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class="vector-toc-numb">2</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Abelian_anyons" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Abelian_anyons"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Abelian anyons</span> </div> </a> <button aria-controls="toc-Abelian_anyons-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Abelian anyons subsection</span> </button> <ul id="toc-Abelian_anyons-sublist" class="vector-toc-list"> <li id="toc-Topological_equivalence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topological_equivalence"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Topological equivalence</span> </div> </a> <ul id="toc-Topological_equivalence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Experiment" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Experiment"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Experiment</span> </div> </a> <ul id="toc-Experiment-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Non-abelian_anyons" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Non-abelian_anyons"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Non-abelian anyons</span> </div> </a> <ul id="toc-Non-abelian_anyons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fusion_of_anyons" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fusion_of_anyons"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Fusion of anyons</span> </div> </a> <ul id="toc-Fusion_of_anyons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topological_basis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Topological_basis"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Topological basis</span> </div> </a> <ul id="toc-Topological_basis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalization_to_higher_dimensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalization_to_higher_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Generalization to higher dimensions</span> </div> </a> <ul id="toc-Generalization_to_higher_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header 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</div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Anyon</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 13 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-13" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">13 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Anyon" title="Anyon – German" lang="de" hreflang="de" data-title="Anyon" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Any%C3%B3n" title="Anyón – Spanish" lang="es" hreflang="es" data-title="Anyón" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A2%D9%86%DB%8C%D9%88%D9%86_(%D8%B4%D8%A8%D9%87%E2%80%8C%D8%B0%D8%B1%D9%87)" title="آنیون (شبه‌ذره) – Persian" lang="fa" hreflang="fa" data-title="آنیون (شبه‌ذره)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Anyon" title="Anyon – French" lang="fr" hreflang="fr" data-title="Anyon" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%95%A0%EB%8B%88%EC%98%A8" title="애니온 – Korean" lang="ko" hreflang="ko" data-title="애니온" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Qualunquone" title="Qualunquone – Italian" lang="it" hreflang="it" data-title="Qualunquone" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A8%E3%83%8B%E3%82%AA%E3%83%B3" title="エニオン – Japanese" lang="ja" hreflang="ja" data-title="エニオン" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Anyon" title="Anyon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Anyon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A7%D9%86%DB%8C%D9%88%D9%86" title="انیون – Western Punjabi" lang="pnb" hreflang="pnb" data-title="انیون" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Anyon" title="Anyon – Polish" lang="pl" hreflang="pl" data-title="Anyon" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%AD%D0%BD%D0%B8%D0%BE%D0%BD" title="Энион – Russian" lang="ru" hreflang="ru" data-title="Энион" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%BD%D1%96%D0%BE%D0%BD" title="Еніон – Ukrainian" lang="uk" hreflang="uk" data-title="Еніон" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BB%BB%E6%84%8F%E5%AD%90" title="任意子 – Chinese" lang="zh" hreflang="zh" data-title="任意子" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q5528#sitelinks-wikipedia" title="Edit 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class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Anyon" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Anyon" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Anyon&oldid=1257855049" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li 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title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Type of two-dimensional quasiparticle</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Anyon_(disambiguation)" class="mw-disambig" title="Anyon (disambiguation)">Anyon (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Anion" class="mw-redirect" title="Anion">anion</a>, a negatively charged ion.</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Increasing_disorder.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Increasing_disorder.svg/220px-Increasing_disorder.svg.png" decoding="async" width="220" height="62" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Increasing_disorder.svg/330px-Increasing_disorder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Increasing_disorder.svg/440px-Increasing_disorder.svg.png 2x" data-file-width="464" data-file-height="131" /></a></span></td></tr><tr><td class="sidebar-content plainlist"> <ul><li><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Kinetic_theory_of_gases" title="Kinetic theory of gases">Kinetic theory</a></li></ul></td> </tr><tr><td class="sidebar-content plainlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><a href="/wiki/Particle_statistics" title="Particle statistics">Particle statistics</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Spin%E2%80%93statistics_theorem" title="Spin–statistics theorem">Spin–statistics theorem</a></li> <li><a href="/wiki/Indistinguishable_particles" title="Indistinguishable particles">Indistinguishable particles</a></li> <li><a href="/wiki/Maxwell%E2%80%93Boltzmann_statistics" title="Maxwell–Boltzmann statistics">Maxwell–Boltzmann</a></li> <li><a href="/wiki/Bose%E2%80%93Einstein_statistics" title="Bose–Einstein statistics">Bose–Einstein</a></li> <li><a href="/wiki/Fermi%E2%80%93Dirac_statistics" title="Fermi–Dirac statistics">Fermi–Dirac</a></li> <li><a href="/wiki/Parastatistics" title="Parastatistics">Parastatistics</a></li> <li><a class="mw-selflink selflink">Anyonic statistics</a></li> <li><a href="/wiki/Braid_statistics" title="Braid statistics">Braid statistics</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content plainlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><a href="/wiki/Statistical_ensemble_(mathematical_physics)" class="mw-redirect" title="Statistical ensemble (mathematical physics)">Thermodynamic ensembles</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><i>NVE</i> <a href="/wiki/Microcanonical_ensemble" title="Microcanonical ensemble">Microcanonical</a></li> <li><i>NVT</i> <a href="/wiki/Canonical_ensemble" title="Canonical ensemble">Canonical</a></li> <li><i>µVT</i> <a href="/wiki/Grand_canonical_ensemble" title="Grand canonical ensemble">Grand canonical</a></li> <li><i>NPH</i> <a href="/wiki/Isoenthalpic%E2%80%93isobaric_ensemble" title="Isoenthalpic–isobaric ensemble">Isoenthalpic–isobaric</a></li> <li><i>NPT</i> <a href="/wiki/Isothermal%E2%80%93isobaric_ensemble" title="Isothermal–isobaric ensemble">Isothermal–isobaric</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content plainlist"> <div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)">Models</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Debye_model" title="Debye model">Debye</a></li> <li><a href="/wiki/Einstein_solid" title="Einstein solid">Einstein</a></li> <li><a href="/wiki/Ising_model" title="Ising model">Ising</a></li> <li><a href="/wiki/Potts_model" title="Potts model">Potts</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content plainlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Internal_energy" title="Internal energy">Internal energy</a></li> <li><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a></li> <li><a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a></li> <li><a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a></li> <li><a href="/wiki/Grand_potential" title="Grand potential">Grand potential / Landau free energy</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content plainlist"> <div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li> <li><a href="/wiki/Richard_C._Tolman" title="Richard C. Tolman">Tolman</a></li> <li><a href="/wiki/Peter_Debye" title="Peter Debye">Debye</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/John_Lighton_Synge" title="John Lighton Synge">Synge</a></li> <li><a href="/wiki/Ernst_Ising" title="Ernst Ising">Ising</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Statistical_mechanics" title="Template:Statistical mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Statistical_mechanics" title="Template talk:Statistical mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Statistical_mechanics" title="Special:EditPage/Template:Statistical mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p> In <a href="/wiki/Physics" title="Physics">physics</a>, an <b>anyon</b> is a type of <a href="/wiki/Quasiparticle" title="Quasiparticle">quasiparticle</a> so far observed only in <a href="/wiki/Two-dimensional" class="mw-redirect" title="Two-dimensional">two-dimensional</a> <a href="/wiki/Physical_system" title="Physical system">systems</a>. In <a href="/wiki/Three-dimensional" class="mw-redirect" title="Three-dimensional">three-dimensional</a> systems, only two kinds of <a href="/wiki/Elementary_particle" title="Elementary particle">elementary particles</a> are seen: <a href="/wiki/Fermion" title="Fermion">fermions</a> and <a href="/wiki/Boson" title="Boson">bosons</a>. Anyons have statistical properties intermediate between fermions and bosons.<sup id="cite_ref-Bartolomei2020_1-0" class="reference"><a href="#cite_note-Bartolomei2020-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In general, the operation of <a href="/wiki/Exchange_symmetry" class="mw-redirect" title="Exchange symmetry">exchanging two identical particles</a>, although it may cause a global phase shift, cannot affect <a href="/wiki/Observables" class="mw-redirect" title="Observables">observables</a>. Anyons are generally classified as <i>abelian</i> or <i>non-abelian</i>. Abelian anyons, detected by two experiments in 2020,<sup id="cite_ref-DISC-20201212_2-0" class="reference"><a href="#cite_note-DISC-20201212-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> play a major role in the <a href="/wiki/Fractional_quantum_Hall_effect" title="Fractional quantum Hall effect">fractional quantum Hall effect</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anyon&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a> of large many-body systems obeys laws described by <a href="/wiki/Maxwell%E2%80%93Boltzmann_statistics" title="Maxwell–Boltzmann statistics">Maxwell–Boltzmann statistics</a>. <a href="/wiki/Quantum_statistics" class="mw-redirect" title="Quantum statistics">Quantum statistics</a> is more complicated because of the different behaviors of two different kinds of particles called <a href="/wiki/Fermions" class="mw-redirect" title="Fermions">fermions</a> and <a href="/wiki/Bosons" class="mw-redirect" title="Bosons">bosons</a>. In two-dimensional systems, however, there is a third type of particle, called an anyon. </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>In the three-dimensional world we live in, there are only two types of particles: "fermions", which repel each other, and "bosons", which like to stick together. A commonly known fermion is the electron, which transports electricity; and a commonly known boson is the photon, which carries light. In the two-dimensional world, however, there is another type of particle, the anyon, which doesn't behave like either a fermion or a boson. </p><div class="templatequotecite">— <cite>"Finally, anyons reveal their exotic quantum properties", Aalto University press release, April 2020<sup id="cite_ref-ExoticQP_3-0" class="reference"><a href="#cite_note-ExoticQP-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></cite></div></blockquote> <p>In a two-dimensional world, two identical anyons change their wavefunction when they swap places in ways that cannot happen in three-dimensional physics: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>...in two dimensions, exchanging identical particles twice is not equivalent to leaving them alone. The particles' wavefunction after swapping places twice may differ from the original one; particles with such unusual exchange statistics are known as anyons. By contrast, in three dimensions, exchanging particles twice cannot change their wavefunction, leaving us with only two possibilities: bosons, whose wavefunction remains the same even after a single exchange, and fermions, whose exchange only changes the sign of their wavefunction. </p><div class="templatequotecite">— <cite>Kirill Shtengel, "A home for anyon?", Nature Physics<sup id="cite_ref-NatPhys2007_4-0" class="reference"><a href="#cite_note-NatPhys2007-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></cite></div></blockquote> <p>This process of exchanging identical particles, or of circling one particle around another, is referred to as "<a href="/wiki/Braid_group#Formal_treatment" title="Braid group">braiding</a>". Braiding two anyons creates a historical record of the event, as their changed wave functions record the number of braids.