CINXE.COM

t-J model - Wikipedia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>t-J model - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"d56dd801-a49b-41e2-8697-80c9f3d157f3","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"T-J_model","wgTitle":"T-J model","wgCurRevisionId":1232260190,"wgRevisionId":1232260190,"wgArticleId":10477190,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Quantum lattice models"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"T-J_model","wgRelevantArticleId":10477190,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0 ,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":9000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q1049381","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options": "loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.quicksurveys.init", "ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&amp;modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&amp;only=styles&amp;skin=vector-2022"> <script async="" src="/w/load.php?lang=en&amp;modules=startup&amp;only=scripts&amp;raw=1&amp;skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&amp;modules=site.styles&amp;only=styles&amp;skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/a/a8/2D-Hubbard-model.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="872"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/a/a8/2D-Hubbard-model.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="581"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="465"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="t-J model - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/T-J_model"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=T-J_model&amp;action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/T-J_model"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&amp;feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-T-J_model rootpage-T-J_model skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page&#039;s font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&amp;utm_medium=sidebar&amp;utm_campaign=C13_en.wikipedia.org&amp;uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&amp;returnto=T-J+model" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&amp;returnto=T-J+model" title="You&#039;re encouraged to log in; however, it&#039;s not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&amp;utm_medium=sidebar&amp;utm_campaign=C13_en.wikipedia.org&amp;uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&amp;returnto=T-J+model" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&amp;returnto=T-J+model" title="You&#039;re encouraged to log in; however, it&#039;s not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><!-- CentralNotice --></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-The_Hamiltonian" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_Hamiltonian"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>The Hamiltonian</span> </div> </a> <ul id="toc-The_Hamiltonian-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">2</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">3</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 class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><i>t</i>-<i>J</i> model</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 7 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-7" 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">7 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D9%85%D9%88%D8%B0%D8%AC_t-J" title="نموذج t-J – Arabic" lang="ar" hreflang="ar" data-title="نموذج t-J" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/T-J_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB" title="T-J модел – Bulgarian" lang="bg" hreflang="bg" data-title="T-J модел" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Model_t-J" title="Model t-J – Catalan" lang="ca" hreflang="ca" data-title="Model t-J" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Mod%C3%A8le_t-J" title="Modèle t-J – French" lang="fr" hreflang="fr" data-title="Modèle t-J" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Modello_t-J" title="Modello t-J – Italian" lang="it" hreflang="it" data-title="Modello t-J" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Model_t-J" title="Model t-J – Polish" lang="pl" hreflang="pl" data-title="Model t-J" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/T-J%E6%A8%A1%E5%9E%8B" title="T-J模型 – Chinese" lang="zh" hreflang="zh" data-title="T-J模型" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1049381#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/T-J_model" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:T-J_model" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/T-J_model"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=T-J_model&amp;action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=T-J_model&amp;action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/T-J_model"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=T-J_model&amp;action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=T-J_model&amp;action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/T-J_model" 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/T-J_model" 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=T-J_model&amp;oldid=1232260190" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=T-J_model&amp;action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&amp;page=T-J_model&amp;id=1232260190&amp;wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&amp;url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FT-J_model"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&amp;url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FT-J_model"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&amp;page=T-J_model&amp;action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=T-J_model&amp;printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1049381" 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"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:2D-Hubbard-model.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/2D-Hubbard-model.png/220px-2D-Hubbard-model.png" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/2D-Hubbard-model.png/330px-2D-Hubbard-model.png 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a8/2D-Hubbard-model.png 2x" data-file-width="355" data-file-height="258" /></a><figcaption>2D Hubbard model. The t-J model is the Hubbard model for <i>U</i> &gt;&gt; <i>t</i></figcaption></figure> <p>In <a href="/wiki/Solid-state_physics" title="Solid-state physics">solid-state physics</a>, the <b><i>t</i>-<i>J</i> model</b> is a model first derived by <a href="/w/index.php?title=J%C3%B3zef_Spa%C5%82ek&amp;action=edit&amp;redlink=1" class="new" title="Józef Spałek (page does not exist)">Józef Spałek</a><sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">&#91;</span>1<span class="cite-bracket">&#93;</span></a></sup> to explain <a href="/wiki/Antiferromagnetism" title="Antiferromagnetism">antiferromagnetic</a> properties of <a href="/wiki/Mott_insulator" title="Mott insulator">Mott insulators</a>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">&#91;</span>2<span class="cite-bracket">&#93;</span></a></sup> taking into account experimental results about the strength of electron-electron repulsion in these materials.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">&#91;</span>3<span class="cite-bracket">&#93;</span></a></sup> </p><p>The <a href="/wiki/Material" title="Material">material</a> is modelled as a <a href="/wiki/Lattice_model_(physics)" title="Lattice model (physics)">lattice</a> with atoms in the sites with conduction <a href="/wiki/Electron" title="Electron">electrons</a> (or holes) moving between them, like in the <a href="/wiki/Hubbard_model" title="Hubbard model">Hubbard model</a>. Unlike the Hubbard model, the electrons are <a href="/wiki/Strongly_correlated_material" title="Strongly correlated material">strongly-correlated</a>, meaning the electrons are sensitive to reciprocal <a href="/wiki/Coulomb%27s_law" title="Coulomb&#39;s law">coulombic repulsion</a>, and so are less likely to occupy lattice sites already occupied by another electron. In the basic Hubbard model, the repulsion, indicated by <i>U</i>, can be small or even zero, and electrons are more free to jump (<i>hopping</i>, parametrized by <i>t</i> as <i>transfer</i> or <a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling"><i>tunnel</i></a>) from one site to another. In the <i>t</i>-<i>J</i> model, instead of <i>U</i>, there is the parameter <i>J</i>, <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of the ratio <i>t</i>/<i>U</i>. </p><p>Like the Hubbard model, it is a prospective microscopic theory of <a href="/wiki/High_temperature_superconductivity" class="mw-redirect" title="High temperature superconductivity">high temperature superconductivity</a> in <a href="/wiki/Cuprate_superconductor" title="Cuprate superconductor">cuprate superconductors</a> which arise from <a href="/wiki/Dopant" title="Dopant">doped</a> antiferromagnets, particularly in the case where the lattice considered is the two-dimensional lattice.