O</a>)</li> </ul> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ady+Stern">Ady Stern</a>, <em>Anyons and the quantum Hall effect – A pedagogical review</em>, Annals of Physics <strong>323</strong> 1 (2008) 204-249 [<a href="https://doi.org/10.1016/j.aop.2007.10.008">doi:10.1016/j.aop.2007.10.008</a>, <a href="https://arxiv.org/abs/0711.4697">arXiv:0711.4697</a>]</p> </li> <li> <p>Menelaos Zikidis: <em>Abelian Anyons and Fractional Quantum Hall Effect</em>, Seminar notes (2017) [<a href="https://www.mathi.uni-heidelberg.de/~walcher/teaching/sose17/atqft/writeups/Menelaos.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Zikidis-AbelianAnyons.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ady+Stern">Ady Stern</a>: <em>Engineering Non-Abelian Quasi-Particles in Fractional Quantum Hall States – A Pedagogical Introduction</em>, Ch. 9 in: <em>Fractional Quantum Hall Effects</em>, World Scientific (2020) 435-486 [<a href="https://doi.org/10.1142/9789811217494_0009">doi:10.1142/9789811217494_0009</a>]</p> </li> <li> <p>D. E. Feldman, <a class="existingWikiWord" href="/nlab/show/Bertrand+Halperin">Bertrand Halperin</a>: <em>Fractional charge and fractional statistics in the quantum Hall effects</em>, Rep. Prog. Phys. <strong>84</strong> (2021) 076501 [<a href="https://doi.org/10.1088/1361-6633/ac03aa">doi:10.1088/1361-6633/ac03aa</a>, <a href="https://arxiv.org/abs/2102.08998">arXiv:2102.08998</a>]</p> </li> </ul> <p>As potential hardware for <a class="existingWikiWord" href="/nlab/show/topological+quantum+computing">topological quantum computing</a>:</p> <ul> <li> <p>D. V. Averin, V. J. Goldman: <em>Quantum computation with quasiparticles of the fractional quantum Hall effect</em>, Solid State Communications <strong>121</strong> 1 (2001) 25-28 [<a href="https://doi.org/10.1016/S0038-1098(01)00447-1">doi:10.1016/S0038-1098(01)00447-1</a>, <a href="https://arxiv.org/abs/cond-mat/0110193">arXiv:cond-mat/0110193</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sankar+Das+Sarma">Sankar Das Sarma</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Freedman">Michael Freedman</a>, <a class="existingWikiWord" href="/nlab/show/Chetan+Nayak">Chetan Nayak</a>: <em>Topologically-Protected Qubits from a Possible Non-Abelian Fractional Quantum Hall State</em>, Phys. Rev. Lett. <strong>94</strong> 166802 (2005) [<a href="https://doi.org/10.1103/PhysRevLett.94.166802">doi:10.1103/PhysRevLett.94.166802</a>, <a href="https://arxiv.org/abs/cond-mat/0412343">arXiv:cond-mat/0412343</a>]</p> </li> <li> <p>Sergey Bravyi: <em>Universal Quantum Computation with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi><mo>=</mo><mn>5</mn><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\nu = 5/2</annotation></semantics></math> Fractional Quantum Hall State</em>, Phys. Rev. A <strong>73</strong> 042313 (2006) [<a href="https://doi.org/10.1103/PhysRevA.73.042313 ">doi:10.1103/PhysRevA.73.042313</a>, <a href="https://arxiv.org/abs/quant-ph/0511178">arXiv:quant-ph/0511178</a>]</p> </li> </ul> <div> <h3 id="ObservationOfAnyonsInFQH">Observation of anyons in fractional quantum Hall systems</h3> <p>While the occurrence of <a class="existingWikiWord" href="/nlab/show/anyon">anyon</a>-excitations in the <a class="existingWikiWord" href="/nlab/show/fractional+quantum+Hall+effect">fractional quantum Hall effect</a> is a robust theoretical prediction (see the references <a href="#AnyonsInTheQuantumHallEffectReferences">above</a>), and while the fractional quantum Hall effect itself has long been established in <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, the actual observation of anyons in these systems is subtle:</p> <p>An early claim of the observation of non-abelian anyons seems to remain unconfirmed:</p> <ul> <li>Sanghun An, P. Jiang, H. Choi, W. Kang, S. H. Simon, L. N. Pfeiffer, K. W. West, K. W. Baldwin, <em>Braiding of Abelian and Non-Abelian Anyons in the Fractional Quantum Hall Effect</em> [<a href="https://arxiv.org/abs/1112.3400">arXiv:1112.3400</a>]</li> </ul> <p>The claimed observation of abelian anyons is more securely established:</p> <p>In <a class="existingWikiWord" href="/nlab/show/gallium+arsenide">gallium arsenide</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GaAs</mi></mrow><annotation encoding="application/x-tex">GaAs</annotation></semantics></math>) heterostructures:</p> <ul> <li> <p>H. Bartolomei, et al.: <em>Fractional statistics in anyon collisions</em>, Science <strong>368</strong> 6487 (2020) 173-177 [<a href="https://doi.org/10.1126/science.aaz5601">doi:10.1126/science.aaz5601</a>, <a href="https://arxiv.org/abs/2006.13157">arXiv:2006.13157</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/James+Nakamura">James Nakamura</a> et al.: <em>Aharonov–Bohm interference of fractional quantum Hall edge modes</em>, Nature Physics <strong>15</strong> 563–569 (2019) [<a href="https://doi.org/10.1038/s41567-019-0441-8">doi:10.1038/s41567-019-0441-8</a>, <a href="https://arxiv.org/abs/1901.08452">arXiv:1901.08452</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/James+Nakamura">James Nakamura</a> et al.: <em>Direct observation of anyonic braiding statistics</em>, Nat. Phys. <strong>16</strong> (2020) 931–936 [<a href="https://doi.org/10.1038/s41567-020-1019-1">doi:10.1038/s41567-020-1019-1</a>, <a href="https://arxiv.org/abs/2006.14115">arXiv:2006.14115</a>]</p> </li> <li> <p>Bob Yirka, <em>Best evidence yet for existence of anyons</em>, PhysOrg News (July 10, 2020) [<a href="https://phys.org/news/2020-07-evidence-anyons.html">phys.org/news/2020-07</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/James+Nakamura">James Nakamura</a> et al.: <em>Impact of bulk-edge coupling on observation of anyonic braiding statistics in quantum Hall interferometers</em>, Nature Communications <strong>13</strong> 344 (2022) [<a href="https://doi.org/10.1038/s41467-022-27958-w">doi:10.1038/s41467-022-27958-w</a>, <a href="https://arxiv.org/abs/2107.02136">arXiv:2107.02136</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/James+Nakamura">James Nakamura</a> et al.: <em>Fabry-Pérot Interferometry at the Fractional Quantum Hall State</em>, Phys. Rev. X <strong>13</strong> (2023) 041012 [<a href="https://doi.org/10.1103/PhysRevX.13.041012">doi:10.1103/PhysRevX.13.041012</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/James+Nakamura">James Nakamura</a>: <em>Fabry-Perot interferometry at the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi><mo>−</mo><mn>2</mn><mo stretchy="false">/</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">\nu-2/5</annotation></semantics></math> fractional quantum Hall state</em>, Phys. Rev. X <strong>13</strong> (2023) 041012 [<a href="https://doi.org/10.1103/PhysRevX.13.041012">doi:10.1103/PhysRevX.13.041012</a>, <a href="https://arxiv.org/abs/2304.12415">arXiv:2304.12415</a>]</p> </li> <li> <p>P. Glidic et al.: <em>Signature of anyonic statistics in the integer quantum Hall regime</em>, Nature Commun. <strong>15</strong> 6578 (2024) 1 [<a href="https://doi.org/10.1038/s41467-024-50820-0">doi:10.1038/s41467-024-50820-0</a>, <a href="https://arxiv.org/abs/2401.06069">arXiv:2401.06069</a>]</p> </li> </ul> <p>and in <a class="existingWikiWord" href="/nlab/show/graphene">graphene</a> heterostructures:</p> <ul> <li> <p>Noah Samuelson et al.: <em>Anyonic statistics and slow quasiparticle dynamics in a graphene fractional quantum Hall interferometer</em> [<a href="https://arxiv.org/abs/2403.19628">arXiv:2403.19628</a>]</p> </li> <li> <p>Jehyun Kim et al.: <em>Aharonov-Bohm Interference in Even-Denominator Fractional Quantum Hall States</em> [<a href="https://arxiv.org/abs/2412.19886">arXiv:2412.19886</a>]</p> </li> </ul> </div></div><div> <h3 id="AnyonsInTopologicalSuperconductorsReferences">Anyons in topological superconductors</h3> <p>On <a class="existingWikiWord" href="/nlab/show/anyon">anyon</a>-excitations in <a class="existingWikiWord" href="/nlab/show/topological+superconductors">topological superconductors</a>.