<sup id="cite_ref-Yirka2020_5-0" class="reference"><a href="#cite_note-Yirka2020-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Microsoft" title="Microsoft">Microsoft</a> has invested in research concerning anyons as a potential basis for <a href="/wiki/Topological_quantum_computing" class="mw-redirect" title="Topological quantum computing">topological quantum computing</a>.<sup id="cite_ref-:0_6-0" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> They may be useful in quantum computing as a form of memory.<sup id="cite_ref-:0_6-1" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Anyons circling each other ("braiding") would encode information in a more robust way than other potential <a href="/wiki/Quantum_computing" title="Quantum computing">quantum computing</a> technologies.<sup id="cite_ref-Welcome2020_7-0" class="reference"><a href="#cite_note-Welcome2020-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Most investment in quantum computing, however, is based on methods that do not use anyons.<sup id="cite_ref-Welcome2020_7-1" class="reference"><a href="#cite_note-Welcome2020-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anyon&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>Like so many deep ideas in physics, the topological underpinnings of anyons can be traced back to <a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a>.</p><div class="templatequotecite">— <cite>Biedenharn et al., The Ancestry of the 'Anyon'<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></cite></div></blockquote> <p>In 1977, two <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physicists</a> working at the <a href="/wiki/University_of_Oslo" title="University of Oslo">University of Oslo</a>, <a href="/wiki/Jon_Magne_Leinaas" title="Jon Magne Leinaas">Jon Magne Leinaas</a> and <a href="/wiki/Jan_Myrheim" title="Jan Myrheim">Jan Myrheim</a>, showed that the traditional classification of particles as either fermions or bosons would not apply if they were restricted to move in only two <a href="/wiki/Dimension" title="Dimension">dimensions</a>.<sup id="cite_ref-WilczekJan2006_9-0" class="reference"><a href="#cite_note-WilczekJan2006-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Hypothetical particles, being neither bosons nor fermions, would be expected to exhibit a diverse range of previously unexpected properties. In 1982, <a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Frank Wilczek</a> published two papers exploring the fractional statistics of quasiparticles in two dimensions, giving them the name "anyons" to indicate that the phase shift upon permutation can take any value.<sup id="cite_ref-SymMag2011_10-0" class="reference"><a href="#cite_note-SymMag2011-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Daniel_Tsui" class="mw-redirect" title="Daniel Tsui">Daniel Tsui</a> and <a href="/wiki/Horst_St%C3%B6rmer" class="mw-redirect" title="Horst Störmer">Horst Störmer</a> discovered the fractional quantum Hall effect in 1982. The mathematics developed by Wilczek proved to be useful to <a href="/wiki/Bertrand_Halperin" title="Bertrand Halperin">Bertrand Halperin</a> at <a href="/wiki/Harvard_University" title="Harvard University">Harvard University</a> in explaining aspects of it.<sup id="cite_ref-Halp1984_11-0" class="reference"><a href="#cite_note-Halp1984-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Frank Wilczek, Dan Arovas, and <a href="/wiki/Robert_Schrieffer" class="mw-redirect" title="Robert Schrieffer">Robert Schrieffer</a> verified this statement in 1985 with an explicit calculation that predicted that particles existing in these systems are in fact anyons.<sup id="cite_ref-1989Khurana_12-0" class="reference"><a href="#cite_note-1989Khurana-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Abelian_anyons">Abelian anyons</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anyon&action=edit&section=3" title="Edit section: Abelian anyons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In quantum mechanics, and some classical stochastic systems, <a href="/wiki/Indistinguishable_particles" title="Indistinguishable particles">indistinguishable particles</a> have the property that exchanging the states of particle <span class="texhtml"><i>i</i></span> with particle <span class="texhtml"><i>j</i></span> (symbolically <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{i}\leftrightarrow \psi _{j}{\text{ for }}i\neq j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">↔</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{i}\leftrightarrow \psi _{j}{\text{ for }}i\neq j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77059d2d8758cbf1d6b44b5a5a139b97398a2bdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.156ex; height:2.843ex;" alt="{\displaystyle \psi _{i}\leftrightarrow \psi _{j}{\text{ for }}i\neq j}"></span>⁠</span>) does not lead to a measurably different many-body state. </p><p>In a quantum mechanical system, for example, a system with two indistinguishable particles, with particle 1 in state <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cfdde1da54e02a016fe2a230c58b25dfcc014d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.509ex;" alt="{\displaystyle \psi _{1}}"></span>⁠</span> and particle 2 in state <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5083a526766f85c4f39ab695791b0b739f06897" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.509ex;" alt="{\displaystyle \psi _{2}}"></span>⁠</span>, has state <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\psi _{1}\psi _{2}\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\psi _{1}\psi _{2}\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f04e164f1a0e87e69c33973fbbbc469ec1ffe41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.686ex; height:2.843ex;" alt="{\displaystyle \left|\psi _{1}\psi _{2}\right\rangle }"></span>⁠</span> in <a href="/wiki/Dirac_notation" class="mw-redirect" title="Dirac notation">Dirac notation</a>. Now suppose we exchange the states of the two particles, then the state of the system would be <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\psi _{2}\psi _{1}\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\psi _{2}\psi _{1}\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cadeb33d99722633cc00e451a900ddc84a9db3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.686ex; height:2.843ex;" alt="{\displaystyle \left|\psi _{2}\psi _{1}\right\rangle }"></span>⁠</span>. These two states should not have a measurable difference, so they should be the same vector, up to a <a href="/wiki/Phase_factor" title="Phase factor">phase factor</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\psi _{1}\psi _{2}\right\rangle =e^{i\theta }\left|\psi _{2}\psi _{1}\right\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mrow> <mo>|</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\psi _{1}\psi _{2}\right\rangle =e^{i\theta }\left|\psi _{2}\psi _{1}\right\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7e6d1548197f399aa906095cddcf061f3277635" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.546ex; height:3.176ex;" alt="{\displaystyle \left|\psi _{1}\psi _{2}\right\rangle =e^{i\theta }\left|\psi _{2}\psi _{1}\right\rangle .}"></span></dd></dl> <p>Here, <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b4b8e67ee479d68e0e5040aaf87eff99214c90f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.654ex; height:2.676ex;" alt="{\displaystyle e^{i\theta }}"></span>⁠</span> is the phase factor. In space of <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three</a> or more dimensions, the phase factor is <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>⁠</span> or <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></span>⁠</span>. Thus, <a href="/wiki/Elementary_particles" class="mw-redirect" title="Elementary particles">elementary particles</a> are either fermions, whose phase factor is <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></span>⁠</span>, or bosons, whose phase factor is <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>⁠</span>. These two types have different <a href="/wiki/Quantum_statistics" class="mw-redirect" title="Quantum statistics">statistical behaviour</a>. Fermions obey <a href="/wiki/Fermi%E2%80%93Dirac_statistics" title="Fermi–Dirac statistics">Fermi–Dirac statistics</a>, while bosons obey <a href="/wiki/Bose%E2%80%93Einstein_statistics" title="Bose–Einstein statistics">Bose–Einstein statistics</a>. In particular, the phase factor is why fermions obey the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a>: If two fermions are in the same state, then we have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\psi \psi \right\rangle =-\left|\psi \psi \right\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mi>ψ</mi> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mi>ψ</mi> </mrow> <mo>⟩</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\psi \psi \right\rangle =-\left|\psi \psi \right\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b53349700ecef38dfbd70bef070d019a561918b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.483ex; height:2.843ex;" alt="{\displaystyle \left|\psi \psi \right\rangle =-\left|\psi \psi \right\rangle .}"></span></dd></dl> <p>The state vector must be zero, which means it is not normalizable, thus it is unphysical. </p><p>In two-dimensional systems, however, <a href="/wiki/Quasiparticle" title="Quasiparticle">quasiparticles</a> can be observed that obey statistics ranging continuously between Fermi–Dirac and Bose–Einstein statistics, as was first shown by <a href="/wiki/Jon_Magne_Leinaas" title="Jon Magne Leinaas">Jon Magne Leinaas</a> and <a href="/wiki/Jan_Myrheim" title="Jan Myrheim">Jan Myrheim</a> of the <a href="/wiki/University_of_Oslo" title="University of Oslo">University of Oslo</a> in 1977.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> In the case of two particles this can be expressed as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\psi _{1}\psi _{2}\right\rangle =e^{i\theta }\left|\psi _{2}\psi _{1}\right\rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mrow> <mo>|</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\psi _{1}\psi _{2}\right\rangle =e^{i\theta }\left|\psi _{2}\psi _{1}\right\rangle ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ded7d8985fcfd233d573f5bc7c06c3c76ff704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.546ex; height:3.176ex;" alt="{\displaystyle \left|\psi _{1}\psi _{2}\right\rangle =e^{i\theta }\left|\psi _{2}\psi _{1}\right\rangle ,}"></span></dd></dl> <p>where <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b4b8e67ee479d68e0e5040aaf87eff99214c90f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.654ex; height:2.676ex;" alt="{\displaystyle e^{i\theta }}"></span>⁠</span> can be other values than just <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></span>⁠</span> or <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>⁠</span>. It is important to note that there is a slight <a href="/wiki/Abuse_of_notation" title="Abuse of notation">abuse of notation</a> in this shorthand expression, as in reality this wave function can be and usually is multi-valued. This expression actually means that when particle 1 and particle 2 are interchanged in a process where each of them makes a counterclockwise half-revolution about the other, the two-particle system returns to its original quantum wave function except multiplied by the complex unit-norm phase factor <span class="nowrap"><i>e</i><sup><i>iθ</i></sup></span>. Conversely, a clockwise half-revolution results in multiplying the wave function by <span class="nowrap"><i>e</i><sup>−<i>iθ</i></sup></span>. Such a theory obviously only makes sense in two-dimensions, where clockwise and counterclockwise are clearly defined directions. </p><p>In the case <span class="nowrap"><i>θ</i> = <i>π</i></span> we recover the Fermi–Dirac statistics (<span class="nowrap"><i>e</i><sup><i>iπ</i></sup> = −1</span>) and in the case <span class="nowrap"><i>θ</i> = 0</span> (or <span class="nowrap"><i>θ</i> = 2<i>π</i></span>) the Bose–Einstein statistics (<span class="nowrap"><i>e</i><sup>2<i>πi</i></sup> = 1</span>). In between we have something different. <a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Frank Wilczek</a> in 1982 explored the behavior of such quasiparticles and coined the term "anyon" to describe them, because they can have any phase when particles are interchanged.