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">&#91;</span>4<span class="cite-bracket">&#93;</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">&#91;</span>5<span class="cite-bracket">&#93;</span></a></sup> Cuprate superconductors are currently (as of 2024) the superconductors with the highest known superconducting transition temperature at ambient pressure, but there is no consensus on the microscopic theory responsible for their superconducting transition. </p> <div class="mw-heading mw-heading2"><h2 id="The_Hamiltonian">The Hamiltonian</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-J_model&amp;action=edit&amp;section=1" title="Edit section: The Hamiltonian"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In quantum physics, system's models are usually based on the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> <a href="/wiki/Operator_(physics)" title="Operator (physics)">operator</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb06de5217295d7fbdbf68fb9c5309a513fc99e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\hat {H}}}"></span>, corresponding to the total energy of that system, including both <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> and <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a>. </p><p>The <i>t</i>-<i>J</i> Hamiltonian can be derived from the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb06de5217295d7fbdbf68fb9c5309a513fc99e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\hat {H}}}"></span> of the Hubbard model using the <a href="/wiki/Schrieffer%E2%80%93Wolff_transformation" title="Schrieffer–Wolff transformation">Schrieffer–Wolff transformation</a>, with the transformation generator depending on <i>t</i>/<i>U</i> and excluding the possibility for electrons to doubly occupy a lattice's site,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">&#91;</span>6<span class="cite-bracket">&#93;</span></a></sup> which results in:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">&#91;</span>7<span class="cite-bracket">&#93;</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}=-t\sum _{\langle ij\rangle ,\sigma }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)+J\sum _{\langle ij\rangle }\left(\mathbf {S} _{i}\cdot \mathbf {S} _{j}-{\frac {n_{i}n_{j}}{4}}\right)+O(t^{3}/U^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>&#x2212;<!-- − --></mo> <mi>t</mi> <munder> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">&#x27E8;<!-- ⟨ --></mo> <mi>i</mi> <mi>j</mi> <mo fence="false" stretchy="false">&#x27E9;<!-- ⟩ --></mo> <mo>,</mo> <mi>&#x03C3;<!-- σ --></mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>&#x03C3;<!-- σ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>&#x2020;<!-- † --></mo> </mrow> </msubsup> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>&#x03C3;<!-- σ --></mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> <mo>.</mo> <mi mathvariant="normal">c</mi> <mo>.</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>J</mi> <munder> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">&#x27E8;<!-- ⟨ --></mo> <mi>i</mi> <mi>j</mi> <mo fence="false" stretchy="false">&#x27E9;<!-- ⟩ --></mo> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>&#x22C5;<!-- ⋅ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>&#x2212;<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=-t\sum _{\langle ij\rangle ,\sigma }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)+J\sum _{\langle ij\rangle }\left(\mathbf {S} _{i}\cdot \mathbf {S} _{j}-{\frac {n_{i}n_{j}}{4}}\right)+O(t^{3}/U^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41547eac904e3d222a9d5be511900db5e8f77cd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:66.502ex; height:7.176ex;" alt="{\displaystyle {\hat {H}}=-t\sum _{\langle ij\rangle ,\sigma }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)+J\sum _{\langle ij\rangle }\left(\mathbf {S} _{i}\cdot \mathbf {S} _{j}-{\frac {n_{i}n_{j}}{4}}\right)+O(t^{3}/U^{2})}"></span></dd></dl> <p>where the term in <i>t</i> corresponds to the kinetic energy and is equal to the one in the Hubbard model. The second one is the potential energy approximated at the second order, because this is an approximation of the Hubbard model in the limit <i>U</i> &gt;&gt; <i>t</i> developed in power of <i>t</i>. Terms at higher order can be added.<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">&#91;</span>1<span class="cite-bracket">&#93;</span></a></sup> </p><p>The parameters are: </p> <ul><li><span class="sfrac nowrap"><span style="display:none;display:inline-block; vertical-align:top; text-align:center;"><span style="display:block;padding:0 0.1em;">Σ</span><span style="display:block; font-size:70%; line-height:1em;padding:0 0.1em;"><span style="position:relative; line-height:1em; margin-top:-0.5em; top:-0.5em;">⟨<i>ij</i>⟩</span></span></span></span> is the sum over nearest-neighbor sites <i>i</i> and <i>j</i>, for all sites, typically on a <a href="/wiki/Two-dimensional" class="mw-redirect" title="Two-dimensional">two-dimensional</a> <a href="/wiki/Square_lattice" title="Square