</p> <p>via <a class="existingWikiWord" href="/nlab/show/Majorana+zero+modes">Majorana zero modes</a>:</p> <p>Original proposal:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nicholas+Read">Nicholas Read</a>, Dmitry Green, <em>Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect</em>, Phys. Rev. B61:10267, 2000 (<a href="https://arxiv.org/abs/cond-mat/9906453">arXiv:cond-mat/9906453</a>)</li> </ul> <p>Review:</p> <ul> <li id="DasSarmaFreedmanNayak15"> <p><a class="existingWikiWord" href="/nlab/show/Sankar+Das+Sarma">Sankar Das Sarma</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Freedman">Michael Freedman</a>, <a class="existingWikiWord" href="/nlab/show/Chetan+Nayak">Chetan Nayak</a>, <em>Majorana Zero Modes and Topological Quantum Computation</em>, npj Quantum Information 1, 15001 (2015) (<a href="https://www.nature.com/articles/npjqi20151">nature:npjqi20151</a>)</p> </li> <li> <p>Nur R. Ayukaryana, Mohammad H. Fauzi, Eddwi H. Hasdeo, <em>The quest and hope of Majorana zero modes in topological superconductor for fault-tolerant quantum computing: an introductory overview</em> (<a href="https://arxiv.org/abs/2009.07764">arXiv:2009.07764</a>)</p> </li> <li> <p>Yusuke Masaki, Takeshi Mizushima, Muneto Nitta, <em>Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors</em> &lbrack;<a href="https://arxiv.org/abs/2301.11614">arXiv:2301.11614</a>&rbrack;</p> </li> </ul> <p>Further developments:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Meng+Cheng">Meng Cheng</a>, Victor Galitski, <a class="existingWikiWord" href="/nlab/show/Sankar+Das+Sarma">Sankar Das Sarma</a>, <em>Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors</em>, Phys. Rev. B 84, 104529 (2011) (<a href="https://arxiv.org/abs/1106.2549">arXiv:1106.2549</a>)</p> </li> <li id="ShabaniEtAl15"> <p><a class="existingWikiWord" href="/nlab/show/Javad+Shabani">Javad Shabani</a> et al., <em>Two-dimensional epitaxial superconductor-semiconductor heterostructures: A platform for topological superconducting networks</em>, Phys. Rev. B <strong>93</strong> 155402 (2016) [<a href="https://doi.org/10.1103/PhysRevB.93.155402">doi:10.1103/PhysRevB.93.155402</a>, <a href="https://arxiv.org/abs/1511.01127">arXiv:1511.01127</a>]</p> </li> <li id="ShabaniEtAl17"> <p><a class="existingWikiWord" href="/nlab/show/Javad+Shabani">Javad Shabani</a> et al., <em>Zero-Energy Modes from Coalescing Andreev States in a Two-Dimensional Semiconductor-Superconductor Hybrid Platform</em>, Phys. Rev. Lett. <strong>119</strong> (2017) 176805 [<a href="https://doi.org/10.1103/PhysRevLett.119.176805">doi:10.1103/PhysRevLett.119.176805</a>, <a href="https://arxiv.org/abs/1703.03699">arXiv:1703.03699</a>]</p> </li> <li id="ShabaniEtAl22"> <p><a class="existingWikiWord" href="/nlab/show/Javad+Shabani">Javad Shabani</a> et al., <em>Fusion of Majorana Bound States with Mini-Gate Control in Two-Dimensional Systems</em>, Nature Communications <strong>13</strong> (2022) 1738-1747 [<a href="https://doi.org/10.1038/s41467-022-29463-6">doi:10.1038/s41467-022-29463-6</a>, <a href="https://arxiv.org/abs/2101.09272">arXiv:2101.09272</a>]</p> </li> <li id="ShabaniEtAl23"> <p><a class="existingWikiWord" href="/nlab/show/Javad+Shabani">Javad Shabani</a> et al., <em>Quasiparticle dynamics in epitaxial Al-InAs planar Josephson junctions</em>, PRX Quantum <strong>4</strong> 030339 (2023) [<a href="https://doi.org/10.1103/PRXQuantum.4.030339">doi:10.1103/PRXQuantum.4.030339</a>, <a href="https://arxiv.org/abs/2303.04784">arXiv:2303.04784</a>]</p> </li> <li> <p>William F. Schiela, Peng Yu, <a class="existingWikiWord" href="/nlab/show/Javad+Shabani">Javad Shabani</a>: <em>Progress in superconductor-semiconductor topological Josephson junctions</em>, PRX Quantum <strong>5</strong> (2024) 030102 [<a href=" https://doi.org/10.1103/PRXQuantum.5.030102">doi:10.1103/PRXQuantum.5.030102</a>, <a href="https://arxiv.org/abs/2408.12749">arXiv:2408.12749</a>]</p> </li> </ul> <p>via <a class="existingWikiWord" href="/nlab/show/Majorana+zero+modes">Majorana zero modes</a> restricted to edges of <a class="existingWikiWord" href="/nlab/show/topological+insulators">topological insulators</a>:</p> <ul> <li>Biao Lian, Xiao-Qi Sun, Abolhassan Vaezi, <a class="existingWikiWord" href="/nlab/show/Xiao-Liang+Qi">Xiao-Liang Qi</a>, Shou-Cheng Zhang: <em>Topological quantum computation based on chiral Majorana fermions</em>, PNA <strong>115</strong> 43 (2018) 10938-10942 [<a href="https://doi.org/10.1073/pnas.1810003115">doi:10.1073/pnas.1810003115</a>]</li> </ul> <p>See also:</p> <ul> <li>Yusuke Masaki, Takeshi Mizushima, <a class="existingWikiWord" href="/nlab/show/Muneto+Nitta">Muneto Nitta</a>, <em>Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors</em> <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://arxiv.org/abs/2301.11614">arXiv:2301.11614</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> </div><div> <h3 id="ReferencesVortexAnyons">Defect anyons</h3> <p>Often the concept of <a class="existingWikiWord" href="/nlab/show/anyons">anyons</a> is introduced as if a generalization of <a class="existingWikiWord" href="/nlab/show/particle+statistics">particle statistics</a> of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative</a> <a class="existingWikiWord" href="/nlab/show/quanta">quanta</a> like fundamental <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a> and <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a>. But many (concepts of) types of anyons are really <em><a class="existingWikiWord" href="/nlab/show/soliton">solitonic</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/defects">defects</a></em> such as <a class="existingWikiWord" href="/nlab/show/vortices">vortices</a> whose <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> phases are <a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">adiabatic</a> <a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a>.</p> <p>The general concept of <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> of <a class="existingWikiWord" href="/nlab/show/defects">defects</a> in <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a>:</p> <ul> <li id="Mermin79"><a class="existingWikiWord" href="/nlab/show/N.+David+Mermin">N. David Mermin</a>, <em>The topological theory of defects in ordered media</em>, Rev. Mod. Phys. <strong>51</strong> (1979) 591 <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/RevModPhys.51.591">doi:10.1103/RevModPhys.51.591</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> <blockquote> <p>(including a review of <a href="Introduction+to+Topology+--+2">basic homotopy theory</a>)</p> </blockquote> </li> </ul> <p>and more specifically for <a class="existingWikiWord" href="/nlab/show/vortices">vortices</a>:</p> <ul> <li id="LoPreskill93"><a class="existingWikiWord" href="/nlab/show/Hoi-Kwong+Lo">Hoi-Kwong Lo</a>, <a class="existingWikiWord" href="/nlab/show/John+Preskill">John Preskill</a>, <em>Non-Abelian vortices and non-Abelian statistics</em>, Phys. Rev. D <strong>48</strong> (1993) 4821 <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/PhysRevD.48.4821">doi:10.1103/PhysRevD.48.4821</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>Explicit discussion as defect <a class="existingWikiWord" href="/nlab/show/anyons">anyons</a>:</p> <ul> <li id="Kitaev06"><a class="existingWikiWord" href="/nlab/show/Alexei+Kitaev">Alexei Kitaev</a>, <em>Anyons in an exactly solved model and beyond</em>, Annals of Physics <strong>321</strong> 1 (2006) 2-111 [<a href="https://doi.org/10.1016/j.aop.2005.10.005">doi:10.1016/j.aop.2005.10.005</a>] <blockquote> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow></mrow></mrow><annotation encoding="application/x-tex">{}</annotation></semantics></math> [p. 4:] “<em>Anyonic particles are best viewed as a kind of topological defects that reveal nontrivial properties of the ground state.</em>”</p> </blockquote> </li> </ul> <p id="ReferencesGenons"> Anyonic defects which act as <strong>genons</strong>, changing the effective <a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus</a> of the ambient 2D <a class="existingWikiWord" href="/nlab/show/surface">surface</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Maissam+Barkeshli">Maissam Barkeshli</a>, <a class="existingWikiWord" href="/nlab/show/Xiao-Liang+Qi">Xiao-Liang Qi</a>: <em>Topological Nematic States and Non-Abelian Lattice Dislocations</em>, Phys. Rev. X <strong>2</strong> 031013 (2012) [<a href="https://doi.org/10.1103/PhysRevX.2.031013">doi:10.1103/PhysRevX.2.031013</a>, <a href="https://arxiv.org/abs/1112.3311">arXiv:1112.3311</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Maissam+Barkeshli">Maissam Barkeshli</a>, <a class="existingWikiWord" href="/nlab/show/Chao-Ming+Jian">Chao-Ming Jian</a>, <a class="existingWikiWord" href="/nlab/show/Xiao-Liang+Qi">Xiao-Liang Qi</a>: <em>Twist defects and projective non-Abelian braiding statistics</em>, Phys. Rev. B <strong>87</strong> (2013) 045130 [<a href="https://doi.org/10.1103/PhysRevB.87.045130">doi:10.1103/PhysRevB.87.045130</a>, <a href="https://arxiv.org/abs/1208.4834">arXiv:1208.4834</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Xiao-Liang+Qi">Xiao-Liang Qi</a>: <em>Defects in topologically ordered states</em>, talk notes (2014) [<a href="https://nationalmaglab.org/media/dlpayc5u/qi_1.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Qi-DefectsInTopologicalOrder.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Andrey+Gromov">Andrey Gromov</a>: <em>Geometric Defects in Quantum Hall States</em>, Phys. Rev. B <strong>94</strong> 085116 (2016) [<a href="https://doi.org/10.1103/PhysRevB.94.085116">doi:10.1103/PhysRevB.94.085116</a>]</p> </li> </ul> <p>see also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Simon+Burton">Simon Burton</a>, Elijah Durso-Sabina, Natalie C. Brown: <em>Genons, Double Covers and Fault-tolerant Clifford Gates</em> [<a href="https://arxiv.org/abs/2406.09951">arXiv:2406.09951</a>]</li> </ul> <p>and their potential experimental realization:</p> <ul> <li>Zhao Liu, Gunnar Möller, Emil J. Bergholtz: <em>Exotic Non-Abelian Topological Defects in Lattice Fractional Quantum Hall States</em>, Phys. Rev. Lett. <strong>119</strong> (2017) 106801 [<a href="https://doi.org/10.1103/PhysRevLett.119.106801">doi:10.1103/PhysRevLett.119.106801</a>]</li> </ul> <p>Concrete <a class="existingWikiWord" href="/nlab/show/vortex">vortex</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo lspace="verythinmathspace" rspace="0em">−</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{-}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/anyons">anyons</a> in <a class="existingWikiWord" href="/nlab/show/Bose-Einstein+condensates">Bose-Einstein condensates</a>:</p> <ul> <li id="PFCZ01"> <p>B. Paredes, P. Fedichev, <a class="existingWikiWord" href="/nlab/show/J.