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Unlike bosons and fermions, anyons have the peculiar property that when they are interchanged twice in the same way (e.g. if anyon 1 and anyon 2 were revolved counterclockwise by half revolution about each other to switch places, and then they were revolved counterclockwise by half revolution about each other again to go back to their original places), the wave function is not necessarily the same but rather generally multiplied by some complex phase (by <span class="nowrap"><i>e</i><sup>2<i>iθ</i></sup></span> in this example). </p><p>We may also use <span class="nowrap"><i>θ</i> = 2<i>πs</i></span> with particle <a href="/wiki/Spin_(physics)#Pauli_exclusion_principle" title="Spin (physics)">spin</a> quantum number <i>s</i>, with <i>s</i> being <a href="/wiki/Integer" title="Integer">integer</a> for bosons, <a href="/wiki/Half-integer" title="Half-integer">half-integer</a> for fermions, so that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\theta }=e^{2i\pi s}=(-1)^{2s},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mi>π</mi> <mi>s</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>s</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\theta }=e^{2i\pi s}=(-1)^{2s},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee8e89b1f1cf056acb9fdb98a9d9e8f041e7bd3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.521ex; height:3.176ex;" alt="{\displaystyle e^{i\theta }=e^{2i\pi s}=(-1)^{2s},}"></span></dd></dl> <p>or </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi _{1}\psi _{2}\rangle =(-1)^{2s}|\psi _{2}\psi _{1}\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>s</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi _{1}\psi _{2}\rangle =(-1)^{2s}|\psi _{2}\psi _{1}\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c3bfb2619eb972e718d3abf7d680ffef89a23ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.723ex; height:3.176ex;" alt="{\displaystyle |\psi _{1}\psi _{2}\rangle =(-1)^{2s}|\psi _{2}\psi _{1}\rangle .}"></span></dd></dl> <p>At an edge, <a href="/wiki/Fractional_quantum_Hall_effect" title="Fractional quantum Hall effect">fractional quantum Hall effect</a> anyons are confined to move in one space dimension. Mathematical models of one-dimensional anyons provide a base of the commutation relations shown above. </p><p>In a three-dimensional position space, the fermion and boson statistics operators (−1 and +1 respectively) are just 1-dimensional representations of the <a href="/wiki/Permutation_group" title="Permutation group">permutation group</a> (S<sub><i>N</i></sub> of <i>N</i> indistinguishable particles) acting on the space of wave functions. In the same way, in two-dimensional position space, the abelian anyonic statistics operators (<span class="nowrap"><i>e</i><sup><i>iθ</i></sup></span>) are just 1-dimensional representations of the <a href="/wiki/Braid_group#Representations" title="Braid group">braid group</a> (<i>B<sub>N</sub></i> of <i>N</i> indistinguishable particles) acting on the space of wave functions. Non-abelian anyonic statistics are higher-dimensional representations of the braid group. Anyonic statistics must not be confused with <a href="/wiki/Parastatistics" title="Parastatistics">parastatistics</a>, which describes statistics of particles whose wavefunctions are higher-dimensional representations of the permutation group.<sup id="cite_ref-Khare2005_16-0" class="reference"><a href="#cite_note-Khare2005-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 22">: 22 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Topological_equivalence">Topological equivalence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anyon&action=edit&section=4" title="Edit section: Topological equivalence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fact that the <a href="/wiki/Homotopy_class" class="mw-redirect" title="Homotopy class">homotopy classes</a> of paths (i.e. notion of <a href="/wiki/Equivalence_principle" title="Equivalence principle">equivalence</a> on <a href="/wiki/Braids" class="mw-redirect" title="Braids">braids</a>) are relevant hints at a more subtle insight. It arises from the <a href="/wiki/Feynman_path_integral" class="mw-redirect" title="Feynman path integral">Feynman path integral</a>, in which all paths from an initial to final point in <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> contribute with an appropriate <a href="/wiki/Phase_factor" title="Phase factor">phase factor</a>. The <a href="/wiki/Feynman_path_integral" class="mw-redirect" title="Feynman path integral">Feynman path integral</a> can be motivated from expanding the propagator using a method called time-slicing,<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> in which time is discretized. </p><p>In non-homotopic paths, one cannot get from any point at one time slice to any other point at the next time slice. This means that we can consider <a href="/wiki/Homotopic" class="mw-redirect" title="Homotopic">homotopic</a> equivalence class of paths to have different weighting factors.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>So it can be seen that the <a href="/wiki/Topological" class="mw-redirect" title="Topological">topological</a> notion of equivalence comes from a study of the <a href="/wiki/Feynman_path_integral" class="mw-redirect" title="Feynman path integral">Feynman path integral</a>.<sup id="cite_ref-Khare2005_16-1" class="reference"><a href="#cite_note-Khare2005-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 28">: 28 </span></sup> </p><p>For a more transparent way of seeing that the homotopic notion of equivalence is the "right" one to use, see <a href="/wiki/Aharonov%E2%80%93Bohm_effect" title="Aharonov–Bohm effect">Aharonov–Bohm effect</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Experiment">Experiment</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anyon&action=edit&section=5" title="Edit section: Experiment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Laughlin_quasiparticle_interferometer_sample_scanning_electron_micrograph.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Laughlin_quasiparticle_interferometer_sample_scanning_electron_micrograph.svg/350px-Laughlin_quasiparticle_interferometer_sample_scanning_electron_micrograph.svg.png" decoding="async" width="350" height="186" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Laughlin_quasiparticle_interferometer_sample_scanning_electron_micrograph.svg/525px-Laughlin_quasiparticle_interferometer_sample_scanning_electron_micrograph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Laughlin_quasiparticle_interferometer_sample_scanning_electron_micrograph.svg/700px-Laughlin_quasiparticle_interferometer_sample_scanning_electron_micrograph.svg.png 2x" data-file-width="281" data-file-height="149" /></a><figcaption><a href="/wiki/Robert_B._Laughlin" title="Robert B. Laughlin">Laughlin</a> quasiparticle interferometer <a href="/wiki/Scanning_electron_microscopy" class="mw-redirect" title="Scanning electron microscopy">scanning electron micrograph</a> of a <a href="/wiki/Semiconductor_device" title="Semiconductor device">semiconductor device</a>. The four light-grey regions are <a href="/wiki/Gold" title="Gold">Au</a>/<a href="/wiki/Titanium" title="Titanium">Ti</a> gates of un<a href="/wiki/Depletion_region" title="Depletion region">depleted</a> <a href="/wiki/Electron" title="Electron">electrons</a>; the blue curves are the edge channels from the <a href="/wiki/Equipotential" title="Equipotential">equipotentials</a> of these undepleted electrons. The dark-grey curves are etched trenches depleted of electrons, the blue dots are the <a href="/wiki/Tunnel_junction" title="Tunnel junction">tunneling junctions</a>, the yellow dots are <a href="/wiki/Ohmic_contact" title="Ohmic contact">Ohmic contacts</a>. The electrons in the device are confined to a 2d plane.<sup id="cite_ref-FractStat_19-0" class="reference"><a href="#cite_note-FractStat-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>In 2020, two teams of scientists (one in Paris, the other at Purdue) announced new experimental evidence for the existence of anyons. Both experiments were featured in <i><a href="/wiki/Discover_Magazine" class="mw-redirect" title="Discover Magazine">Discover Magazine</a></i><span class="nowrap" style="padding-left:0.1em;">'</span>s 2020 annual "state of science" issue.<sup id="cite_ref-DISC-20201212_2-1" class="reference"><a href="#cite_note-DISC-20201212-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In April, 2020, researchers from the <a href="/wiki/%C3%89cole_normale_sup%C3%A9rieure_(Paris)" title="École normale supérieure (Paris)">École normale supérieure (Paris)</a> and the <a href="/wiki/Centre_for_Nanosciences_and_Nanotechnologies" title="Centre for Nanosciences and Nanotechnologies">Centre for Nanosciences and Nanotechnologies</a> (C2N) reported results from a tiny "particle collider" for anyons. They detected properties that matched predictions by theory for anyons.<sup id="cite_ref-Bartolomei2020_1-1" class="reference"><a href="#cite_note-Bartolomei2020-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Paris_20-0" class="reference"><a href="#cite_note-Paris-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-QMag2020_21-0" class="reference"><a href="#cite_note-QMag2020-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>In July, 2020, scientists at Purdue University detected anyons using a different setup. The team's interferometer routes the electrons through a specific maze-like etched nanostructure made of <a href="/wiki/Gallium_arsenide" title="Gallium arsenide">gallium arsenide</a> and <a href="/wiki/Aluminium_gallium_arsenide" title="Aluminium gallium arsenide">aluminium gallium arsenide</a>. "In the case of our anyons the phase generated by braiding was 2π/3", he said. "That's different than what's been seen in nature before."<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Nakamura2020_23-0" class="reference"><a href="#cite_note-Nakamura2020-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>As of 2023, this remains an active area of research; using a superconducting processor, Google Quantum AI reported on the first braiding of non-Abelian anyon-like particles in an arXiv article by Andersen <i>et al.</i> in October 2022,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> later published in Nature.<sup id="cite_ref-non-abelian2_25-0" class="reference"><a href="#cite_note-non-abelian2-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> In an arXiv article released in May 2023, Quantinuum reported on non-abelian braiding using a trapped-ion processor.<sup id="cite_ref-non-abelian_26-0" class="reference"><a href="#cite_note-non-abelian-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Non-abelian_anyons">Non-abelian anyons</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anyon&action=edit&section=6" title="Edit section: Non-abelian anyons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> <div role="note" aria-labelledby="unsolved-label-physics" class="unsolved"> <div><span class="unsolved-label" id="unsolved-label-physics">Unsolved problem in physics</span>:</div> <div class="unsolved-body">Is <a href="/wiki/Topological_order" title="Topological order">topological order</a> stable at non-zero <a href="/wiki/Temperature" title="Temperature">temperature</a>?</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_physics" title="List of unsolved problems in physics">(more unsolved problems in physics)</a></div> </div> <p>In 1988, <a href="/wiki/J%C3%BCrg_Fr%C3%B6hlich" title="Jürg Fröhlich">Jürg Fröhlich</a> showed that it was valid under the <a href="/wiki/Spin%E2%80%93statistics_theorem" title="Spin–statistics theorem">spin–statistics theorem</a> for the particle exchange to be monoidal (non-abelian statistics).<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> In particular, this can be achieved when the system exhibits some degeneracy, so that multiple distinct states of the system have the same configuration of particles. Then an exchange of particles can contribute not just a phase change, but can send the system into a different state with the same particle configuration. Particle exchange then corresponds to a linear transformation on this subspace of degenerate states. When there is no degeneracy, this subspace is one-dimensional and so all such linear transformations commute (because they are just multiplications by a phase factor). When there is degeneracy and this subspace has higher dimension, then these linear transformations need not commute (just as matrix multiplication does not). </p><p><a href="/wiki/Greg_Moore_(physicist)" title="Greg Moore (physicist)">Gregory Moore</a>, <a href="/wiki/Nicholas_Read" title="Nicholas Read">Nicholas Read</a>, and <a href="/wiki/Xiao-Gang_Wen" title="Xiao-Gang Wen">Xiao-Gang Wen</a> pointed out that non-Abelian statistics can be realized in the <a href="/wiki/Fractional_quantum_Hall_effect" title="Fractional quantum Hall effect">fractional quantum Hall effect</a> (FQHE).<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> While at first non-abelian anyons were generally considered a mathematical curiosity, physicists began pushing toward their discovery when <a href="/wiki/Alexei_Kitaev" title="Alexei Kitaev">Alexei Kitaev</a> showed that non-abelian anyons could be used to construct a <a href="/wiki/Topological_quantum_computer" title="Topological quantum computer">topological quantum computer</a>. As of 2012, no experiment has conclusively demonstrated the existence of non-abelian anyons although promising hints are emerging in the study of the ν = 5/2 FQHE state.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers#Chronological_items" title="Wikipedia:Manual of Style/Dates and numbers"><span title="See new info in page introduction re: Google Quantum AI and Quantinuum 2022-2023 research. (May 2023)">needs update</span></a></i>]</sup><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Experimental evidence of non-abelian anyons, although not yet conclusive and currently contested,<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> was presented in October, 2013.