lattice">square lattice</a>,</li> <li><i>c</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">†</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>iσ</i></sub></span></span>, <i>c</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>iσ</i></sub></span></span> are the <a href="/wiki/Fermion" title="Fermion">fermionic</a> <a href="/wiki/Creation_and_annihilation_operator" class="mw-redirect" title="Creation and annihilation operator">creation and annihilation operators</a> at site <i>i</i>,</li> <li><i>σ</i> is the <a href="/wiki/Spin_polarization" title="Spin polarization">spin polarization</a>,</li> <li><i>t</i> is the <a href="/wiki/Tight_binding#Second_quantization" title="Tight binding">hopping integral</a>,</li> <li><i>J</i> is the antiferromagnetic <a href="/wiki/Exchange_interaction" title="Exchange interaction">exchange coupling</a>, <i>J</i> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">&#8288;<span class="tion"><span class="num">4<i>t</i><sup>2</sup></span><span class="sr-only">/</span><span class="den"><i>U</i></span></span>&#8288;</span>,</li> <li><i>U</i> is the on-site <a href="/wiki/Coulomb%27s_law" title="Coulomb&#39;s law">coulombic repulsion</a>, that must satisfy the condition for <i>U</i> &gt;&gt; <i>t</i>,</li> <li><i>n<sub>i</sub></i> = <span class="sfrac nowrap"><span style="display:none;display:inline-block; vertical-align:top; text-align:center;"><span style="display:block;padding:0 0.1em;">Σ</span><span style="display:block; font-size:70%; line-height:1em;padding:0 0.1em;"><span style="position:relative; line-height:1em; margin-top:-0.5em; top:-0.5em;"><i>σ</i></span></span></span></span><i>c</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">†</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>iσ</i></sub></span></span><i>c</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>iσ</i></sub></span></span> is the particle number at site <i>i</i> and can be maximum 1, so that double occupancy is forbidden (in the Hubbard model is possible),</li> <li><b>S</b><sub><i>i</i></sub> and <b>S</b><sub><i>j</i></sub> are the <a href="/wiki/Spin_(physics)#Mathematical_formulation" title="Spin (physics)">spins</a> on sites <i>i</i> and <i>j</i>,</li> <li>h. c. stands for <a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">Hermitian conjugate</a>,</li></ul> <p>If <i>n<sub>i</sub></i> = 1, that is when in the <a href="/wiki/Ground_state" title="Ground state">ground state</a>, there is just one electron per lattice's site (half-filling), the model reduces to the <a href="/wiki/Quantum_Heisenberg_model" title="Quantum Heisenberg model">Heisenberg model</a> and the ground state reproduce a <a href="/wiki/Dielectric" title="Dielectric">dielectric</a> antiferromagnets (<a href="/wiki/Mott_insulator" title="Mott insulator">Mott insulator</a>).<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">&#91;</span>8<span class="cite-bracket">&#93;</span></a></sup> </p><p>The model can be further extended considering also the next-nearest-neighbor sites and the <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</a> to set the ground state in function of the total number of particles:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">&#91;</span>9<span class="cite-bracket">&#93;</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">&#91;</span>10<span class="cite-bracket">&#93;</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {\hat {H}}}=t_{1}\sum \limits _{\langle i,j\rangle }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)\ +\ t_{2}\sum \limits _{\langle \langle i,j\rangle \rangle }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)\ +\ J\sum \limits _{\langle i,j\rangle }\left(\mathbf {S} _{i}\cdot \mathbf {S} _{j}-{\frac {n_{i}n_{j}}{4}}\right)-\ \mu \sum \limits _{i}n_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> <mo class="MJX-tex-caligraphic" mathvariant="script" stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> </mrow> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <munder> <mo movablelimits="false">&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">&#x27E8;<!-- ⟨ --></mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo fence="false" stretchy="false">&#x27E9;<!-- ⟩ --></mo> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>&#x03C3;<!-- σ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>&#x2020;<!-- † --></mo> </mrow> </msubsup> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>&#x03C3;<!-- σ --></mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> <mo>.</mo> <mi mathvariant="normal">c</mi> <mo>.</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mtext>&#xA0;</mtext> <mo>+</mo> <mtext>&#xA0;</mtext> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <munder> <mo movablelimits="false">&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">&#x27E8;<!