+Ignacio+Cirac">J. Ignacio Cirac</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Zoller">Peter Zoller</a>: <em>1/2-Anyons in small atomic Bose-Einstein condensates</em>, Phys. Rev. Lett. <strong>87</strong> (2001) 010402 [<a href="https://doi.org/10.1103/PhysRevLett.87.010402">doi:10.1103/PhysRevLett.87.010402</a>, <a href="https://arxiv.org/abs/cond-mat/0103251">arXiv:cond-mat/0103251</a>]</p> </li> <li> <p>Julien Garaud, Jin Dai, <a class="existingWikiWord" href="/nlab/show/Antti+J.+Niemi">Antti J. Niemi</a>, <em>Vortex precession and exchange in a Bose-Einstein condensate</em>, J. High Energ. Phys. <strong>2021</strong> 157 (2021) [<a href="https://arxiv.org/abs/2010.04549">arXiv:2010.04549</a>]</p> </li> <li id="MPSS19"> <p>Thomas Mawson, Timothy Petersen, <a class="existingWikiWord" href="/nlab/show/Joost+Slingerland">Joost Slingerland</a>, <a class="existingWikiWord" href="/nlab/show/Tapio+Simula">Tapio Simula</a>, <em>Braiding and fusion of non-Abelian vortex anyons</em>, Phys. Rev. Lett. <strong>123</strong> (2019) 140404 [<a href="https://doi.org/10.1103/PhysRevLett.123.140404">doi:10.1103/PhysRevLett.123.140404</a>]</p> </li> </ul> <p>and in (other) <a class="existingWikiWord" href="/nlab/show/superfluids">superfluids</a>:</p> <ul> <li id="MMN21">Yusuke Masaki, Takeshi Mizushima, Muneto Nitta, <em>Non-Abelian Half-Quantum Vortices in 3P2 Topological Superfluids</em> [<a href="https://arxiv.org/abs/2107.02448">arXiv:2107.02448</a>]</li> </ul> <p>and in condensates <em>of</em> non-defect anyons:</p> <ul> <li id="CDLR19">Michele Correggi, Romain Duboscq, Douglas Lundholm, Nicolas Rougerie: <em>Vortex patterns in the almost-bosonic anyon gas</em>, Europhys. Lett. <strong>126</strong> (2019) 20005 [<a href="https://doi.org/10.1209/0295-5075/126/20005">doi:10.1209/0295-5075/126/20005</a>, <a href="https://arxiv.org/abs/1901.10739">arXiv:1901.10739</a>]</li> </ul> <p>On analog behaviour in liquid crystals:</p> <ul> <li>Alexander Mietke, Jörn Dunkel: <em>Anyonic defect braiding and spontaneous chiral symmetry breaking in dihedral liquid crystals</em>, Phys. Rev. X <strong>12</strong> (2022) 011027 [<a href="https://arxiv.org/abs/2011.04648">arXiv:2011.04648</a>, <a href="https://doi.org/10.1103/PhysRevX.12.011027">doi:10.1103/PhysRevX.12.011027</a>]</li> </ul> <p>See also <a href="anyon#AhnParkYang19">Ahn, Park & Yang 19</a> who refer to the band nodes in the <a class="existingWikiWord" href="/nlab/show/Brillouin+torus">Brillouin torus</a> of a <a class="existingWikiWord" href="/nlab/show/semi-metal">semi-metal</a> as “vortices in momentum space”.</p> <p>And see at <em><a class="existingWikiWord" href="/nlab/show/defect+brane">defect brane</a></em>.</p> </div><div> <h3 id="ReferencesAnyonicBraidingInMomentumSpace">Anyons in momentum-space</h3> <p>On non-trivial <a class="existingWikiWord" href="/nlab/show/braid+group+statistics">braiding</a> of nodal points in the <a class="existingWikiWord" href="/nlab/show/Brillouin+torus">Brillouin torus</a> of <a class="existingWikiWord" href="/nlab/show/topological+semi-metals">topological semi-metals</a> (“braiding in momentum space”):</p> <ul> <li id="AhnParkYang19"> <p>Junyeong Ahn, Sungjoon Park, Bohm-Jung Yang, <em>Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle</em>, Phys. Rev. X <strong>9</strong> (2019) 021013 [<a href="https://doi.org/10.1103/PhysRevX.9.021013">doi:10.1103/PhysRevX.9.021013</a>, <a href="https://arxiv.org/abs/1808.05375">arXiv:1808.05375</a>]</p> <blockquote> <p>“here are band crossing points, henceforth called <a class="existingWikiWord" href="/nlab/show/vortices">vortices</a>”</p> </blockquote> </li> <li> <p>QuanSheng Wu, Alexey A. Soluyanov, <a class="existingWikiWord" href="/nlab/show/Tom%C3%A1%C5%A1+Bzdu%C5%A1ek">Tomáš Bzdušek</a>, <em>Non-Abelian band topology in noninteracting metals</em>, Science <strong>365</strong> (2019) 1273-1277 [<a href="https://doi.org/10.1126/science.aau8740">doi:10.1126/science.aau8740</a>, <a href="https://arxiv.org/abs/1808.07469">arXiv:1808.07469</a>]</p> <blockquote> <p>“<a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> of <em><a class="existingWikiWord" href="/nlab/show/complement">complement</a></em> of nodal points/lines considered above (3)”</p> </blockquote> </li> <li id="TiwariBzdusek20"> <p>Apoorv Tiwari, <a class="existingWikiWord" href="/nlab/show/Tom%C3%A1%C5%A1+Bzdu%C5%A1ek">Tomáš Bzdušek</a>, <em>Non-Abelian topology of nodal-line rings in PT-symmetric systems</em>, Phys. Rev. B <strong>101</strong> (2020) 195130 [<a href="https://doi.org/10.1103/PhysRevB.101.195130">doi:10.1103/PhysRevB.101.195130</a>]</p> <blockquote> <p>“a new type non-Abelian ‘braiding’ of nodal-line rings inside the momentum space”</p> </blockquote> </li> <li id="BWSWYB20"> <p><a class="existingWikiWord" href="/nlab/show/Adrien+Bouhon">Adrien Bouhon</a>, QuanSheng Wu, <a class="existingWikiWord" href="/nlab/show/Robert-Jan+Slager">Robert-Jan Slager</a>, Hongming Weng, Oleg V. Yazyev, <a class="existingWikiWord" href="/nlab/show/Tom%C3%A1%C5%A1+Bzdu%C5%A1ek">Tomáš Bzdušek</a>, <em>Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe</em>, Nature Physics <strong>16</strong> (2020) 1137-1143 [<a href="https://doi.org/10.1038/s41567-020-0967-9">doi:10.1038/s41567-020-0967-9</a>, <a href="https://arxiv.org/abs/1907.10611">arXiv:1907.10611</a>]</p> <blockquote> <p>“Here we report that Weyl points in three-dimensional (3D) systems with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mn>2</mn></msub><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}_2\mathcal{T}</annotation></semantics></math> symmetry carry non-Abelian topological charges. These charges are transformed via non-trivial phase factors that arise upon braiding the nodes inside the reciprocal momentum space.”</p> </blockquote> </li> </ul> <p id="InTwistedBilayerGraphene"> Braiding of Dirac points in twisted bilayer <a class="existingWikiWord" href="/nlab/show/graphene">graphene</a>:</p> <ul> <li> <p>Jian Kang, Oskar Vafek: <em>Non-Abelian Dirac node braiding and near-degeneracy of correlated phases at odd integer filling in magic angle twisted bilayer graphene</em>, Phys. Rev. B <strong>102</strong> (2020) 035161 [<a href="https://doi.org/10.1103/PhysRevB.102.035161">doi:10.1103/PhysRevB.102.035161</a>, <a href="https://arxiv.org/abs/2002.10360">arXiv:2002.10360</a>]</p> </li> <li id="JBLZHLSJ21"> <p>Bin Jiang, <a class="existingWikiWord" href="/nlab/show/Adrien+Bouhon">Adrien Bouhon</a>, Zhi-Kang Lin, Xiaoxi Zhou, Bo Hou, Feng Li, <a class="existingWikiWord" href="/nlab/show/Robert-Jan+Slager">Robert-Jan Slager</a>, Jian-Hua Jiang: <em>Experimental observation of non-Abelian topological acoustic semimetals and their phase transitions</em>, Nature Physics <strong>17</strong> (2021) 1239-1246 [<a href="https://doi.org/10.1038/s41567-021-01340-x">doi:10.1038/s41567-021-01340-x</a>, <a href="https://arxiv.org/abs/2104.13397">arXiv:2104.13397</a>]</p> <p>(analog realization in <a class="existingWikiWord" href="/nlab/show/phononic+crystals">phononic crystals</a>)</p> <blockquote> <p>Here, we consider an exotic type of topological phases beyond the above paradigms that, instead, depend on topological charge conversion processes when band nodes are braided with respect to each other in momentum space or recombined over the Brillouin zone. The braiding of band nodes is in some sense the reciprocal space analog of the non-Abelian braiding of particles in real space. […] we experimentally observe non-Abelian topological semimetals and their evolutions using acoustic Bloch bands in kagome acoustic metamaterials. By tuning the geometry of the metamaterials, we experimentally confirm the creation, annihilation, moving, merging and splitting of the topological band nodes in multiple bandgaps and the associated non-Abelian topological phase transitions</p> </blockquote> </li> <li id="ParkWongZhangOh21"> <p>Haedong Park, Stephan Wong, Xiao Zhang, and Sang Soon Oh, <em>Non-Abelian Charged Nodal Links in a Dielectric Photonic Crystal</em>, ACS Photonics <strong>8</strong> (2021) 2746–2754 &lbrack;<a href="https://doi.org/10.1021/acsphotonics.1c00876">doi:10.1021/acsphotonics.1c00876</a>&rbrack;</p> </li> <li id="CBSM22"> <p>Siyu Chen, <a class="existingWikiWord" href="/nlab/show/Adrien+Bouhon">Adrien Bouhon</a>, <a class="existingWikiWord" href="/nlab/show/Robert-Jan+Slager">Robert-Jan Slager</a>, <a class="existingWikiWord" href="/nlab/show/Bartomeu+Monserrat">Bartomeu Monserrat</a>, <em>Non-Abelian braiding of Weyl nodes via symmetry-constrained phase transitions</em> (formerly: <em>Manipulation and braiding of Weyl nodes using symmetry-constrained phase transitions</em>), Phys. Rev. B <strong>105</strong> (2022) L081117 [<a href="https://doi.org/10.1103/PhysRevB.105.L081117">doi:10.1103/PhysRevB.105.L081117</a>, <a href="https://arxiv.org/abs/2108.10330">arXiv:2108.10330</a>]</p> <blockquote> <p>“Our work opens up routes to readily manipulate Weyl nodes using only slight external parameter changes, paving the way for the practical realization of reciprocal space braiding.”