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers#Chronological_items" title="Wikipedia:Manual of Style/Dates and numbers"><span title="The date of the event predicted near this tag has passed. (May 2023)">needs update</span></a></i>]</sup><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Recent works claim the creation of non-abelian topological order and anyons on a trapped-ion processor <sup id="cite_ref-non-abelian_26-1" class="reference"><a href="#cite_note-non-abelian-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> and demonstration of non-abelian braiding of graph vertices in a superconducting processor.<sup id="cite_ref-non-abelian2_25-1" class="reference"><a href="#cite_note-non-abelian2-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Fusion_of_anyons">Fusion of anyons</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anyon&action=edit&section=7" title="Edit section: Fusion of anyons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In much the same way that two fermions (e.g. both of spin 1/2) can be looked at together as a composite boson (with total spin in a <a href="/wiki/Clebsch%E2%80%93Gordan_coefficients" title="Clebsch–Gordan coefficients">superposition</a> of 0 and 1), two or more anyons together make up a composite anyon (possibly a boson or fermion). The composite anyon is said to be the result of the <a href="/wiki/Fusion_of_anyons" title="Fusion of anyons">fusion</a> of its components. </p><p>If <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span>⁠</span> identical abelian anyons each with individual statistics <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>⁠</span> (that is, the system picks up a phase <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dee801d5ef9c81a538f38ecb87dbdc586edae699" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.935ex; height:2.676ex;" alt="{\displaystyle e^{i\alpha }}"></span>⁠</span> when two individual anyons undergo adiabatic counterclockwise exchange) all fuse together, they together have statistics <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N^{2}\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N^{2}\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43b78ea61b6a39ec7328b6dcdd66a7a6a3b8892f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.665ex; height:2.676ex;" alt="{\displaystyle N^{2}\alpha }"></span>⁠</span>. This can be seen by noting that upon counterclockwise rotation of two composite anyons about each other, there are <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe131b76af8a2bc86e01b14a7ba843db69c1a164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.177ex; height:2.676ex;" alt="{\displaystyle N^{2}}"></span>⁠</span> pairs of individual anyons (one in the first composite anyon, one in the second composite anyon) that each contribute a phase <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dee801d5ef9c81a538f38ecb87dbdc586edae699" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.935ex; height:2.676ex;" alt="{\displaystyle e^{i\alpha }}"></span>⁠</span>. An analogous analysis applies to the fusion of non-identical abelian anyons. The statistics of the composite anyon is uniquely determined by the statistics of its components. </p><p>Non-abelian anyons have more complicated fusion relations. As a rule, in a system with non-abelian anyons, there is a composite particle whose statistics label is not uniquely determined by the statistics labels of its components, but rather exists as a quantum superposition (this is completely analogous to how two fermions known to have spin 1/2 are together in quantum superposition of total spin 1 and 0). If the overall statistics of the fusion of all of several anyons is known, there is still ambiguity in the fusion of some subsets of those anyons, and each possibility is a unique quantum state. These multiple states provide a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> on which quantum computation can be done.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Topological_basis">Topological basis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anyon&action=edit&section=8" title="Edit section: Topological basis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:308px;max-width:308px"><div class="trow"><div class="tsingle" style="width:152px;max-width:152px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Particle_exchange_2d_anticlockwise.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Particle_exchange_2d_anticlockwise.gif/150px-Particle_exchange_2d_anticlockwise.gif" decoding="async" width="150" height="296" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Particle_exchange_2d_anticlockwise.gif/225px-Particle_exchange_2d_anticlockwise.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/6/67/Particle_exchange_2d_anticlockwise.gif 2x" data-file-width="243" data-file-height="480" /></a></span></div><div class="thumbcaption">Anticlockwise rotation</div></div><div class="tsingle" style="width:152px;max-width:152px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Particle_exchange_2d_clockwise.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Particle_exchange_2d_clockwise.gif/150px-Particle_exchange_2d_clockwise.gif" decoding="async" width="150" height="296" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Particle_exchange_2d_clockwise.gif/225px-Particle_exchange_2d_clockwise.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/a/ae/Particle_exchange_2d_clockwise.gif 2x" data-file-width="243" data-file-height="480" /></a></span></div><div class="thumbcaption">Clockwise rotation</div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Exchange of two particles in 2+1 spacetime by rotation. The rotations are inequivalent, since one cannot be deformed into the other (without the worldlines leaving the plane, an impossibility in 2d space).</div></div></div></div> <p>In more than two dimensions, the <a href="/wiki/Spin%E2%80%93statistics_theorem" title="Spin–statistics theorem">spin–statistics theorem</a> states that any multiparticle state of <a href="/wiki/Indistinguishable_particles" title="Indistinguishable particles">indistinguishable particles</a> has to obey either Bose–Einstein or Fermi–Dirac statistics. For any <var>d</var> > 2, the <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> <a href="/wiki/Orthogonal_group" title="Orthogonal group">SO(<var>d</var>,1)</a> (which generalizes the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a>) and <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré(<var>d</var>,1)</a> have <b>Z</b><sub>2</sub> as their <a href="/wiki/First_homotopy_group" class="mw-redirect" title="First homotopy group">first homotopy group</a>. Because the <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> <b>Z</b><sub>2</sub> is composed of two elements, only two possibilities remain. (The details are more involved than that, but this is the crucial point.) </p><p>The situation changes in two dimensions. Here the first homotopy group of SO(2,1), and also Poincaré(2,1), is <b>Z</b> (infinite cyclic). This means that Spin(2,1) is not the <a href="/wiki/Universal_covering_group" class="mw-redirect" title="Universal covering group">universal cover</a>: it is not <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a>. In detail, there are <a href="/wiki/Projective_representation" title="Projective representation">projective representations</a> of the <a href="/wiki/Generalized_orthogonal_group" class="mw-redirect" title="Generalized orthogonal group">special orthogonal group</a> SO(2,1) which do not arise from <a href="/wiki/Linear_representation" class="mw-redirect" title="Linear representation">linear representations</a> of SO(2,1), or of its <a href="/wiki/Double_covering_group" class="mw-redirect" title="Double covering group">double cover</a>, the <a href="/wiki/Spin_group" title="Spin group">spin group</a> Spin(2,1). Anyons are evenly complementary representations of spin polarization by a charged particle. </p><p>This concept also applies to nonrelativistic systems. The relevant part here is that the spatial rotation group SO(2) has an infinite first homotopy group. </p><p>This fact is also related to the <a href="/wiki/Braid_group" title="Braid group">braid groups</a> well known in <a href="/wiki/Knot_theory" title="Knot theory">knot theory</a>. The relation can be understood when one considers the fact that in two dimensions the group of permutations of two particles is no longer the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> <i>S</i><sub>2</sub> (with two elements) but rather the braid group <i>B</i><sub>2</sub> (with an infinite number of elements). The essential point is that one braid can wind around the other one, an operation that can be performed infinitely often, and clockwise as well as counterclockwise. </p><p>A very different approach to the stability-decoherence problem in <a href="/wiki/Quantum_computer" class="mw-redirect" title="Quantum computer">quantum computing</a> is to create a <a href="/wiki/Topological_quantum_computer" title="Topological quantum computer">topological quantum computer</a> with anyons, quasi-particles used as threads and relying on <a href="/wiki/Braid_theory" class="mw-redirect" title="Braid theory">braid theory</a> to form stable <a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">quantum logic gates</a>.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Generalization_to_higher_dimensions">Generalization to higher dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anyon&action=edit&section=9" title="Edit section: Generalization to higher dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fractionalized excitations as point particles can be bosons, fermions or anyons in 2+1 spacetime dimensions. It is known that point particles can be only either bosons or fermions in 3+1 and higher spacetime dimensions. However, the loop- (or string-) or membrane-like excitations are extended objects that can have fractionalized statistics. </p><p>Current research shows that the loop- and string-like excitations exist for <a href="/wiki/Topological_order" title="Topological order">topological orders</a> in the 3+1 dimensional spacetime, and their multi-loop/string-braiding statistics are the key signatures for identifying 3+1‑dimensional topological orders.<sup id="cite_ref-1403.7437_37-0" class="reference"><a href="#cite_note-1403.7437-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-1404.7854_38-0" class="reference"><a href="#cite_note-1404.7854-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-1612.09298_39-0" class="reference"><a href="#cite_note-1612.09298-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> The multi-loop/string-braiding statistics of 3+1‑dimensional topological orders can be captured by the link invariants of particular <a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">topological quantum field theories</a> in 4 spacetime dimensions.<sup id="cite_ref-1612.09298_39-1" class="reference"><a href="#cite_note-1612.09298-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> Explained in a colloquial manner, the extended objects (loop, string, or membrane, etc.) can be potentially anyonic in 3+1 and higher spacetime dimensions in the long-range <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">entangled systems</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anyon&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/anyon" class="extiw" title="wiktionary:Special:Search/anyon">anyon</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><a href="/wiki/Anyonic_Lie_algebra" title="Anyonic Lie algebra">Anyonic Lie algebra</a> – Graded vector space equipped with a bilinear operator</li> <li><a href="/wiki/Flux_tube" title="Flux tube">Flux tube</a> – Tube-like region of space with constant magnet flux along its length</li> <li><a href="/wiki/Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau theory</a> – Superconductivity theory</li> <li><a href="/wiki/Husimi_Q_representation" title="Husimi Q representation">Husimi Q representation</a> – Computational physics simulation tool</li> <li><a href="/wiki/Josephson_effect" title="Josephson effect">Josephson effect</a> – Quantum physical phenomenon</li> <li><a href="/wiki/Macroscopic_quantum_phenomena" title="Macroscopic quantum phenomena">Macroscopic quantum phenomena</a> – Macroscopic processes showing quantum behavior</li> <li><a href="/wiki/Magnetic_domain" title="Magnetic domain">Magnetic domain</a> – Region of a magnetic material in which the magnetization has uniform direction</li> <li><a href="/wiki/Magnetic_flux_quantum" title="Magnetic flux quantum">Magnetic flux quantum</a> – Quantized unit of magnetic flux</li> <li><a href="/wiki/Meissner_effect" title="Meissner effect">Meissner effect</a> – Expulsion of a magnetic field from a superconductor</li> <li><a href="/wiki/Plekton" class="mw-redirect" title="Plekton">Plekton</a> – Possible statistical behavior of particles in quantum statistical mechanics<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Quantum_vortex" title="Quantum vortex">Quantum vortex</a> – Quantized flux circulation of some physical quantity</li> <li><a href="/wiki/Random_matrix" title="Random matrix">Random matrix</a> – Matrix-valued random variable</li> <li><a href="/wiki/Topological_defect" title="Topological defect">Topological defect</a> – Topologically stable solution of a partial differential equation.