</p> </blockquote> </li> <li id="PBSM22"> <p><a class="existingWikiWord" href="/nlab/show/Bo+Peng">Bo Peng</a>, <a class="existingWikiWord" href="/nlab/show/Adrien+Bouhon">Adrien Bouhon</a>, <a class="existingWikiWord" href="/nlab/show/Robert-Jan+Slager">Robert-Jan Slager</a>, <a class="existingWikiWord" href="/nlab/show/Bartomeu+Monserrat">Bartomeu Monserrat</a>, <em>Multi-gap topology and non-Abelian braiding of phonons from first principles</em>, Phys. Rev. B <strong>105</strong> (2022) 085115 [<a href="https://arxiv.org/abs/2111.05872">arXiv:2111.05872</a>, <a href="https://doi.org/10.1103/PhysRevB.105.085115">doi:10.1103/PhysRevB.105.085115</a>]</p> <p>(analog realization in <a class="existingWikiWord" href="/nlab/show/phononic+crystals">phononic crystals</a>)</p> <blockquote> <p>new opportunities for exploring non-Abelian braiding of band crossing points (nodes) in reciprocal space, providing an alternative to the real space braiding exploited by other strategies.</p> <p>Real space braiding is practically constrained to boundary states, which has made experimental observation and manipulation difficult; instead, reciprocal space braiding occurs in the bulk states of the band structures and we demonstrate in this work that this provides a straightforward platform for non-Abelian braiding.</p> </blockquote> </li> <li id="PBMS22"> <p><a class="existingWikiWord" href="/nlab/show/Bo+Peng">Bo Peng</a>, <a class="existingWikiWord" href="/nlab/show/Adrien+Bouhon">Adrien Bouhon</a>, <a class="existingWikiWord" href="/nlab/show/Bartomeu+Monserrat">Bartomeu Monserrat</a>, <a class="existingWikiWord" href="/nlab/show/Robert-Jan+Slager">Robert-Jan Slager</a>, <em>Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates</em>, Nature Communications <strong>13</strong> 423 (2022) [<a href="https://doi.org/10.1038/s41467-022-28046-9">doi:10.1038/s41467-022-28046-9</a>]</p> <p>(analog realization in <a class="existingWikiWord" href="/nlab/show/phononic+crystals">phononic crystals</a>)</p> <blockquote> <p>it is possible to controllably braid Kagome band nodes in monolayer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">Si</mi> <mn>2</mn></msub><msub><mi mathvariant="normal">O</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{Si}_2 \mathrm{O}_3</annotation></semantics></math> using strain and/or an external electric field.</p> </blockquote> </li> <li id="ParkGaoZhangOh22"> <p>Haedong Park, Wenlong Gao, Xiao Zhang, Sang Soon Oh, <em>Nodal lines in momentum space: topological invariants and recent realizations in photonic and other systems</em>, Nanophotonics <strong>11</strong> 11 (2022) 2779–2801 [<a href="https://doi.org/10.1515/nanoph-2021-0692">doi:10.1515/nanoph-2021-0692</a>]</p> <p>(analog realization in <a class="existingWikiWord" href="/nlab/show/photonic+crystals">photonic crystals</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Adrien+Bouhon">Adrien Bouhon</a>, <a class="existingWikiWord" href="/nlab/show/Robert-Jan+Slager">Robert-Jan Slager</a>, <em>Multi-gap topological conversion of Euler class via band-node braiding: minimal models, PT-linked nodal rings, and chiral heirs</em> [<a href="https://arxiv.org/abs/2203.16741">arXiv:2203.16741</a>]</p> </li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Robert-Jan+Slager">Robert-Jan Slager</a>, <a class="existingWikiWord" href="/nlab/show/Adrien+Bouhon">Adrien Bouhon</a>, <a class="existingWikiWord" href="/nlab/show/Fatma+Nur+%C3%9Cnal">Fatma Nur Ünal</a>, <em>Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase</em>, Nature Communications <strong>15</strong> 1144 (2024) [<a href="https://arxiv.org/abs/2208.12824">arXiv:2208.12824</a>, <a href="https://doi.org/10.1038/s41467-024-45302-2">doi:10.1038/s41467-024-45302-2</a>]</p> </li> <li> <p>Huahui Qiu et al., <em>Minimal non-abelian nodal braiding in ideal metamaterials</em>, Nature Communications <strong>14</strong> 1261 (2023) [<a href="https://doi.org/10.1038/s41467-023-36952-9">doi:10.1038/s41467-023-36952-9</a>]</p> </li> <li> <p>Wojciech J. Jankowski, Mohammedreza Noormandipour, <a class="existingWikiWord" href="/nlab/show/Adrien+Bouhon">Adrien Bouhon</a>, <a class="existingWikiWord" href="/nlab/show/Robert-Jan+Slager">Robert-Jan Slager</a>, <em>Disorder-induced topological quantum phase transitions in Euler semimetals</em> [<a href="https://arxiv.org/abs/2306.13084">arXiv:2306.13084</a>]</p> </li> <li> <p>Seung Hun Lee, Yuting Qian, Bohm-Jung Yang, <em>Euler band topology in spin-orbit coupled magnetic systems</em> [<a href="https://arxiv.org/abs/2404.16383">arXiv:2404.16383</a>]</p> <blockquote> <p>“Based on first-principles calculations, we report that such nodal point braiding in 2D electronic bands can be realized in a MSWI candidate, the bilayer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ZrTe</mi> <mn>5</mn></msub></mrow><annotation encoding="application/x-tex">ZrTe_5</annotation></semantics></math> with in-plane ferromagnetism under pressure. […] one can expect that the braiding of nodes can be achieved in 2D bilayer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ZrTe</mi> <mn>5</mn></msub></mrow><annotation encoding="application/x-tex">ZrTe_5</annotation></semantics></math> under the influence of an external in-plane Zeeman field.”</p> </blockquote> </li> </ul> <p><br /></p> <p>Incidentally, references indicating that the required toroidal (or yet higher genus) geometry for anyonic topological order in position space is dubious (as opposed to the evident toroidal geometry of the momentum-space <a class="existingWikiWord" href="/nlab/show/Brillouin+torus">Brillouin torus</a>): <a href="Laughlin+state#Lan19">Lan 19, p. 1</a>, ….</p> <p><strong>Knotted nodal lines in 3d semimetals</strong></p> <p>Beware that various authors consider <a class="existingWikiWord" href="/nlab/show/braids">braids</a>/<a class="existingWikiWord" href="/nlab/show/knots">knots</a> formed by nodal <em>lines</em> in <em>3d</em> semimetals, i.e. knotted nodal lines in 3 spatial dimensions, as opposed to <a class="existingWikiWord" href="/nlab/show/worldlines">worldlines</a> (in 2+1 spacetime dimensions) of nodal points in effectively 2d semimetals needed for the <a class="existingWikiWord" href="/nlab/show/anyon">anyon</a>-<a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> considered above.</p> <p>An argument that these nodal lines in 3d space, nevertheless, may be controlled by <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>:</p> <ul> <li id="LianVafaVafaZhang17"><a class="existingWikiWord" href="/nlab/show/Biao+Lian">Biao Lian</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <a class="existingWikiWord" href="/nlab/show/Farzan+Vafa">Farzan Vafa</a>, <a class="existingWikiWord" href="/nlab/show/Shou-Cheng+Zhang">Shou-Cheng Zhang</a>, <em>Chern-Simons theory and Wilson loops in the Brillouin zone</em>, Phys. Rev. B <strong>95</strong> (2017) 094512 [<a href="https://doi.org/10.1103/PhysRevB.95.094512">doi:10.1103/PhysRevB.95.094512</a>]</li> </ul> </div><div> <h3 id="TopologicalQuantumComputationWithAnyons">Topological quantum computation with anyons</h3> <p>The idea of <a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a> via a <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> with <a class="existingWikiWord" href="/nlab/show/anyon">anyon</a> <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> <a class="existingWikiWord" href="/nlab/show/defects">defects</a> is due to:</p> <ul> <li id="Kitaev03"> <p><a class="existingWikiWord" href="/nlab/show/Alexei+Kitaev">Alexei Kitaev</a>, <em>Fault-tolerant quantum computation by anyons</em>, Annals Phys. 303 (2003) 2-30 [<a href="https://doi.org/10.1016/S0003-4916(02)00018-0">doi:10.1016/S0003-4916(02)00018-0</a>, <a href="https://arxiv.org/abs/quant-ph/9707021">arXiv:quant-ph/9707021</a>]</p> </li> <li id="Freedman98"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Freedman">Michael Freedman</a>, <em>P/NP, and the quantum field computer</em>, Proc. Nat. Acad. Sci. <strong>95</strong> 1 (1998) 98-101 [<a href="https://doi.org/10.1073/pnas.95.1.9">doi:10.1073/pnas.95.1.9</a>]</p> </li> <li id="FreedmanKitaevLarsenWang03"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Freedman">Michael Freedman</a>, <a class="existingWikiWord" href="/nlab/show/Alexei+Kitaev">Alexei Kitaev</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Larsen">Michael Larsen</a>, <a class="existingWikiWord" href="/nlab/show/Zhenghan+Wang">Zhenghan Wang</a>, <em>Topological quantum computation</em>, Bull. Amer. Math. Soc. <strong>40</strong> (2003), 31-38 (<a href="https://arxiv.org/abs/quant-ph/0101025">arXiv:quant-ph/0101025</a>, <a href="https://doi.org/10.1090/S0273-0979-02-00964-3">doi:10.1090/S0273-0979-02-00964-3</a>, <a href="http://www.ams.org/journals/bull/2003-40-01/S0273-0979-02-00964-3/S0273-0979-02-00964-3.pdf">pdf</a>)</p> </li> <li id="FreedmanLarsenWang02"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Freedman">Michael Freedman</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Larsen">Michael Larsen</a>, <a class="existingWikiWord" href="/nlab/show/Zhenghan+Wang">Zhenghan Wang</a>, <em>A modular functor which is universal for quantum computation</em>, Communications in Mathematical Physics. <strong>227</strong> 3 (2002) 605-622 [<a href="https://doi.org/10.1007/s002200200645">doi:10.1007/s002200200645</a>, <a href="http://arxiv.org/abs/quant-ph/0001108">arXiv:quant-ph/0001108</a>]</p> <blockquote> <p>(specifically via <a class="existingWikiWord" href="/nlab/show/su%282%29-anyons">su(2)-anyons</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chetan+Nayak">Chetan Nayak</a>, <a class="existingWikiWord" href="/nlab/show/Steven+H.