</li> <li><a href="/wiki/Topological_quantum_computer" title="Topological quantum computer">Topological quantum computing</a> – Hypothetical fault-tolerant quantum computer based on topological condensed matter</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anyon&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-Bartolomei2020-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Bartolomei2020_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Bartolomei2020_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBartolomei,_H.Kumar,_M.Bisognin,_R.2020" class="citation cs2">Bartolomei, H.; Kumar, M.; Bisognin, R.; et al. (10 April 2020), <a rel="nofollow" class="external text" href="https://doi.org/10.1126/science.aaz5601">"Fractional statistics in anyon collisions"</a>, <i><a href="/wiki/Science_(journal)" title="Science (journal)">Science</a></i>, <b>368</b> (6487): 173–177, <q>Elementary particles in three dimensions are either bosons or fermions, depending on their spin. In two dimensions, it is in principle possible to have particles that lie somewhere in between, but detecting the statistics of these so-called anyons directly is tricky.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Fractional+statistics+in+anyon+collisions&rft.volume=368&rft.issue=6487&rft.pages=173-177&rft.date=2020-04-10&rft.au=Bartolomei%2C+H.&rft.au=Kumar%2C+M.&rft.au=Bisognin%2C+R.&rft_id=https%3A%2F%2Fdoi.org%2F10.1126%2Fscience.aaz5601&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-DISC-20201212-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-DISC-20201212_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-DISC-20201212_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOrnes2020" class="citation news cs1">Ornes, Stephen (12 December 2020). <a rel="nofollow" class="external text" href="https://www.discovermagazine.com/the-sciences/physicists-prove-anyons-exist-a-third-type-of-particle-in-the-universe">"Physicists Prove Anyons Exist, a Third Type of Particle in the Universe – Physicists give us an early view of a third kingdom of quasiparticles that only arise in two dimensions"</a>. <i><a href="/wiki/Discover_(magazine)" title="Discover (magazine)">Discover</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">12 December</span> 2020</span>. <q>This year brought two solid confirmations of the quasiparticles. The first arrived in April, in a paper featured on the cover of <i>Science</i>, from a group of researchers at the École Normale Supérieure in Paris ... The second confirmation came in July, when a group at Purdue University in Indiana used an experimental setup on an etched chip that screened out interactions that might obscure the anyon behavior.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discover&rft.atitle=Physicists+Prove+Anyons+Exist%2C+a+Third+Type+of+Particle+in+the+Universe+%E2%80%93+Physicists+give+us+an+early+view+of+a+third+kingdom+of+quasiparticles+that+only+arise+in+two+dimensions.&rft.date=2020-12-12&rft.aulast=Ornes&rft.aufirst=Stephen&rft_id=https%3A%2F%2Fwww.discovermagazine.com%2Fthe-sciences%2Fphysicists-prove-anyons-exist-a-third-type-of-particle-in-the-universe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-ExoticQP-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-ExoticQP_3-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://sciencex.com/wire-news/347971706/finally-anyons-reveal-their-exotic-quantum-properties.html">"Finally, anyons reveal their exotic quantum properties"</a>. Aalto University. 7 December 2018<span class="reference-accessdate">. Retrieved <span class="nowrap">24 September</span> 2020</span>. <q>They were first proposed in the late 1970s, but direct experimental evidence of their quantum statistics hasn't been conclusively shown until now.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Finally%2C+anyons+reveal+their+exotic+quantum+properties&rft.pub=Aalto+University&rft.date=2018-12-07&rft_id=https%3A%2F%2Fsciencex.com%2Fwire-news%2F347971706%2Ffinally-anyons-reveal-their-exotic-quantum-properties.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-NatPhys2007-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-NatPhys2007_4-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShtengel2007" class="citation journal cs1">Shtengel, Kirilli (2007). <a rel="nofollow" class="external text" href="https://www.nature.com/articles/nphys767">"A home for anyon?"</a>. <i>Nature Physics</i>. <b>3</b> (11): 763. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnphys767">10.1038/nphys767</a></span><span class="reference-accessdate">. Retrieved <span class="nowrap">30 November</span> 2020</span>. <q>From a physicist's point of view, having two spatial dimensions is special: a pair of particles trading places behave very differently in two dimensions than they do in three. In three dimensions, any two sets of paths taken by two identical particles in the process of exchanging their positions can be continuously morphed into one another. But in two dimensions, particles can wind around each other in two distinct ways, clockwise or anticlockwise. A profound consequence of this observation for quantum mechanics is that in two dimensions, exchanging identical particles twice is not equivalent to leaving them alone.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Physics&rft.atitle=A+home+for+anyon%3F&rft.volume=3&rft.issue=11&rft.pages=763&rft.date=2007&rft_id=info%3Adoi%2F10.1038%2Fnphys767&rft.aulast=Shtengel&rft.aufirst=Kirilli&rft_id=https%3A%2F%2Fwww.nature.com%2Farticles%2Fnphys767&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-Yirka2020-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Yirka2020_5-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYirka2020" class="citation web cs1">Yirka, Bob (10 July 2020). <a rel="nofollow" class="external text" href="https://phys.org/news/2020-07-evidence-anyons.html">"Best evidence yet for existence of anyons"</a>. Phys.org News<span class="reference-accessdate">. Retrieved <span class="nowrap">30 November</span> 2020</span>. <q>If a fermion or a boson were dragged around another of its kind, theory suggests, the action would not produce a record of what had occurred. But because anyons alter wave functions, they would create such a record.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Best+evidence+yet+for+existence+of+anyons&rft.pub=Phys.org+News&rft.date=2020-07-10&rft.aulast=Yirka&rft.aufirst=Bob&rft_id=https%3A%2F%2Fphys.org%2Fnews%2F2020-07-evidence-anyons.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-:0-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilczek2021" class="citation book cs1">Wilczek, Frank (2021). <i>Fundamentals : Ten Keys to Reality</i>. New York, New York: Penguin Press. pp. 89–90. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780735223790" title="Special:BookSources/9780735223790"><bdi>9780735223790</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2020020086">2020020086</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+%3A+Ten+Keys+to+Reality&rft.place=New+York%2C+New+York&rft.pages=89-90&rft.pub=Penguin+Press&rft.date=2021&rft_id=info%3Alccn%2F2020020086&rft.isbn=9780735223790&rft.aulast=Wilczek&rft.aufirst=Frank&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-Welcome2020-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Welcome2020_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Welcome2020_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCastelvecchi2020" class="citation journal cs1">Castelvecchi, Davide (3 July 2020). <a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fd41586-020-01988-0">"Welcome anyons! Physicists find best evidence yet for long-sought 2D structures"</a>. <i>Nature</i>. <b>583</b> (7815): 176–177. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020Natur.583..176C">2020Natur.583..176C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fd41586-020-01988-0">10.1038/d41586-020-01988-0</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/32620884">32620884</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:220336025">220336025</a>. <q>Simon and others have developed elaborate theories that use anyons as the platform for quantum computers. Pairs of the quasiparticle could encode information in their memory of how they have circled around one another. And because the fractional statistics is 'topological' – it depends on the number of times one anyon went around another, and not on slight changes to its path – it is unaffected by tiny perturbations. This robustness could make topological quantum computers easier to scale up than are current quantum-computing technologies, which are error-prone.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Welcome+anyons%21+Physicists+find+best+evidence+yet+for+long-sought+2D+structures&rft.volume=583&rft.issue=7815&rft.pages=176-177&rft.date=2020-07-03&rft_id=info%3Adoi%2F10.1038%2Fd41586-020-01988-0&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A220336025%23id-name%3DS2CID&rft_id=info%3Apmid%2F32620884&rft_id=info%3Abibcode%2F2020Natur.583..176C&rft.aulast=Castelvecchi&rft.aufirst=Davide&rft_id=https%3A%2F%2Fdoi.org%2F10.1038%252Fd41586-020-01988-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBiedenharn,_L.Lieb,_E.Simon,_B.Wilczek,_F.1990" class="citation journal cs1"><a href="/wiki/Lawrence_Biedenharn" title="Lawrence Biedenharn">Biedenharn, L.</a>; <a href="/wiki/Elliott_H._Lieb" title="Elliott H. Lieb">Lieb, E.</a>; <a href="/wiki/Barry_Simon" title="Barry Simon">Simon, B.</a>; Wilczek, F. (August 1990). "The Ancestry of the Anyon". <i>Physics Today</i>. <b>43</b> (8): 90–91. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1990PhT....43h..90B">1990PhT....43h..90B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.2810672">10.1063/1.2810672</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=The+Ancestry+of+the+Anyon&rft.volume=43&rft.issue=8&rft.pages=90-91&rft.date=1990-08&rft_id=info%3Adoi%2F10.1063%2F1.2810672&rft_id=info%3Abibcode%2F1990PhT....43h..90B&rft.au=Biedenharn%2C+L.&rft.au=Lieb%2C+E.&rft.au=Simon%2C+B.&rft.au=Wilczek%2C+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-WilczekJan2006-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-WilczekJan2006_9-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilczek2006" class="citation journal cs1">Wilczek, Frank (January 2006). <a rel="nofollow" class="external text" href="https://physicsworld.com/a/from-electronics-to-anyonics/">"From electronics to anyonics"</a>. <i>Physics World</i>. <b>19</b>: 22–23. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F2058-7058%2F19%2F1%2F31">10.1088/2058-7058/19/1/31</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0953-8585">0953-8585</a>. <q>In the early 1980s I named the hypothetical new particles 'anyons', the idea being that anything goes – but I did not lose much sleep anticipating their discovery. Very soon afterwards, however, Bert Halperin at Harvard University found the concept of anyons useful in understanding certain aspects of the fractional quantum Hall effect, which describes the modifications that take place in electronics at low temperatures in strong magnetic fields.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+World&rft.atitle=From+electronics+to+anyonics&rft.volume=19&rft.pages=22-23&rft.date=2006-01&rft_id=info%3Adoi%2F10.1088%2F2058-7058%2F19%2F1%2F31&rft.issn=0953-8585&rft.aulast=Wilczek&rft.aufirst=Frank&rft_id=https%3A%2F%2Fphysicsworld.com%2Fa%2Ffrom-electronics-to-anyonics%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-SymMag2011-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-SymMag2011_10-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.symmetrymagazine.org/breaking/2011/08/31/anyons-anyone">"Anyons, anyone?"</a>. <i>Symmetry Magazine</i>. 31 August 2011<span class="reference-accessdate">. Retrieved <span class="nowrap">24 September</span> 2020</span>. <q>In 1982 physicist Frank Wilczek gave these interstitial particles the name anyon ... 'Any anyon can be anything between a boson or a fermion', Keilmann says. 'Wilczek is a funny guy.'<span class="cs1-kern-right"></span></q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Symmetry+Magazine&rft.atitle=Anyons%2C+anyone%3F&rft.date=2011-08-31&rft_id=https%3A%2F%2Fwww.symmetrymagazine.org%2Fbreaking%2F2011%2F08%2F31%2Fanyons-anyone&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-Halp1984-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Halp1984_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalperin1984" class="citation journal cs1">Halperin, B. I. (1984). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevLett.52.1583">"Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States"</a>. <i>Phys. Rev. Lett</i>. <b>52</b> (18). American Physical Society: 1583–1586. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1984PhRvL..52.1583H">1984PhRvL..52.1583H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.52.1583">10.1103/PhysRevLett.52.1583</a>. <q>The appearance of fractional statistics in the present context is strongly reminiscent of the fractional statistics introduced by Wilczek to describe charged particles tied to "magnetic flux tubes" in two dimensions.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.+Lett.&rft.atitle=Statistics+of+Quasiparticles+and+the+Hierarchy+of+Fractional+Quantized+Hall+States&rft.volume=52&rft.issue=18&rft.pages=1583-1586&rft.date=1984&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.52.1583&rft_id=info%3Abibcode%2F1984PhRvL..52.1583H&rft.aulast=Halperin&rft.aufirst=B.