+Simon">Steven H. Simon</a>, <a class="existingWikiWord" href="/nlab/show/Ady+Stern">Ady Stern</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Freedman">Michael Freedman</a>, <a class="existingWikiWord" href="/nlab/show/Sankar+Das+Sarma">Sankar Das Sarma</a>, <em>Non-Abelian Anyons and Topological Quantum Computation</em>, Rev. Mod. Phys. <strong>80</strong> 1083 (2008) [<a href="https://doi.org/10.1103/RevModPhys.80.1083">doi:10.1103/RevModPhys.80.1083</a>, <a href="http://arxiv.org/abs/0707.1889">arXiv:0707.1888</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dmitry+Melnikov">Dmitry Melnikov</a>, <a class="existingWikiWord" href="/nlab/show/Andrei+Mironov">Andrei Mironov</a>, <a class="existingWikiWord" href="/nlab/show/Sergey+Mironov">Sergey Mironov</a>, <a class="existingWikiWord" href="/nlab/show/Alexei+Morozov">Alexei Morozov</a>, <a class="existingWikiWord" href="/nlab/show/Andrey+Morozov">Andrey Morozov</a>, <em>Towards topological quantum computer</em>, Nucl. Phys. B926 (2018) 491-508 (<a href="https://arxiv.org/abs/1703.00431">arXiv:1703.00431</a>, <a href="https://doi.org/10.1016/j.nuclphysb.2017.11.016">doi:10.1016/j.nuclphysb.2017.11.016</a>)</p> </li> </ul> <p>and via a <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> (like <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> but with <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete</a> <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>):</p> <ul> <li> <p>R. Walter Ogburn, <a class="existingWikiWord" href="/nlab/show/John+Preskill">John Preskill</a>, <em>Topological Quantum Computation</em>, in: <em>Quantum Computing and Quantum Communications</em>, Lecture Notes in Computer Science <strong>1509</strong>, Springer (1998) [<a href="https://doi.org/10.1007/3-540-49208-9_31">doi:10.1007/3-540-49208-9_31</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Mochon">Carlos Mochon</a>, <em>Anyons from non-solvable finite groups are sufficient for universal quantum computation</em>, Phys. Rev. A <strong>67</strong> 022315 (2003) [<a href="https://doi.org/10.1103/PhysRevA.67.022315">doi:10.1103/PhysRevA.67.022315</a>, <a href="https://arxiv.org/abs/quant-ph/0206128">arXiv:quant-ph/0206128</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Mochon">Carlos Mochon</a>, <em>Anyon computers with smaller groups</em>, Phys. Rev. A <strong>69</strong> 032306 (2004) [<a href="https://doi.org/10.1103/PhysRevA.69.032306">doi:10.1103/PhysRevA.69.032306</a>, <a href="https://arxiv.org/abs/quant-ph/0306063">arXiv:quant-ph/0306063</a>]</p> </li> </ul> <p>Proposals more specifically for topological quantum gates implemented in <a class="existingWikiWord" href="/nlab/show/fractional+quantum+Hall+systems">fractional quantum Hall systems</a>:</p> <ul> <li> <p>D. V. Averin, V.J. Goldman: <em>Quantum computation with quasiparticles of the fractional quantum Hall effect</em>, Solid State Communications <strong>121</strong> 1 (2001) 25-28 [<a href="https://doi.org/10.1016/S0038-1098(01)00447-1">doi:10.1016/S0038-1098(01)00447-1</a>, <a href="https://arxiv.org/abs/cond-mat/0110193">arXiv:cond-mat/0110193</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sankar+Das+Sarma">Sankar Das Sarma</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Freedman">Michael Freedman</a>, <a class="existingWikiWord" href="/nlab/show/Chetan+Nayak">Chetan Nayak</a>: <em>Topologically-Protected Qubits from a Possible Non-Abelian Fractional Quantum Hall State</em>, Phys. Rev. Lett. <strong>94</strong> 166802 (2005) [<a href="https://doi.org/10.1103/PhysRevLett.94.166802">doi:10.1103/PhysRevLett.94.166802</a>, <a href="https://arxiv.org/abs/cond-mat/0412343">arXiv:cond-mat/0412343</a>]</p> </li> <li> <p>Sergey Bravyi: <em>Universal Quantum Computation with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi><mo>=</mo><mn>5</mn><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\nu = 5/2</annotation></semantics></math> Fractional Quantum Hall State</em>, Phys. Rev. A <strong>73</strong> 042313 (2006) [<a href="https://doi.org/10.1103/PhysRevA.73.042313 ">doi:10.1103/PhysRevA.73.042313</a>, <a href="https://arxiv.org/abs/quant-ph/0511178">arXiv:quant-ph/0511178</a>]</p> </li> </ul> <p>Monographs:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Zhenghan+Wang">Zhenghan Wang</a>, <em>Topological Quantum Computation</em>, CBMS Regional Conference Series in Mathematics <strong>112</strong>, AMS 2010 (<a href="https://bookstore.ams.org/cbms-112">ISBN-13: 978-0-8218-4930-9</a>, <a href="http://web.math.ucsb.edu/~zhenghwa/data/course/cbms.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jiannis+K.+Pachos">Jiannis K. Pachos</a>, <em>Introduction to Topological Quantum Computation</em>, Cambridge University Press (2012) [<a href="https://doi.org/10.1017/CBO9780511792908">doi:10.1017/CBO9780511792908</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tudor+D.+Stanescu">Tudor D. Stanescu</a>, Part IV 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> <li id="Simon23"> <p><a class="existingWikiWord" href="/nlab/show/Steven+H.+Simon">Steven H. Simon</a>, <em>Topological Quantum</em>, Oxford University Press (2023) [<a href="https://global.oup.com/academic/product/topological-quantum-9780198886723">ISBN:9780198886723</a>, <a href="http://www-thphys.physics.ox.ac.uk/people/SteveSimon/topological2021/TopoBook-Sep28-2021.pdf">pdf</a>, <a href="http://www-thphys.physics.ox.ac.uk/people/SteveSimon/topological2021/topocourse2021.html">webpage</a>]</p> </li> </ul> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Louis+Kauffman">Louis Kauffman</a>, <em>Quantum Topology and Quantum Computing</em>, in: <a class="existingWikiWord" href="/nlab/show/Samuel+J.+Lomonaco">Samuel J. Lomonaco</a> (ed.), <em>Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium</em>, Proceedings of Symposia in Applied Mathematics <strong>58</strong>, AMS (2002) [<a href="https://redirect.cs.umbc.edu/~lomonaco/ams/lecturenotes/Kauffman.pdf">pdf</a>, <a href="https://doi.org/10.1090/psapm/058">doi:10.1090/psapm/058</a>]</p> <blockquote> <p>(in relation to <a class="existingWikiWord" href="/nlab/show/quantum+topology">quantum topology</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Preskill">John Preskill</a>: <em>Topological Quantum Computation</em>, Chapter 9 in: <em>Quantum Computation</em>, lecture notes (since 2004) [<a href="http://theory.caltech.edu/~preskill/ph219/topological.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Preskill-TQClecture.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gavin+K.+Brennen">Gavin K. Brennen</a>, <a class="existingWikiWord" href="/nlab/show/Jiannis+K.+Pachos">Jiannis K. Pachos</a>, <em>Why should anyone care about computing with anyons?</em>, Proc. R. Soc. A <strong>464</strong> (2008) 1-24 [<a href="https://doi.org/10.1098/rspa.2007.0026">doi:10.1098/rspa.2007.0026</a>, <a href="https://arxiv.org/abs/0704.2241">arXiv:0704.2241</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ady+Stern">Ady Stern</a>, Netanel H. Lindner, <em>Topological Quantum Computation – From Basic Concepts to First Experiments</em>, Science <strong>339</strong> 6124 (2013) 1179-1184 [<a href="https://doi.org/10.1126/science.1231473">doi:10.1126/science.1231473</a>, <a href="https://inspirehep.net/literature/2748124">spire:2748124</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+C.+Rowell">Eric C. Rowell</a>, <em>An Invitation to the Mathematics of Topological Quantum Computation</em>, J. Phys.: Conf. Ser. <strong>698</strong> (2016) 012012 [<a href="https://iopscience.iop.org/article/10.1088/1742-6596/698/1/012012">doi:10.1088/1742-6596/698/1/012012</a>, <a href="https://arxiv.org/abs/1601.05288">arXiv:1601.05288</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ananda+Roy">Ananda Roy</a>, <a class="existingWikiWord" href="/nlab/show/David+P.+DiVincenzo">David P. DiVincenzo</a>, <em>Topological Quantum Computing</em>, Lecture notes of the <a href="https://www.fz-juelich.de/en/pgi/expertise/schools-and-courses/iff-spring-school/the-iff-spring-school-topics-and-history/springschool2017">48th IFF Spring School</a> (2017) [<a href="https://arxiv.org/abs/1701.05052">arXiv:1701.05052</a>]</p> </li> <li id="LahtinenPachos17"> <p><a class="existingWikiWord" href="/nlab/show/Ville+Lahtinen">Ville Lahtinen</a>, <a class="existingWikiWord" href="/nlab/show/Jiannis+K.+Pachos">Jiannis K. Pachos</a>, <em>A Short Introduction to Topological Quantum Computation</em>, SciPost Phys. <strong>3</strong> 021 (2017) [<a href="https://scipost.org/SciPostPhys.3.3.021">doi: 10.21468/SciPostPhys.3.3.021</a>, <a href="https://arxiv.org/abs/1705.04103">arXiv:1705.04103</a>]</p> </li> <li id="RowellWang18"> <p><a class="existingWikiWord" href="/nlab/show/Eric+C.+Rowell">Eric C. Rowell</a>, <a class="existingWikiWord" href="/nlab/show/Zhenghan+Wang">Zhenghan Wang</a>, <em>Mathematics of Topological Quantum Computing</em>, Bull. Amer. Math. Soc. 55 (2018), 183-238 (<a href="https://arxiv.org/abs/1705.06206">arXiv:1705.06206</a>, <a href="https://doi.org/10.1090/bull/1605">doi:10.1090/bull/1605</a>)</p> </li> <li> <p>Bernard Field, <a class="existingWikiWord" href="/nlab/show/Tapio+Simula">Tapio Simula</a>, <em>Introduction to topological quantum computation with non-Abelian anyons</em>, Quantum Science and Technology <strong>3</strong> (2018) 045004 [<a href="https://iopscience.iop.org/article/10.1088/2058-9565/aacad2">doi:10.1088/2058-9565/aacad2</a>, <a href="https://arxiv.org/abs/1802.06176">arXiv:1802.06176</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Muhammad+Ilyas">Muhammad Ilyas</a>, <em>Quantum Field Theories, Topological Materials, and Topological Quantum Computing</em> [<a href="https://arxiv.org/abs/2208.09707">arXiv:2208.09707</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+C.+Rowell">Eric C. Rowell</a>, <em>Braids, Motions and Topological Quantum Computing</em> [<a href="https://arxiv.org/abs/2208.11762">arXiv:2208.11762</a>, <a href="https://inspirehep.net/literature/2141848">spire:2141848</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Sachin+Valera">Sachin Valera</a>: <em>Topological Quantum Computing</em>, <a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a> <strong>4</strong> (2025) 325-345 [<a href="https://doi.org/10.1016/B978-0-323-95703-8.00262-7">doi:10.1016/B978-0-323-95703-8.00262-7</a>]</p> </li> </ul> <p id="FocusOnAbelianAnyons"> Focus on <em><a class="existingWikiWord" href="/nlab/show/abelian+anyons">abelian anyons</a></em>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jiannis+K.+Pachos">Jiannis K. Pachos</a>, <em>Quantum computation with abelian anyons on the honeycomb lattice</em>, International Journal of Quantum Information <strong>4</strong> 6 (2006) 947-954 [<a href="https://doi.org/10.1142/S0219749906002328">doi:10.1142/S0219749906002328</a>, <a href="https://arxiv.org/abs/quant-ph/0511273">arXiv:quant-ph/0511273</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/James+Robin+Wootton">James Robin Wootton</a>, <em>Dissecting Topological Quantum Computation</em>, PhD thesis, Leeds (2010) [<a href="https://etheses.whiterose.ac.uk/1163">ethesis:1163</a>, <a href="https://etheses.whiterose.ac.uk/1163/1/main.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/WoottonDissectingTQC.pdf" title="pdf">pdf</a>]</p> <blockquote> <p>“non-Abelian anyons are usually assumed to be better suited to the task. Here we challenge this view, demonstrating that Abelian anyon models have as much potential as some simple non-Abelian models. […] Though universal non-Abelian models are admittedly the holy grail of topological quantum computation, and rightly so, this thesis has shown that Abelian models are just as useful as non-universal non-Abelian models. […] Abelian models are a computationally powerful, fault-tolerant and experimentally realistic prospect for quantum computation.”</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Seth+Lloyd">Seth Lloyd</a>, <em>Quantum computation with abelian anyons</em>, Quantum Information Processing <strong>1</strong> 1/2 (2002) [<a href="https://doi.org/10.1023/A:1019649101654">doi:10.1023/A:1019649101654</a>, <a href="https://arxiv.org/abs/quant-ph/0004010">arXiv:quant-ph/0004010</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/James+R.+Wootton">James R. Wootton</a>, <a class="existingWikiWord" href="/nlab/show/Jiannis+K.+Pachos">Jiannis K. Pachos</a>: <em>Universal Quantum Computation with Abelian Anyon Models</em>, Electronic Notes in Theoretical Computer Science <strong>270</strong> 2 (2011) 209-218 [<a href="https://doi.org/10.1016/j.entcs.2011.01.032">doi:10.1016/j.entcs.2011.01.032</a>, <a href="https://arxiv.org/abs/0904.4373">arXiv:0904.4373</a>]</p> </li> </ul> <p>see also:</p> <ul> <li> <p>Menelaos Zikidis: <em>Abelian Anyons and Fractional Quantum Hall Effect</em>, Seminar notes (2017) [<a href="https://www.mathi.uni-heidelberg.de/~walcher/teaching/sose17/atqft/writeups/Menelaos.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Zikidis-AbelianAnyons.pdf" title="pdf">pdf</a>]</p> </li> <li> <p>Wade Bloomquist, <a class="existingWikiWord" href="/nlab/show/Zhenghan+Wang">Zhenghan Wang</a>, <em>On Topological Quantum Computing With Mapping Class Group Representations</em>, J. Phys. A: Math. Theor. <strong>52</strong> (2019) 015301 [<a href="https://iopscience.iop.org/article/10.1088/1751-8121/aaeea1">doi:10.1088/1751-8121/aaeea1</a>, <a href="https://arxiv.org/abs/1805.04622">arXiv:1805.04622</a>]</p> </li> <li> <p>Yichen Hu, Biao Lian: <em>Chiral Sachdev-Ye model: Integrability and chaos of anyons in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>+</mo><mn>1</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">1+1d</annotation></semantics></math></em>, Phys. Rev. B <strong>105</strong> (2022) 125109 [<a href="https://doi.org/10.1103/PhysRevB.105.125109">doi:10.1103/PhysRevB.105.125109</a>]</p> </li> </ul> <p id="ReferencesRealizationInExperiment"> Realization in <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a> (so far via <a class="existingWikiWord" href="/nlab/show/quantum+simulation">quantum simulation</a> of anyons on non-topological quantum hardware, cf. <a href="#FF24">FF24, Fig 5</a>, as in “topological codes” for <a class="existingWikiWord" href="/nlab/show/quantum+error+correction">quantum error correction</a>):</p> <p>on <a class="existingWikiWord" href="/nlab/show/superconductor">superconducting</a> <a class="existingWikiWord" href="/nlab/show/qbits">qbits</a>:</p> <ul> <li>T. Andersen et al.: <em>Non-Abelian braiding of graph vertices in a superconducting processor</em>, Nature <strong>618</strong> (2023) 264–269 [<a href="https://arxiv.org/abs/2210.10255">arXiv:2210.10255</a>, <a href="https://doi.org/10.1038/s41586-023-05954-4">doi:10.1038/s41586-023-05954-4</a>]</li> </ul> <p>on <a class="existingWikiWord" href="/nlab/show/trapped-ion+quantum+hardware">trapped-ion quantum hardware</a>:</p> <ul> <li> <p>Daniel Nigg, Markus Mueller, Esteban A. Martinez, Philipp Schindler, Markus Hennrich, Thomas Monz, Miguel A. Martin-Delgado, Rainer Blatt: <em>Experimental Quantum Computations on a Topologically Encoded Qubit</em>, Science 18 Jul 2014: Vol. 345, Issue 6194, pp. 302-305 (<a href="https://arxiv.org/abs/1403.5426">arXiv:1403.5426</a>, <a href="https://science.sciencemag.org/content/345/6194/302">doi:10.1126/science.1253742</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mohsin+Iqbal">Mohsin Iqbal</a>, <a class="existingWikiWord" href="/nlab/show/Nathanan+Tantivasadakarn">Nathanan Tantivasadakarn</a>: <em>Topological Order from Measurements and Feed-Forward on a Trapped Ion Quantum Computer</em>, Nature Communications Physics <strong>7</strong> (2024) 205 [<a href="https://doi.org/10.1038/s42005-024-01698-3">doi:10.1038/s42005-024-01698-3</a>, <a href="https://arxiv.org/abs/2302.01917">arXiv:2302.01917</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mohsin+Iqbal">Mohsin Iqbal</a>, <a class="existingWikiWord" href="/nlab/show/Nathanan+Tantivasadakarn">Nathanan Tantivasadakarn</a>, R. Verresen et al., Figure 5 in : <em>Non-Abelian topological order and anyons on a trapped-ion processor</em>, Nature <strong>626</strong> (2024) 505–511 [<a href="https://doi.org/10.1038/s41586-023-06934-4">doi:10.1038/s41586-023-06934-4</a>]</p> <p>Nature research briefing: <em>Topological matter created on a quantum chip produces quasiparticles with computing power</em> [<a href="https://doi.org/10.1038/d41586-023-04126-8">doi:10.1038/d41586-023-04126-8</a>]</p> </li> <li id="FF24"> <p>Michael Foss-Feig, Guido Pagano, Andrew C. Potter, Norman Y. Yao: <em>Progress in Trapped-Ion Quantum Simulation</em>, Annual Reviews of Condensed Matter Physics (2024) [<a href="https://arxiv.org/abs/2409.02990">arXiv:2409.02990</a> <a href="https://doi.org/10.1146/annurev-conmatphys-032822-045619">doi:10.1146/annurev-conmatphys-032822-045619</a>]</p> </li> </ul> <p>Discussion of anyon braid gates via <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/David+Jaz+Myers">David Jaz Myers</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <em><a class="existingWikiWord" href="/schreiber/show/Topological+Quantum+Gates+in+Homotopy+Type+Theory">Topological Quantum Gates in Homotopy Type Theory</a></em>, Comm. Math. Phys. <strong>405</strong> 172 (2024) [<a href="https://arxiv.org/abs/2303.02382">arXiv:2303.02382</a>, <a href="https://doi.org/10.1007/s00220-024-05020-8">doi:10.1007/s00220-024-05020-8</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/David+Jaz+Myers">David Jaz Myers</a>: <em>Topological Quantum Gates</em>, <a href="CQTS#MyersApr2024">talk at</a> <em><a href="CQTS#RunningHoTT2024">Running HoTT 2024</a></em>, <a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a>@NYUAD (April 2024) [video:<a href="https://cdnapisec.kaltura.com/html5/html5lib/v2.73.2/mwEmbedFrame.php/p/1674401/uiconf_id/23435151/entry_id/1_k89ytmg4?wid=_1674401&iframeembed=true&playerId=kaltura_player&entry_id=1_k89ytmg4">kt</a>]</p> </li> </ul> <div> <h3 