+I.&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevLett.52.1583&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-1989Khurana-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-1989Khurana_12-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKhurana2018" class="citation journal cs1">Khurana, Anil (7 December 2018). <a rel="nofollow" class="external text" href="https://doi.org/10.1063/1.2811205">"Bosons Condense and Fermions 'Exclude', But Anyons ...?"</a>. <i>Physics Today</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.2811205">10.1063/1.2811205</a><span class="reference-accessdate">. Retrieved <span class="nowrap">26 November</span> 2020</span>. <q>In 1984, two years after Wilczek discussed this seemingly arcane possibility, Bertrand Halperin (Harvard University) suggested that the excitations in the theory of fractional quantum Hall effect discussed by Robert Laughlin (Stanford University) behave like anyons. Later Wilczek, Daniel Arovas (University of California, San Diego) and Robert Schrieffer (University of California, Santa Barbara) confirmed the idea.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=Bosons+Condense+and+Fermions+%27Exclude%27%2C+But+Anyons+...%3F&rft.date=2018-12-07&rft_id=info%3Adoi%2F10.1063%2F1.2811205&rft.aulast=Khurana&rft.aufirst=Anil&rft_id=https%3A%2F%2Fdoi.org%2F10.1063%2F1.2811205&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArovasSchriefferWilczek1984" class="citation cs2">Arovas, D.; Schrieffer, J. R.; Wilczek, Frank (1984), <a rel="nofollow" class="external text" href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.53.722">"Fractional Statistics and the Quantum Hall Effect"</a>, <i>Physical Review Letters</i>, <b>53</b>: 722</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Fractional+Statistics+and+the+Quantum+Hall+Effect&rft.volume=53&rft.pages=722&rft.date=1984&rft.aulast=Arovas&rft.aufirst=D.&rft.au=Schrieffer%2C+J.+R.&rft.au=Wilczek%2C+Frank&rft_id=https%3A%2F%2Fjournals.aps.org%2Fprl%2Fabstract%2F10.1103%2FPhysRevLett.53.722&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeinaasMyrheim1977" class="citation journal cs1"><a href="/wiki/Jon_Magne_Leinaas" title="Jon Magne Leinaas">Leinaas, Jon Magne</a>; <a href="/wiki/Jan_Myrheim" title="Jan Myrheim">Myrheim, Jan</a> (11 January 1977). <a rel="nofollow" class="external text" href="http://www.ifi.unicamp.br/~cabrera/teaching/referencia.pdf">"On the theory of identical particles"</a> <span class="cs1-format">(PDF)</span>. <i>Il Nuovo Cimento B</i>. <b>37</b> (1): 1–23. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1977NCimB..37....1L">1977NCimB..37....1L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02727953">10.1007/BF02727953</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:117277704">117277704</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Il+Nuovo+Cimento+B&rft.atitle=On+the+theory+of+identical+particles&rft.volume=37&rft.issue=1&rft.pages=1-23&rft.date=1977-01-11&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A117277704%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF02727953&rft_id=info%3Abibcode%2F1977NCimB..37....1L&rft.aulast=Leinaas&rft.aufirst=Jon+Magne&rft.au=Myrheim%2C+Jan&rft_id=http%3A%2F%2Fwww.ifi.unicamp.br%2F~cabrera%2Fteaching%2Freferencia.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilczek1982" class="citation journal cs1"><a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Wilczek, Frank</a> (4 October 1982). <a rel="nofollow" class="external text" href="http://www.ifi.unicamp.br/~mtamash/f689_mecquant_i/prl49_957.pdf">"Quantum Mechanics of Fractional-Spin Particles"</a> <span class="cs1-format">(PDF)</span>. <i>Physical Review Letters</i>. <b>49</b> (14): 957–959. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1982PhRvL..49..957W">1982PhRvL..49..957W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.49.957">10.1103/PhysRevLett.49.957</a>. <q>If there is a generalized spin-statistics connection, we must expect that the flux-tube-particle composites have unusual statistics, interpolating between bosons and fermions. Since interchange of two of these particles can give <i>any</i> phase, I will call them generically anyons.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Quantum+Mechanics+of+Fractional-Spin+Particles&rft.volume=49&rft.issue=14&rft.pages=957-959&rft.date=1982-10-04&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.49.957&rft_id=info%3Abibcode%2F1982PhRvL..49..957W&rft.aulast=Wilczek&rft.aufirst=Frank&rft_id=http%3A%2F%2Fwww.ifi.unicamp.br%2F~mtamash%2Ff689_mecquant_i%2Fprl49_957.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-Khare2005-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-Khare2005_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Khare2005_16-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKhare2005" class="citation book cs1">Khare, Avinash (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rBi7zTpvjaAC&q=Fractional+Statistics+and+Quantum+Theory"><i>Fractional Statistics and Quantum Theory</i></a>. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-256-160-2" title="Special:BookSources/978-981-256-160-2"><bdi>978-981-256-160-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fractional+Statistics+and+Quantum+Theory&rft.pub=World+Scientific&rft.date=2005&rft.isbn=978-981-256-160-2&rft.aulast=Khare&rft.aufirst=Avinash&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrBi7zTpvjaAC%26q%3DFractional%2BStatistics%2Band%2BQuantum%2BTheory&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLancasterBlundell2014" class="citation book cs1">Lancaster, Tom; Blundell, Stephen J. (17 June 2014). <i>Quantum Field Theory for the Gifted Amateur</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-969932-2" title="Special:BookSources/978-0-19-969932-2"><bdi>978-0-19-969932-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Field+Theory+for+the+Gifted+Amateur&rft.pub=Oxford+University+Press&rft.date=2014-06-17&rft.isbn=978-0-19-969932-2&rft.aulast=Lancaster&rft.aufirst=Tom&rft.au=Blundell%2C+Stephen+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchulman1981" class="citation book cs1">Schulman, L. S. (February 1981). <i>Techniques and Applications of Path Integration</i>. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-76450-7" title="Special:BookSources/0-471-76450-7"><bdi>0-471-76450-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Techniques+and+Applications+of+Path+Integration&rft.pub=Dover+Publications&rft.date=1981-02&rft.isbn=0-471-76450-7&rft.aulast=Schulman&rft.aufirst=L.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-FractStat-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-FractStat_19-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCaminoZhouGoldman2005" class="citation journal cs1">Camino, Fernando E.; Zhou, Wei; Goldman, Vladimir J. (17 August 2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150619152736/http://quantum.physics.sunysb.edu/MyPapers/PDF/prbFracStat.pdf">"Realization of a Laughlin quasiparticle interferometer: Observation of fractional statistics"</a> <span class="cs1-format">(PDF)</span>. <i>Physical Review B</i>. <b>72</b> (7): 075342. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0502406">cond-mat/0502406</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005PhRvB..72g5342C">2005PhRvB..72g5342C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevB.72.075342">10.1103/PhysRevB.72.075342</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:52245802">52245802</a>. Archived from <a rel="nofollow" class="external text" href="http://quantum.physics.sunysb.edu/MyPapers/PDF/prbFracStat.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 19 June 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+B&rft.atitle=Realization+of+a+Laughlin+quasiparticle+interferometer%3A+Observation+of+fractional+statistics&rft.volume=72&rft.issue=7&rft.pages=075342&rft.date=2005-08-17&rft_id=info%3Aarxiv%2Fcond-mat%2F0502406&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A52245802%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevB.72.075342&rft_id=info%3Abibcode%2F2005PhRvB..72g5342C&rft.aulast=Camino&rft.aufirst=Fernando+E.&rft.au=Zhou%2C+Wei&rft.au=Goldman%2C+Vladimir+J.&rft_id=http%3A%2F%2Fquantum.physics.sunysb.edu%2FMyPapers%2FPDF%2FprbFracStat.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span>, see <i>fig. 2.B</i>.</span> </li> <li id="cite_note-Paris-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-Paris_20-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYirka2020" class="citation web cs1">Yirka, Bob (10 April 2020). <a rel="nofollow" class="external text" href="https://phys.org/news/2020-04-anyon-evidence-tiny-collider.html">"Anyon evidence observed using tiny anyon collider"</a>. Phys.org<span class="reference-accessdate">. Retrieved <span class="nowrap">12 December</span> 2020</span>. <q>The work involved creating a very tiny 2-D anyon collider—so small they had to use an electron microscope to observe the action inside of it. The collider consisted of a 2-D plane set between another layered material. More specifically, the collider held a quantum Hall liquid that was kept inside of a strong magnetic field.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Anyon+evidence+observed+using+tiny+anyon+collider&rft.pub=Phys.org&rft.date=2020-04-10&rft.aulast=Yirka&rft.aufirst=Bob&rft_id=https%3A%2F%2Fphys.org%2Fnews%2F2020-04-anyon-evidence-tiny-collider.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-QMag2020-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-QMag2020_21-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNajjar2020" class="citation web cs1">Najjar, Dana (12 May 2020). <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/milestone-evidence-for-anyons-a-third-kingdom-of-particles-20200512/">"<span class="cs1-kern-left"></span>'Milestone' Evidence for Anyons, a Third Kingdom of Particles"</a>. <i>Quanta Magazine</i><span class="reference-accessdate">. Retrieved <span class="nowrap">12 December</span> 2020</span>. <q>In 2016, three physicists described an experimental setup that resembles a tiny particle collider in two dimensions. Fève and his colleagues built something similar and used it to smash anyons together. By measuring the fluctuations of the currents in the collider, they were able to show that the behavior of the anyons corresponds exactly with theoretical predictions.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Quanta+Magazine&rft.atitle=%27Milestone%27+Evidence+for+Anyons%2C+a+Third+Kingdom+of+Particles&rft.date=2020-05-12&rft.aulast=Najjar&rft.aufirst=Dana&rft_id=https%3A%2F%2Fwww.quantamagazine.org%2Fmilestone-evidence-for-anyons-a-third-kingdom-of-particles-20200512%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTally2020" class="citation news cs1">Tally, Steve (4 September 2020). <a rel="nofollow" class="external text" href="https://phys.org/news/2020-09-evidence-quantum-world-stranger-thought.html">"New evidence that the quantum world is even stranger than we thought"</a>. Phys.org. <q>One characteristic difference between fermions and bosons is how the particles act when they are looped, or braided, around each other. Fermions respond in one straightforward way, and bosons in another expected and straightforward way. Anyons respond as if they have a fractional charge, and even more interestingly, create a nontrivial phase change as they braid around one another. This can give the anyons a type of "memory" of their interaction.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=New+evidence+that+the+quantum+world+is+even+stranger+than+we+thought&rft.date=2020-09-04&rft.aulast=Tally&rft.aufirst=Steve&rft_id=https%3A%2F%2Fphys.org%2Fnews%2F2020-09-evidence-quantum-world-stranger-thought.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-Nakamura2020-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-Nakamura2020_23-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNakamuraLiangGardnerManfra2020" class="citation journal cs1">Nakamura, J.; Liang, S.; Gardner, G. C.; Manfra, M. J. (September 2020). <a rel="nofollow" class="external text" href="https://www.nature.com/articles/s41567-020-1019-1">"Direct observation of anyonic braiding statistics"</a>. <i>Nature Physics</i>. <b>16</b> (9): 931–936. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2006.14115">2006.14115</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020NatPh..16..931N">2020NatPh..16..931N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41567-020-1019-1">10.1038/s41567-020-1019-1</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1745-2481">1745-2481</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:220055512">220055512</a>. <q>Anyons are quasiparticles that, unlike fermions and bosons, show fractional statistics when two of them are exchanged.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Physics&rft.atitle=Direct+observation+of+anyonic+braiding+statistics&rft.volume=16&rft.issue=9&rft.pages=931-936&rft.date=2020-09&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A220055512%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2020NatPh..16..931N&rft_id=info%3Aarxiv%2F2006.14115&rft.issn=1745-2481&rft_id=info%3Adoi%2F10.1038%2Fs41567-020-1019-1&rft.aulast=Nakamura&rft.aufirst=J.&rft.au=Liang%2C+S.&rft.au=Gardner%2C+G.+C.&rft.au=Manfra%2C+M.+J.