id="braid_group_representations_as_topological_quantum_gates">Braid group representations (as topological quantum gates)</h3> <p>On <a class="existingWikiWord" href="/nlab/show/linear+representations">linear representations</a> of braid groups (see also at <em><a class="existingWikiWord" href="/nlab/show/braid+group+statistics">braid group statistics</a></em> and interpretation as <a class="existingWikiWord" href="/nlab/show/quantum+gates">quantum gates</a> in <em><a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a></em>):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ivan+Marin">Ivan Marin</a>, <em>On the representation theory of braid groups</em>, Annales mathématiques Blaise Pascal, <strong>20</strong> 2 (2013) 193-260 (<a href="https://arxiv.org/abs/math/0502118">arXiv:math/0502118</a>, <a href="https://eudml.org/doc/275607">dml:275607</a>)</li> </ul> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chen+Ning+Yang">Chen Ning Yang</a>, M. L. Ge (eds.). <em>Braid Group, Knot Theory and Statistical Mechanics</em>, Advanced Series in Mathematical Physics <strong>9</strong>, World Scientific (1991) <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/10.1142/0796">doi:10.1142/0796</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></p> <blockquote> <p>(focus on <a class="existingWikiWord" href="/nlab/show/quantum+Yang-Baxter+equation">quantum Yang-Baxter equation</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Camilo+Arias+Abad">Camilo Arias Abad</a>, <em>Introduction to representations of braid groups</em>, Rev. colomb. mat. vol.49 no.1 (2015) (<a href="https://arxiv.org/abs/1404.0724">arXiv:1404.0724</a>, <a href="https://doi.org/10.15446/recolma.v49n1.54160">doi:10.15446/recolma.v49n1.54160</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Toshitake+Kohno">Toshitake Kohno</a>, <em>Introduction to representation theory of braid groups</em>, Peking 2018 (<a href="https://www.math.pku.edu.cn/misc/puremath/summerschool/Peking_SummerSchool_kohno.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/KohnoBraidRepresentations.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>in relation to <a class="existingWikiWord" href="/nlab/show/modular+tensor+categories">modular tensor categories</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Colleen+Delaney">Colleen Delaney</a>, <em>Lecture notes on modular tensor categories and braid group representations</em>, 2019 (<a href="http://web.math.ucsb.edu/~cdelaney/MTC_Notes.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/DelaneyModularTensorCategories.pdf" title="pdf">pdf</a>)</li> </ul> <p>Braid representations from the <a class="existingWikiWord" href="/nlab/show/monodromy">monodromy</a> of the <a class="existingWikiWord" href="/nlab/show/Knizhnik-Zamolodchikov+connection">Knizhnik-Zamolodchikov connection</a> on bundles of <a class="existingWikiWord" href="/nlab/show/conformal+blocks">conformal blocks</a> over <a class="existingWikiWord" href="/nlab/show/configuration+spaces+of+points">configuration spaces of points</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ivan+Todorov">Ivan Todorov</a>, <a class="existingWikiWord" href="/nlab/show/Ludmil+Hadjiivanov">Ludmil Hadjiivanov</a>, <em>Monodromy Representations of the Braid Group</em>, Phys. Atom. Nucl. <strong>64</strong> (2001) 2059-2068; Yad.Fiz. <strong>64</strong> (2001) 2149-2158 <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://arxiv.org/abs/hep-th/0012099">arXiv:hep-th/0012099</a>, <a href="https://doi.org/10.1134/1.1432899">doi:10.1134/1.1432899</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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ivan+Marin">Ivan Marin</a>, <em>Sur les représentations de Krammer génériques</em>, Annales de l’Institut Fourier, <strong>57</strong> 6 (2007) 1883-1925 <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="http://www.numdam.org/item/AIF_2007__57_6_1883_0">numdam:AIF_2007__57_6_1883_0</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></p> </li> </ul> <p>and understood in terms of <a class="existingWikiWord" href="/nlab/show/anyon+statistics">anyon statistics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Xia+Gu">Xia Gu</a>, <a class="existingWikiWord" href="/nlab/show/Babak+Haghighat">Babak Haghighat</a>, <a class="existingWikiWord" href="/nlab/show/Yihua+Liu">Yihua Liu</a>, <em>Ising- and Fibonacci-Anyons from KZ-equations</em>, J. High Energ. Phys. <strong>2022</strong> 15 (2022) [<a href="https://doi.org/10.1007/JHEP09(2022)015">doi:10.1007/JHEP09(2022)015</a>, <a href="https://arxiv.org/abs/2112.07195">arXiv:2112.07195</a>]</li> </ul> <p>Braid representations seen inside the <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> of the <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a>‘s <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alejandro+Adem">Alejandro Adem</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+C.+Cohen">Daniel C. Cohen</a>, <a class="existingWikiWord" href="/nlab/show/Frederick+R.+Cohen">Frederick R. Cohen</a>, <em>On representations and K-theory of the braid groups</em>, Math. Ann. <strong>326</strong> (2003) 515-542 (<a href="https://arxiv.org/abs/math/0110138">arXiv:math/0110138</a>, <a href="https://doi.org/10.1007/s00208-003-0435-8">doi:10.1007/s00208-003-0435-8</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frederick+R.+Cohen">Frederick R. Cohen</a>, Section 3 of: <em>On braid groups, homotopy groups, and modular forms</em>, in: J.M. Bryden (ed.), <em>Advances in Topological Quantum Field Theory</em>, Kluwer 2004, 275–288 (<a href="https://link.springer.com/content/pdf/10.1007/978-1-4020-2772-7_11.pdf">pdf</a>)</p> </li> </ul> <p>See also:</p> <ul> <li>R. B. Zhang, <em>Braid group representations arising from quantum supergroups with arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> and link polynomials</em>, Journal of Mathematical Physics 33, 3918 (1992) (<a href="https://doi.org/10.1063/1.529840">doi:10.1063/1.529840</a>)</li> </ul> <p>As <a class="existingWikiWord" href="/nlab/show/quantum+gates">quantum gates</a> for <a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a> with <a class="existingWikiWord" href="/nlab/show/anyons">anyons</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Louis+H.+Kauffman">Louis H. Kauffman</a>, <a class="existingWikiWord" href="/nlab/show/Samuel+J.+Lomonaco">Samuel J. Lomonaco</a>, <em>Braiding Operators are Universal Quantum Gates</em>, New Journal of Physics, Volume 6, January 2004 (<a href="https://arxiv.org/abs/quant-ph/0401090">arXiv:quant-ph/0401090</a>, <a href="https://iopscience.iop.org/article/10.1088/1367-2630/6/1/134">doi:10.1088/1367-2630/6/1/134</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Samuel+J.+Lomonaco">Samuel J. Lomonaco</a>, <a class="existingWikiWord" href="/nlab/show/Louis+Kauffman">Louis Kauffman</a>, <em>Topological Quantum Computing and the Jones Polynomial</em>, Proc. SPIE 6244, Quantum Information and Computation IV, 62440Z (2006) (<a href="https://arxiv.org/abs/quant-ph/0605004">arXiv:quant-ph/0605004</a>)</p> <blockquote> <p>(braid group representation serving as a <a class="existingWikiWord" href="/nlab/show/topological+quantum+computer">topological</a> <a class="existingWikiWord" href="/nlab/show/quantum+gate">quantum gate</a> to compute the <a class="existingWikiWord" href="/nlab/show/Jones+polynomial">Jones polynomial</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Louis+H.+Kauffman">Louis H. Kauffman</a>, <a class="existingWikiWord" href="/nlab/show/Samuel+J.+Lomonaco">Samuel J. Lomonaco</a>, <em>Topological quantum computing and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(2)</annotation></semantics></math> braid group representations</em>, Proceedings Volume 6976, Quantum Information and Computation VI; 69760M (2008) (<a href="https://doi.org/10.1117/12.778068">doi:10.1117/12.778068</a>, <a href="https://www.researchgate.net/publication/228451452">rg:228451452</a>)</p> </li> <li> <p>C.-L. Ho, A.I. Solomon, C.-H.Oh, <em>Quantum entanglement, unitary braid representation and Temperley-Lieb algebra</em>, EPL 92 (2010) 30002 (<a href="https://arxiv.org/abs/1011.6229">arXiv:1011.6229</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Louis+H.+Kauffman">Louis H. Kauffman</a>, <em>Majorana Fermions and Representations of the Braid Group</em>, International Journal of Modern Physics AVol. 33, No. 23, 1830023 (2018) (<a href="https://arxiv.org/abs/1710.04650">arXiv:1710.04650</a>, <a href="https://doi.org/10.1142/S0217751X18300235">doi:10.1142/S0217751X18300235</a>)</p> </li> <li> <p>David Lovitz, <em>Universal Braiding Quantum Gates</em> &lbrack;<a href="https://arxiv.org/abs/2304.00710">arXiv:2304.00710</a>&rbrack;</p> </li> </ul> <p>Introduction and review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Colleen+Delaney">Colleen Delaney</a>, <a class="existingWikiWord" href="/nlab/show/Eric+C.+Rowell">Eric C. Rowell</a>, <a class="existingWikiWord" href="/nlab/show/Zhenghan+Wang">Zhenghan Wang</a>, <em>Local unitary representations of the braid group and their applications to quantum computing</em>, Revista Colombiana de Matemáticas(2017), 50 (2):211 (<a href="https://arxiv.org/abs/1604.06429">arXiv:1604.06429</a>, <a href="http://dx.doi.org/10.15446/recolma.v50n2.62211">doi:10.15446/recolma.v50n2.62211</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+C.