&rft_id=https%3A%2F%2Fwww.nature.com%2Farticles%2Fs41567-020-1019-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndersenLenskyKechedzhiDrozdov2022" class="citation arxiv cs1">Andersen, Trond I.; et al. (19 October 2022). "Observation of non-Abelian exchange statistics on a superconducting processor". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2210.10255">2210.10255</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/quant-ph">quant-ph</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Observation+of+non-Abelian+exchange+statistics+on+a+superconducting+processor&rft.date=2022-10-19&rft_id=info%3Aarxiv%2F2210.10255&rft.aulast=Andersen&rft.aufirst=Trond+I.&rft.au=Lensky%2C+Yuri+D.&rft.au=Kechedzhi%2C+Kostyantyn&rft.au=Drozdov%2C+Ilya&rft.au=Bengtsson%2C+Andreas&rft.au=Hong%2C+Sabrina&rft.au=Morvan%2C+Alexis&rft.au=Mi%2C+Xiao&rft.au=Opremcak%2C+Alex&rft.au=Acharya%2C+Rajeev&rft.au=Allen%2C+Richard&rft.au=Ansmann%2C+Markus&rft.au=Arute%2C+Frank&rft.au=Arya%2C+Kunal&rft.au=Asfaw%2C+Abraham&rft.au=Atalaya%2C+Juan&rft.au=Babbush%2C+Ryan&rft.au=Bacon%2C+Dave&rft.au=Bardin%2C+Joseph+C.&rft.au=Bortoli%2C+Gina&rft.au=Bourassa%2C+Alexandre&rft.au=Bovaird%2C+Jenna&rft.au=Brill%2C+Leon&rft.au=Broughton%2C+Michael&rft.au=Buckley%2C+Bob+B.&rft.au=Buell%2C+David+A.&rft.au=Burger%2C+Tim&rft.au=Burkett%2C+Brian&rft.au=Bushnell%2C+Nicholas&rft.au=Chen%2C+Zijun&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-non-abelian2-25"><span class="mw-cite-backlink">^ <a href="#cite_ref-non-abelian2_25-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-non-abelian2_25-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.nature.com/articles/s41586-023-05954-4">"Non-Abelian braiding of graph vertices in a superconducting processor"</a>. Nature. 11 May 2023<span class="reference-accessdate">. Retrieved <span class="nowrap">17 May</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Non-Abelian+braiding+of+graph+vertices+in+a+superconducting+processor&rft.pub=Nature&rft.date=2023-05-11&rft_id=https%3A%2F%2Fwww.nature.com%2Farticles%2Fs41586-023-05954-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-non-abelian-26"><span class="mw-cite-backlink">^ <a href="#cite_ref-non-abelian_26-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-non-abelian_26-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIqbalTantivasadakarnVerresenCampbell2023" class="citation arxiv cs1">Iqbal, Mohsin; Tantivasadakarn, Nathanan; Verresen, Ruben; Campbell, Sara L.; Dreiling, Joan M.; Figgatt, Caroline; Gaebler, John P.; Johansen, Jacob; Mills, Michael; Moses, Steven A.; Pino, Juan M.; Ransford, Anthony; Rowe, Mary; Siegfried, Peter; Stutz, Russell P.; Foss-Feig, Michael; Vishwanath, Ashvin; Dreyer, Henrik (7 May 2023). "Creation of Non-Abelian Topological Order and Anyons on a Trapped-Ion Processors". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2305.03766">2305.03766</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/quant-ph">quant-ph</a>]. <q>Non-Abelian topological order (TO) is a coveted state of matter with remarkable properties, including quasiparticles that can remember the sequence in which they are exchanged. These anyonic excitations are promising building blocks of fault-tolerant quantum computers. However, despite extensive efforts, non-Abelian TO and its excitations have remained elusive, unlike the simpler quasiparticles or defects in Abelian TO. In this work, we present the first unambiguous realization of non-Abelian TO and demonstrate control of its anyons.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Creation+of+Non-Abelian+Topological+Order+and+Anyons+on+a+Trapped-Ion+Processors&rft.date=2023-05-07&rft_id=info%3Aarxiv%2F2305.03766&rft.aulast=Iqbal&rft.aufirst=Mohsin&rft.au=Tantivasadakarn%2C+Nathanan&rft.au=Verresen%2C+Ruben&rft.au=Campbell%2C+Sara+L.&rft.au=Dreiling%2C+Joan+M.&rft.au=Figgatt%2C+Caroline&rft.au=Gaebler%2C+John+P.&rft.au=Johansen%2C+Jacob&rft.au=Mills%2C+Michael&rft.au=Moses%2C+Steven+A.&rft.au=Pino%2C+Juan+M.&rft.au=Ransford%2C+Anthony&rft.au=Rowe%2C+Mary&rft.au=Siegfried%2C+Peter&rft.au=Stutz%2C+Russell+P.&rft.au=Foss-Feig%2C+Michael&rft.au=Vishwanath%2C+Ashvin&rft.au=Dreyer%2C+Henrik&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFröhlich1988" class="citation book cs1">Fröhlich, Jürg (1988). "Statistics of Fields, the Yang–Baxter Equation, and the Theory of Knots and Links". <i>Nonperturbative Quantum Field Theory</i>. NATO ASI Series. Vol. 185. New York: Springer. pp. 71–100. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4613-0729-7_4">10.1007/978-1-4613-0729-7_4</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4612-8053-2" title="Special:BookSources/1-4612-8053-2"><bdi>1-4612-8053-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Statistics+of+Fields%2C+the+Yang%E2%80%93Baxter+Equation%2C+and+the+Theory+of+Knots+and+Links&rft.btitle=Nonperturbative+Quantum+Field+Theory&rft.place=New+York&rft.series=NATO+ASI+Series&rft.pages=71-100&rft.pub=Springer&rft.date=1988&rft_id=info%3Adoi%2F10.1007%2F978-1-4613-0729-7_4&rft.isbn=1-4612-8053-2&rft.aulast=Fr%C3%B6hlich&rft.aufirst=J%C3%BCrg&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMooreRead1991" class="citation journal cs1"><a href="/wiki/Greg_Moore_(physicist)" title="Greg Moore (physicist)">Moore, Gregory</a>; <a href="/wiki/Nicholas_Read" title="Nicholas Read">Read, Nicholas</a> (19 August 1991). <a rel="nofollow" class="external text" href="http://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf">"Nonabelions in the fractional quantum hall effect"</a> <span class="cs1-format">(PDF)</span>. <i>Nuclear Physics B</i>. <b>360</b> (2–3): 362–396. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991NuPhB.360..362M">1991NuPhB.360..362M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0550-3213%2891%2990407-O">10.1016/0550-3213(91)90407-O</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nuclear+Physics+B&rft.atitle=Nonabelions+in+the+fractional+quantum+hall+effect&rft.volume=360&rft.issue=2%E2%80%933&rft.pages=362-396&rft.date=1991-08-19&rft_id=info%3Adoi%2F10.1016%2F0550-3213%2891%2990407-O&rft_id=info%3Abibcode%2F1991NuPhB.360..362M&rft.aulast=Moore&rft.aufirst=Gregory&rft.au=Read%2C+Nicholas&rft_id=http%3A%2F%2Fwww.physics.rutgers.edu%2F~gmoore%2FMooreReadNonabelions.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWen1991" class="citation journal cs1"><a href="/wiki/Xiao-Gang_Wen" title="Xiao-Gang Wen">Wen, Xiao-Gang</a> (11 February 1991). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150326181929/http://dao.mit.edu/~wen/pub/nab.pdf">"Non-Abelian statistics in the fractional quantum Hall states"</a> <span class="cs1-format">(PDF)</span>. <i>Physical Review Letters</i>. <b>66</b> (6): 802–805. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991PhRvL..66..802W">1991PhRvL..66..802W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.66.802">10.1103/PhysRevLett.66.802</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10043904">10043904</a>. 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"Non-Abelian states of matter". <i>Nature</i>. <b>464</b> (7286): 187–193. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010Natur.464..187S">2010Natur.464..187S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature08915">10.1038/nature08915</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/20220836">20220836</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4362827">4362827</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Non-Abelian+states+of+matter&rft.volume=464&rft.issue=7286&rft.pages=187-193&rft.date=2010&rft_id=info%3Adoi%2F10.1038%2Fnature08915&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4362827%23id-name%3DS2CID&rft_id=info%3Apmid%2F20220836&rft_id=info%3Abibcode%2F2010Natur.464..187S&rft.aulast=Stern&rft.aufirst=Ady&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnJiangChoiKang2011" class="citation arxiv cs1">An, Sanghun; Jiang, P.; Choi, H.; Kang, W.; Simon, S. H.; Pfeiffer, L. N.; West, K. W.; Baldwin, K. W. (15 December 2011). "Braiding of Abelian and Non-Abelian Anyons in the Fractional Quantum Hall Effect". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1112.3400">1112.3400</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cond-mat.mes-hall">cond-mat.mes-hall</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Braiding+of+Abelian+and+Non-Abelian+Anyons+in+the+Fractional+Quantum+Hall+Effect&rft.date=2011-12-15&rft_id=info%3Aarxiv%2F1112.3400&rft.aulast=An&rft.aufirst=Sanghun&rft.au=Jiang%2C+P.&rft.au=Choi%2C+H.&rft.au=Kang%2C+W.&rft.au=Simon%2C+S.+H.&rft.au=Pfeiffer%2C+L.+N.&rft.au=West%2C+K.+W.&rft.au=Baldwin%2C+K.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_KeyserlingSimonBernd2015" class="citation journal cs1">von Keyserling, Curt; Simon, S. H.; Bernd, Rosenow (2015). "Enhanced Bulk-Edge Coulomb Coupling in Fractional Fabry-Perot Interferometers". <i>Physical Review Letters</i>. <b>115</b> (12): 126807. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1411.4654">1411.4654</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015PhRvL.115l6807V">2015PhRvL.115l6807V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.115.126807">10.1103/PhysRevLett.115.126807</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26431008">26431008</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:20103218">20103218</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Enhanced+Bulk-Edge+Coulomb+Coupling+in+Fractional+Fabry-Perot+Interferometers&rft.volume=115&rft.issue=12&rft.pages=126807&rft.date=2015&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A20103218%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2015PhRvL.115l6807V&rft_id=info%3Aarxiv%2F1411.4654&rft_id=info%3Apmid%2F26431008&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.115.126807&rft.aulast=von+Keyserling&rft.aufirst=Curt&rft.au=Simon%2C+S.+H.&rft.au=Bernd%2C+Rosenow&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWillettNayakPfeifferWest2013" class="citation journal cs1">Willett, R. 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"Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory". <i>Physical Review B</i>. <b>91</b> (3): 035134. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1404.7854">1404.7854</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevB.91.035134">10.1103/PhysRevB.91.035134</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2469-9969">2469-9969</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13893760">13893760</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+B&rft.atitle=Non-Abelian+String+and+Particle+Braiding+in+Topological+Order%3A+Modular+SL%283%2CZ%29+Representation+and+3%2B1D+Twisted+Gauge+Theory&rft.volume=91&rft.issue=3&rft.pages=035134&rft.date=2015-01-15&rft_id=info%3Aarxiv%2F1404.7854&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13893760%23id-name%3DS2CID&rft.issn=2469-9969&rft_id=info%3Adoi%2F10.1103%2FPhysRevB.91.035134&rft.aulast=Wang&rft.aufirst=Juven&rft.au=Wen%2C+Xiao-Gang&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> <li id="cite_note-1612.09298-39"><span class="mw-cite-backlink">^ <a href="#cite_ref-1612.09298_39-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-1612.09298_39-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPutrovWangYau2017" class="citation journal cs1">Putrov, Pavel; Wang, Juven; Yau, Shing-Tung (September 2017). "Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions". <i>Annals of Physics</i>. <b>384C</b>: 254–287. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1612.09298">1612.09298</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017AnPhy.384..254P">2017AnPhy.384..254P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.aop.2017.06.019">10.1016/j.aop.2017.06.019</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119578849">119578849</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Physics&rft.atitle=Braiding+Statistics+and+Link+Invariants+of+Bosonic%2FFermionic+Topological+Quantum+Matter+in+2%2B1+and+3%2B1+dimensions&rft.volume=384C&rft.pages=254-287&rft.date=2017-09&rft_id=info%3Aarxiv%2F1612.09298&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119578849%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.aop.2017.06.019&rft_id=info%3Abibcode%2F2017AnPhy.384..254P&rft.aulast=Putrov&rft.aufirst=Pavel&rft.au=Wang%2C+Juven&rft.au=Yau%2C+Shing-Tung&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Anyon&action=edit&section=12" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNayakSimonSternFreedman2008" class="citation journal cs1">Nayak, Chetan; Simon, Steven H.; Stern, Ady; Freedman, Michael; Das Sarma, Sankar (2008). "Non-Abelian anyons and topological quantum computation". <i>Reviews of Modern Physics</i>. <b>80</b> (3): 1083. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0707.1889">0707.1889</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008RvMP...80.1083N">2008RvMP...80.1083N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.80.1083">10.1103/RevModPhys.80.1083</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119628297">119628297</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Non-Abelian+anyons+and+topological+quantum+computation&rft.volume=80&rft.issue=3&rft.pages=1083&rft.date=2008&rft_id=info%3Aarxiv%2F0707.1889&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119628297%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.80.1083&rft_id=info%3Abibcode%2F2008RvMP...80.1083N&rft.aulast=Nayak&rft.aufirst=Chetan&rft.au=Simon%2C+Steven+H.&rft.au=Stern%2C+Ady&rft.au=Freedman%2C+Michael&rft.au=Das+Sarma%2C+Sankar&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWen2002" class="citation journal cs1">Wen, Xiao-Gang (15 April 2002). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110609204800/http://dao.mit.edu/~wen/pub/qosl.pdf">"Quantum orders and symmetric spin liquids"</a> <span class="cs1-format">(PDF)</span>. <i>Physical Review B</i>. <b>65</b> (16): 165113. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0107071">cond-mat/0107071</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002PhRvB..65p5113W">2002PhRvB..65p5113W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevB.65.165113">10.1103/PhysRevB.65.165113</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119061254">119061254</a>. Archived from <a rel="nofollow" class="external text" href="http://dao.mit.edu/~wen/pub/qosl.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 9 June 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+B&rft.atitle=Quantum+orders+and+symmetric+spin+liquids&rft.volume=65&rft.issue=16&rft.pages=165113&rft.date=2002-04-15&rft_id=info%3Aarxiv%2Fcond-mat%2F0107071&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119061254%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevB.65.165113&rft_id=info%3Abibcode%2F2002PhRvB..65p5113W&rft.aulast=Wen&rft.aufirst=Xiao-Gang&rft_id=http%3A%2F%2Fdao.mit.edu%2F~wen%2Fpub%2Fqosl.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStern2008" class="citation journal cs1">Stern, Ady (2008). <a rel="nofollow" class="external text" href="http://pitp.physics.ubc.ca/confs/7pines2009/readings/arovas_Stern_2007.pdf">"Anyons and the quantum Hall effect—A pedagogical review"</a> <span class="cs1-format">(PDF)</span>. <i>Annals of Physics</i>. <b>323</b> (1): 204–249. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0711.4697">0711.4697</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008AnPhy.323..204S">2008AnPhy.323..204S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.aop.2007.10.008">10.1016/j.aop.2007.10.008</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15582782">15582782</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Physics&rft.atitle=Anyons+and+the+quantum+Hall+effect%E2%80%94A+pedagogical+review&rft.volume=323&rft.issue=1&rft.pages=204-249&rft.date=2008&rft_id=info%3Aarxiv%2F0711.4697&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15582782%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.aop.2007.10.008&rft_id=info%3Abibcode%2F2008AnPhy.323..204S&rft.aulast=Stern&rft.aufirst=Ady&rft_id=http%3A%2F%2Fpitp.physics.ubc.ca%2Fconfs%2F7pines2009%2Freadings%2Farovas_Stern_2007.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNajjar2020" class="citation journal cs1">Najjar, Dana (2020). <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/milestone-evidence-for-anyons-a-third-kingdom-of-particles-20200512/">"<span class="cs1-kern-left"></span>'Milestone' Evidence for Anyons, a Third Kingdom of Particles"</a>. <i>Quanta Magazine</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quanta+Magazine&rft.atitle=%27Milestone%27+Evidence+for+Anyons%2C+a+Third+Kingdom+of+Particles&rft.date=2020&rft.aulast=Najjar&rft.aufirst=Dana&rft_id=https%3A%2F%2Fwww.quantamagazine.org%2Fmilestone-evidence-for-anyons-a-third-kingdom-of-particles-20200512%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnyon" class="Z3988"></span></li></ul> </div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid 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style="width:1%;font-weight:normal; text-align: center;"><a href="/wiki/Gaugino" title="Gaugino">Gauginos</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gluino" title="Gluino">Gluino</a></li> <li><a href="/wiki/Gravitino" title="Gravitino">Gravitino</a></li> <li><a href="/wiki/Photino" title="Photino">Photino</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;">Others</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axino" title="Axino">Axino</a></li> <li><a href="/wiki/Chargino" title="Chargino">Chargino</a></li> <li><a href="/wiki/Higgsino" title="Higgsino">Higgsino</a></li> <li><a href="/wiki/Neutralino" title="Neutralino">Neutralino</a></li> <li><a href="/wiki/Sfermion" title="Sfermion">Sfermion</a> (<a href="/wiki/Stop_squark" title="Stop squark">Stop squark</a>)</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;">Others</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axion" title="Axion">Axion</a></li> <li><a href="/wiki/Curvaton" title="Curvaton">Curvaton</a></li> <li><a href="/wiki/Dilaton" title="Dilaton">Dilaton</a></li> <li><a href="/wiki/Dual_graviton" title="Dual graviton">Dual graviton</a></li> <li><a href="/wiki/Graviphoton" title="Graviphoton">Graviphoton</a></li> <li><a href="/wiki/Graviton" title="Graviton">Graviton</a></li> <li><a href="/wiki/Inflaton" title="Inflaton">Inflaton</a></li> <li><a href="/wiki/Leptoquark" title="Leptoquark">Leptoquark</a></li> <li><a href="/wiki/Magnetic_monopole" title="Magnetic monopole">Magnetic monopole</a></li> <li><a href="/wiki/Majoron" title="Majoron">Majoron</a></li> <li><a href="/wiki/Majorana_fermion" title="Majorana fermion">Majorana fermion</a></li> <li><a href="/wiki/Dark_photon" title="Dark photon">Dark photon</a></li> <li><a href="/wiki/Preon" title="Preon">Preon</a></li> <li><a href="/wiki/Sterile_neutrino" title="Sterile neutrino">Sterile neutrino</a></li> <li><a href="/wiki/Tachyon" title="Tachyon">Tachyon</a></li> <li><a href="/wiki/W%E2%80%B2_and_Z%E2%80%B2_bosons" title="W′ and Z′ bosons">W′ and Z′ bosons</a></li> <li><a href="/wiki/X_and_Y_bosons" title="X and Y bosons">X and Y bosons</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Bound_state" title="Bound state">Composite</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Hadrons" scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Hadron" title="Hadron">Hadrons</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;"><a href="/wiki/Baryon" title="Baryon">Baryons</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nucleon" title="Nucleon">Nucleon</a> <ul><li><a href="/wiki/Proton" title="Proton">Proton</a></li> <li><a href="/wiki/Antiproton" title="Antiproton">Antiproton</a></li> <li><a href="/wiki/Neutron" title="Neutron">Neutron</a></li> <li><a href="/wiki/Antineutron" title="Antineutron">Antineutron</a></li></ul></li> <li><a href="/wiki/Delta_baryon" title="Delta baryon">Delta baryon</a></li> <li><a href="/wiki/Lambda_baryon" title="Lambda baryon">Lambda baryon</a></li> <li><a href="/wiki/Sigma_baryon" title="Sigma baryon">Sigma baryon</a></li> <li><a href="/wiki/Xi_baryon" title="Xi baryon">Xi baryon</a></li> <li><a href="/wiki/Omega_baryon" title="Omega baryon">Omega baryon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;"><a href="/wiki/Meson" title="Meson">Mesons</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pion" title="Pion">Pion</a></li> <li><a href="/wiki/Rho_meson" title="Rho meson">Rho meson</a></li> <li><a href="/wiki/Eta_meson" class="mw-redirect" title="Eta meson">Eta and eta prime mesons</a></li> <li><a href="/wiki/Bottom_eta_meson" title="Bottom eta meson">Bottom eta meson</a></li> <li><a href="/wiki/Phi_meson" title="Phi meson">Phi meson</a></li> <li><a href="/wiki/J/psi_meson" title="J/psi meson">J/psi meson</a></li> <li><a href="/wiki/Omega_meson" title="Omega meson">Omega meson</a></li> <li><a href="/wiki/Upsilon_meson" title="Upsilon meson">Upsilon meson</a></li> <li><a href="/wiki/Kaon" title="Kaon">Kaon</a></li> <li><a href="/wiki/B_meson" title="B meson">B meson</a></li> <li><a href="/wiki/D_meson" title="D meson">D meson</a></li> <li><a href="/wiki/Quarkonium" title="Quarkonium">Quarkonium</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;"><a href="/wiki/Exotic_hadron" title="Exotic hadron">Exotic hadrons</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tetraquark" title="Tetraquark">Tetraquark</a> (<a href="/wiki/Double-charm_tetraquark" title="Double-charm tetraquark">Double-charm tetraquark</a>)</li> <li><a href="/wiki/Pentaquark" title="Pentaquark">Pentaquark</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;;font-weight:normal; text-align: center;">Others</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atomic_nucleus" title="Atomic nucleus">Atomic nuclei</a></li> <li><a href="/wiki/Atom" title="Atom">Atoms</a></li> <li><a href="/wiki/Exotic_atom" title="Exotic atom">Exotic atoms</a> <ul><li><a href="/wiki/Positronium" title="Positronium">Positronium</a></li> <li><a href="/wiki/Muonium" title="Muonium">Muonium</a></li> <li><a href="/wiki/Tauonium" class="mw-redirect" title="Tauonium">Tauonium</a></li> <li><a href="/wiki/Onium" title="Onium">Onia</a></li> <li><a href="/wiki/Pionium" title="Pionium">Pionium</a></li> <li><a href="/wiki/Protonium" title="Protonium">Protonium</a></li></ul></li> <li><a href="/wiki/Superatom" title="Superatom">Superatoms</a></li> <li><a href="/wiki/Molecule" title="Molecule">Molecules</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Category:Hypothetical_composite_particles" title="Category:Hypothetical composite particles">Hypothetical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Baryons</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hexaquark" title="Hexaquark">Hexaquark</a></li> <li><a href="/wiki/Heptaquark" title="Heptaquark">Heptaquark</a></li> <li><a href="/wiki/Skyrmion" title="Skyrmion">Skyrmion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mesons</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Glueball" title="Glueball">Glueball</a></li> <li><a href="/wiki/Theta_meson" title="Theta meson">Theta meson</a></li> <li><a href="/wiki/T_meson" title="T meson">T meson</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal; text-align: center;;font-weight:normal; text-align: center;">Others</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mesonic_molecule" title="Mesonic molecule">Mesonic molecule</a></li> <li><a href="/wiki/Pomeron" title="Pomeron">Pomeron</a></li> <li><a href="/wiki/Diquark" title="Diquark">Diquark</a></li> <li><a href="/wiki/R-hadron" title="R-hadron">R-hadron</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%"><a href="/wiki/Quasiparticle" title="Quasiparticle">Quasiparticles</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Anyon</a></li> <li><a href="/wiki/Davydov_soliton" title="Davydov soliton">Davydov soliton</a></li> <li><a href="/wiki/Dropleton" title="Dropleton">Dropleton</a></li> <li><a href="/wiki/Exciton" title="Exciton">Exciton</a></li> <li><a href="/wiki/Fracton_(subdimensional_particle)" title="Fracton (subdimensional particle)">Fracton</a></li> <li><a href="/wiki/Electron_hole" title="Electron hole">Hole</a></li> <li><a href="/wiki/Magnon" title="Magnon">Magnon</a></li> <li><a href="/wiki/Phonon" title="Phonon">Phonon</a></li> <li><a href="/wiki/Plasmaron" title="Plasmaron">Plasmaron</a></li> <li><a href="/wiki/Plasmon" title="Plasmon">Plasmon</a></li> <li><a href="/wiki/Polariton" title="Polariton">Polariton</a></li> <li><a href="/wiki/Polaron" title="Polaron">Polaron</a></li> <li><a href="/wiki/Roton" title="Roton">Roton</a></li> <li><a href="/wiki/Trion_(physics)" title="Trion (physics)">Trion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_baryons" title="List of baryons">Baryons</a></li> <li><a href="/wiki/List_of_mesons" title="List of mesons">Mesons</a></li> <li><a href="/wiki/List_of_particles" title="List of particles">Particles</a></li> <li><a href="/wiki/List_of_quasiparticles" title="List of quasiparticles">Quasiparticles</a></li> <li><a href="/wiki/Timeline_of_particle_discoveries" title="Timeline of particle discoveries">Timeline of particle discoveries</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_subatomic_physics" title="History of subatomic physics">History of subatomic physics</a> <ul><li><a href="/wiki/Timeline_of_atomic_and_subatomic_physics" title="Timeline of atomic and subatomic physics">timeline</a></li></ul></li> <li><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> <ul><li><a href="/wiki/Mathematical_formulation_of_the_Standard_Model" title="Mathematical formulation of the Standard Model">mathematical formulation</a></li></ul></li> <li><a href="/wiki/Subatomic_particle" title="Subatomic particle">Subatomic particles</a></li> <li><a href="/wiki/Particle" title="Particle">Particles</a></li> <li><a href="/wiki/Antiparticle" title="Antiparticle">Antiparticles</a></li> <li><a href="/wiki/Nuclear_physics" title="Nuclear physics">Nuclear physics</a></li> <li><a href="/wiki/Eightfold_way_(physics)" title="Eightfold way (physics)">Eightfold way</a> <ul><li><a href="/wiki/Quark_model" title="Quark model">Quark model</a></li></ul></li> <li><a href="/wiki/Exotic_matter" title="Exotic matter">Exotic matter</a></li> <li><a href="/wiki/Massless_particle" title="Massless particle">Massless particle</a></li> <li><a href="/wiki/Relativistic_particle" title="Relativistic particle">Relativistic particle</a></li> <li><a href="/wiki/Virtual_particle" title="Virtual particle">Virtual particle</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li> <li><a href="/wiki/Particle_chauvinism" title="Particle chauvinism">Particle chauvinism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align: center;"><div><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Symbol_portal_class.svg" class="mw-file-description" title="Portal"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span> <b><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></b></div></td></tr></tbody></table></div> <div class="navbox-styles"><link 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