+Rowell">Eric C. Rowell</a>, <em>Braids, Motions and Topological Quantum Computing</em> <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://arxiv.org/abs/2208.11762">arXiv:2208.11762</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></p> </li> </ul> <p>Realization of <a class="existingWikiWord" href="/nlab/show/Fibonacci+anyons">Fibonacci anyons</a> on <a class="existingWikiWord" href="/nlab/show/quasicrystal">quasicrystal</a>-states:</p> <ul> <li>Marcelo Amaral, <a class="existingWikiWord" href="/nlab/show/David+Chester">David Chester</a>, Fang Fang, Klee Irwin, <em>Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing</em>, Symmetry <strong>14</strong> 9 (2022) 1780 <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://arxiv.org/abs/2207.08928">arXiv:2207.08928</a>, <a href="https://doi.org/10.3390/sym14091780">doi:10.3390/sym14091780</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>Realization on <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetric</a> <a class="existingWikiWord" href="/nlab/show/spin+chains">spin chains</a>:</p> <ul> <li>Indrajit Jana, Filippo Montorsi, Pramod Padmanabhan, Diego Trancanelli, <em>Topological Quantum Computation on Supersymmetric Spin Chains</em> <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://arxiv.org/abs/2209.03822">arXiv:2209.03822</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> </div> <p>See also:</p> <ul> <li>Yuanjie Ren, <a class="existingWikiWord" href="/nlab/show/Peter+Shor">Peter Shor</a>, <em>Topological quantum computation assisted by phase transitions</em> [<a href="https://arxiv.org/abs/2311.00103">arXiv:2311.00103</a>]</li> </ul> <p><br /></p> <h3 id="CompilationToBraidGates">Compilation to braid gate circuits</h3> <p>On approximating (cf. the <a class="existingWikiWord" href="/nlab/show/Solovay-Kitaev+theorem">Solovay-Kitaev theorem</a>) given <a class="existingWikiWord" href="/nlab/show/quantum+gates">quantum gates</a> by (i.e. compiling them to) <a class="existingWikiWord" href="/nlab/show/quantum+circuit">cicuits</a> of <a class="existingWikiWord" href="/nlab/show/anyon">anyon</a> <a class="existingWikiWord" href="/nlab/show/braid+representation">braid</a> <a class="existingWikiWord" href="/nlab/show/quantum+gates">gates</a> (generally considered for <a class="existingWikiWord" href="/nlab/show/su%282%29-anyons">su(2)-anyons</a> and here mostly for universal <a class="existingWikiWord" href="/nlab/show/Fibonacci+anyons">Fibonacci anyons</a>, to some extent also for non-universal <a class="existingWikiWord" href="/nlab/show/Majorana+anyons">Majorana anyons</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Nicholas+E.+Bonesteel">Nicholas E. Bonesteel</a>, <a class="existingWikiWord" href="/nlab/show/Layla+Hormozi">Layla Hormozi</a>, <a class="existingWikiWord" href="/nlab/show/Georgios+Zikos">Georgios Zikos</a>, <a class="existingWikiWord" href="/nlab/show/Steven+H.+Simon">Steven H. Simon</a>, <em>Braid Topologies for Quantum Computation</em>, Phys. Rev. Lett. <strong>95</strong> 140503 (2005) [<a href="https://doi.org/10.1103/PhysRevLett.95.140503">doi:10.1103/PhysRevLett.95.140503</a>, <a href="https://arxiv.org/abs/quant-ph/0505065">arXiv:quant-ph/0505065</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Layla+Hormozi">Layla Hormozi</a>, <a class="existingWikiWord" href="/nlab/show/Georgios+Zikos">Georgios Zikos</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+E.+Bonesteel">Nicholas E. Bonesteel</a>, <a class="existingWikiWord" href="/nlab/show/Steven+H.+Simon">Steven H. Simon</a>, <em>Topological Quantum Compiling</em>, Phys. Rev. B <strong>75</strong> 165310 (2007) [<a href="https://doi.org/10.1103/PhysRevB.75.165310">doi:10.1103/PhysRevB.75.165310</a>, <a href="https://arxiv.org/abs/quant-ph/0610111">arXiv:quant-ph/0610111</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Layla+Hormozi">Layla Hormozi</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+E.+Bonesteel">Nicholas E. Bonesteel</a>, <a class="existingWikiWord" href="/nlab/show/Steven+H.+Simon">Steven H. Simon</a>, <em>Topological Quantum Computing with Read-Rezayi States</em>, Phys. Rev. Lett. <strong>103</strong> 160501 (2009) [<a href="https://doi.org/10.1103/PhysRevLett.103.160501">doi:10.1103/PhysRevLett.103.160501</a>, <a href="https://arxiv.org/abs/0903.2239">arXiv:0903.2239</a>]</p> </li> <li> <p>M. Baraban, <a class="existingWikiWord" href="/nlab/show/Nicholas+E.+Bonesteel">Nicholas E. Bonesteel</a>, <a class="existingWikiWord" href="/nlab/show/Steven+H.+Simon">Steven H. Simon</a>, <em>Resources required for topological quantum factoring</em>, Phys. Rev. A <strong>81</strong> 062317 (2010) [<a href="https://doi.org/10.1103/PhysRevA.81.062317">doi:10.1103/PhysRevA.81.062317</a>, <a href="https://arxiv.org/abs/1002.0537">arXiv:1002.0537</a>]</p> <blockquote> <p>(focus on compiling <a class="existingWikiWord" href="/nlab/show/Shor%27s+algorithm">Shor's algorithm</a>)</p> </blockquote> </li> <li> <p>Vadym Kliuchnikov, Alex Bocharov, Krysta M. Svore, <em>Asymptotically Optimal Topological Quantum Compiling</em>, Phys. Rev. Lett. <strong>112</strong> 140504 (2014) [<a href="https://doi.org/10.1103/PhysRevLett.112.140504">doi:10.1103/PhysRevLett.112.140504</a>, <a href="https://arxiv.org/abs/1310.4150">arXiv:1310.4150</a>, talk recording: <a href="https://doi.org/10.48660/13100129">doi:10.48660/13100129</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Joren+W.+Brunekreef">Joren W. Brunekreef</a>, <em>Topological Quantum Computation and Quantum Compilation</em>, Utrecht (2014) [<a href="https://studenttheses.uu.nl/handle/20.500.12932/17738">hdl:20.500.12932/17738</a>]</p> </li> <li> <p>Yuan-Hang Zhang, Pei-Lin Zheng, Yi Zhang, Dong-Ling Deng, <em>Topological Quantum Compiling with Reinforcement Learning</em>, Phys. Rev. Lett. <strong>125</strong> 170501 (2020) [<a href="https://doi.org/10.1103/PhysRevLett.125.170501">doi:10.1103/PhysRevLett.125.170501</a>, <a href="https://arxiv.org/abs/2004.04743">arXiv:2004.04743</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Emil+G%C3%A9netay-Johansen">Emil Génetay-Johansen</a>, <a class="existingWikiWord" href="/nlab/show/Tapio+Simula">Tapio Simula</a>, Section IV of: <em>Fibonacci anyons versus Majorana fermions – A Monte Carlo Approach to the Compilation of Braid Circuits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mo stretchy="false">)</mo> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">SU(2)_k</annotation></semantics></math> Anyon Models</em>, PRX Quantum <strong>2</strong> 010334 (2021) [<a href="https://doi.org/10.1103/PRXQuantum.2.010334">doi:10.1103/PRXQuantum.2.010334</a>, <a href="https://arxiv.org/abs/2008.10790">arXiv:2008.10790</a>]</p> </li> <li> <p>Cheng-Qian Xu, D. L. Zhou, <em>Quantum teleportation using Ising anyons</em>, Phys. Rev. A <strong>106</strong> 012413 (2022) [<a href="https://doi.org/10.1103/PhysRevA.106.012413">doi:10.1103/PhysRevA.106.012413</a>, <a href="https://arxiv.org/abs/2201.11923">arXiv:2201.11923</a>]</p> <blockquote> <p>(focus on implemening the <a class="existingWikiWord" href="/nlab/show/quantum+teleportation">quantum teleportation</a>-protocol with <a class="existingWikiWord" href="/nlab/show/Ising+anyons">Ising anyons</a>)</p> </blockquote> </li> </ul> <p>Approximating all topological quantum gates by just the <strong>weaves</strong> among all <a class="existingWikiWord" href="/nlab/show/braids">braids</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Steven+H.+Simon">Steven H. Simon</a>, <a class="existingWikiWord" href="/nlab/show/Nick+E.+Bonesteel">Nick E. Bonesteel</a>, <a class="existingWikiWord" href="/nlab/show/Michael+H.+Freedman">Michael H. Freedman</a>, N. Petrovic, <a class="existingWikiWord" href="/nlab/show/Layla+Hormozi">Layla Hormozi</a>, <em>Topological Quantum Computing with Only One Mobile Quasiparticle</em>, Phys. Rev. Lett. 96 (2006) 070503 (<a href="https://arxiv.org/abs/quant-ph/0509175">arXiv:quant-ph/0509175</a>, <a href="https://doi.org/10.1103/PhysRevLett.96.070503">doi:10.1103/PhysRevLett.96.070503</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Layla+Hormozi">Layla Hormozi</a>, <a class="existingWikiWord" href="/nlab/show/Georgios+Zikos">Georgios Zikos</a>, <a class="existingWikiWord" href="/nlab/show/Nick+E.+Bonesteel">Nick E. Bonesteel</a>, <a class="existingWikiWord" href="/nlab/show/Steven+H.+Simon">Steven H. Simon</a>, <em>Topological quantum compiling</em>, Phys. Rev. B 75, 165310 (<a href="https://journals.aps.org/prb/abstract/10.1103/PhysRevB.75.165310">doi:10.1103/PhysRevB.75.165310</a>, <a href="https://arxiv.org/abs/quant-ph/0610111">arXiv:quant-ph/0610111</a>)</p> </li> <li id="Rouabah20"> <p>Mohamed Taha Rouabah, <em>Compiling single-qubit braiding gate for Fibonacci anyons topological quantum computation</em> (<a href="https://arxiv.org/abs/2008.03542">arXiv:2008.03542</a>)</p> </li> <li> <p><a href="#Simon23">Simon 2023, Sec. 11.4.1</a></p> </li> </ul> </div></body></html> </div> <div class="revisedby"> <p> Last revised on January 27, 2025 at 07:14:59. 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