-- ⟨ --></mo> <mo fence="false" stretchy="false">&#x27E8;<!-- ⟨ --></mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo fence="false" stretchy="false">&#x27E9;<!-- ⟩ --></mo> <mo fence="false" stretchy="false">&#x27E9;<!-- ⟩ --></mo> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>&#x03C3;<!-- σ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>&#x2020;<!-- † --></mo> </mrow> </msubsup> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>&#x03C3;<!-- σ --></mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> <mo>.</mo> <mi mathvariant="normal">c</mi> <mo>.</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mtext>&#xA0;</mtext> <mo>+</mo> <mtext>&#xA0;</mtext> <mi>J</mi> <munder> <mo movablelimits="false">&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">&#x27E8;<!-- ⟨ --></mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo fence="false" stretchy="false">&#x27E9;<!-- ⟩ --></mo> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>&#x22C5;<!-- ⋅ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>&#x2212;<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>&#x2212;<!-- − --></mo> <mtext>&#xA0;</mtext> <mi>&#x03BC;<!-- μ --></mi> <munder> <mo movablelimits="false">&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {\hat {H}}}=t_{1}\sum \limits _{\langle i,j\rangle }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)\ +\ t_{2}\sum \limits _{\langle \langle i,j\rangle \rangle }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)\ +\ J\sum \limits _{\langle i,j\rangle }\left(\mathbf {S} _{i}\cdot \mathbf {S} _{j}-{\frac {n_{i}n_{j}}{4}}\right)-\ \mu \sum \limits _{i}n_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c792e7dda07d103adbf8006fdf09302bd6b1c59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:92.591ex; height:7.176ex;" alt="{\displaystyle {\mathcal {\hat {H}}}=t_{1}\sum \limits _{\langle i,j\rangle }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)\ +\ t_{2}\sum \limits _{\langle \langle i,j\rangle \rangle }\left(c_{i\sigma }^{\dagger }c_{j\sigma }+\mathrm {h.c.} \right)\ +\ J\sum \limits _{\langle i,j\rangle }\left(\mathbf {S} _{i}\cdot \mathbf {S} _{j}-{\frac {n_{i}n_{j}}{4}}\right)-\ \mu \sum \limits _{i}n_{i},}"></span></dd></dl> <p>where ⟨...⟩ and ⟨⟨...⟩⟩ denote the nearest and next-nearest neighbors, respectively, with two different values for the hopping integral (<i>t</i><sub>1</sub> and <i>t</i><sub>2</sub>) and <i>μ</i> is the chemical potential. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-J_model&amp;action=edit&amp;section=2" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .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="CITEREFChaoSpałekOleś1978" class="citation journal cs1">Chao, K. A.; Spałek, J.; Oleś, A. M. (1978-10-01). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevB.18.3453">"Canonical perturbation expansion of the Hubbard model"</a>. <i>Physical Review B</i>. <b>18</b> (7): 3453–3464. <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/1978PhRvB..18.3453C">1978PhRvB..18.3453C</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.18.3453">10.1103/PhysRevB.18.3453</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Physical+Review+B&amp;rft.atitle=Canonical+perturbation+expansion+of+the+Hubbard+model&amp;rft.volume=18&amp;rft.issue=7&amp;rft.pages=3453-3464&amp;rft.date=1978-10-01&amp;rft_id=info%3Adoi%2F10.1103%2FPhysRevB.18.3453&amp;rft_id=info%3Abibcode%2F1978PhRvB..18.3453C&amp;rft.aulast=Chao&amp;rft.aufirst=K.+A.&amp;rft.au=Spa%C5%82ek%2C+J.&amp;rft.au=Ole%C5%9B%2C+A.+M.&amp;rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevB.18.3453&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnderson1959" class="citation journal cs1">Anderson, P. W. (1959-07-01). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRev.115.2">"New Approach to the Theory of Superexchange Interactions"</a>. <i>Physical Review</i>. <b>115</b> (1): 2–13. <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/1959PhRv..115....2A">1959PhRv..115....2A</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%2FPhysRev.115.2">10.1103/PhysRev.115.2</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Physical+Review&amp;rft.atitle=New+Approach+to+the+Theory+of+Superexchange+Interactions&amp;rft.volume=115&amp;rft.issue=1&amp;rft.pages=2-13&amp;rft.date=1959-07-01&amp;rft_id=info%3Adoi%2F10.1103%2FPhysRev.115.2&amp;rft_id=info%3Abibcode%2F1959PhRv..115....2A&amp;rft.aulast=Anderson&amp;rft.aufirst=P.+W.&amp;rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRev.115.2&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNagaoka1966" class="citation journal cs1">Nagaoka, Yosuke (1966-07-08). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRev.147.392">"Ferromagnetism in a Narrow, Almost Half-Filled s Band"</a>. <i>Physical Review</i>. <b>147</b> (1): 392–405. <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/1966PhRv..147..392N">1966PhRv..147..392N</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%2FPhysRev.147.392">10.1103/PhysRev.147.392</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Physical+Review&amp;rft.atitle=Ferromagnetism+in+a+Narrow%2C+Almost+Half-Filled+s+Band&amp;rft.volume=147&amp;rft.issue=1&amp;rft.pages=392-405&amp;rft.date=1966-07-08&amp;rft_id=info%3Adoi%2F10.1103%2FPhysRev.147.392&amp;rft_id=info%3Abibcode%2F1966PhRv..147..392N&amp;rft.aulast=Nagaoka&amp;rft.aufirst=Yosuke&amp;rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRev.147.392&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpalek2007" class="citation journal cs1">Spalek, Jozef (2007-06-28). "t-J model then and now: A personal perspective from the pioneering times". <i>Acta Physica Polonica A</i>. <b>111</b> (4): 409. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0706.4236">0706.4236</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007AcPPA.111..409S">2007AcPPA.111..409S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.12693%2FAPhysPolA.111.409">10.12693/APhysPolA.111.409</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:53117123">53117123</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Acta+Physica+Polonica+A&amp;rft.atitle=t-J+model+then+and+now%3A+A+personal+perspective+from+the+pioneering+times&amp;rft.volume=111&amp;rft.issue=4&amp;rft.pages=409&amp;rft.date=2007-06-28&amp;rft_id=info%3Aarxiv%2F0706.4236&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A53117123%23id-name%3DS2CID&amp;rft_id=info%3Adoi%2F10.12693%2FAPhysPolA.111.409&amp;rft_id=info%3Abibcode%2F2007AcPPA.111..409S&amp;rft.aulast=Spalek&amp;rft.aufirst=Jozef&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRømerMaierKreiselEremin2020" class="citation journal cs1">Rømer, Astrid T.; Maier, Thomas A.; Kreisel, Andreas; Eremin, Ilya; Hirschfeld, P. J.; Andersen, Brian M. (2020-01-31). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevResearch.2.013108">"Pairing in the two-dimensional Hubbard model from weak to strong coupling"</a>. <i>Physical Review Research</i>. <b>2</b> (1): 013108. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1909.00627">1909.00627</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020PhRvR...2a3108R">2020PhRvR...2a3108R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevResearch.2.013108">10.1103/PhysRevResearch.2.013108</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:202540002">202540002</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Physical+Review+Research&amp;rft.atitle=Pairing+in+the+two-dimensional+Hubbard+model+from+weak+to+strong+coupling&amp;rft.volume=2&amp;rft.issue=1&amp;rft.pages=013108&amp;rft.date=2020-01-31&amp;rft_id=info%3Aarxiv%2F1909.00627&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A202540002%23id-name%3DS2CID&amp;rft_id=info%3Adoi%2F10.1103%2FPhysRevResearch.2.013108&amp;rft_id=info%3Abibcode%2F2020PhRvR...2a3108R&amp;rft.aulast=R%C3%B8mer&amp;rft.aufirst=Astrid+T.&amp;rft.au=Maier%2C+Thomas+A.&amp;rft.au=Kreisel%2C+Andreas&amp;rft.au=Eremin%2C+Ilya&amp;rft.au=Hirschfeld%2C+P.+J.&amp;rft.au=Andersen%2C+Brian+M.&amp;rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevResearch.2.013108&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">This is done using a <a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">projector quantum operator</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a46a8cf7bc789e8c4f8de7ca0d9946a46fb8842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.812ex; height:2.843ex;" alt="{\displaystyle {\hat {P}}}"></span> that project <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb044130844bbac0e4a80636cf4058a4eaddccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.89ex; height:3.176ex;" alt="{\displaystyle {\hat {H}}_{t}}"></span> on the subspace where fermionic operators can not add an electron on a site already occupied (see next note)</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEckle2019" class="citation book cs1">Eckle, Hans-Peter (2019). <a rel="nofollow" class="external text" href="http://physics.gu.se/~tfkhj/HubbardEckle.pdf">"8.8.1 From the Hubbard to the t–J model: non-half filled band case"</a> <span class="cs1-format">(PDF)</span>. <i>Models of Quantum Matter</i>. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/9780199678839" title="Special:BookSources/9780199678839"><bdi>9780199678839</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.atitle=8.8.1+From+the+Hubbard+to+the+t%E2%80%93J+model%3A+non-half+filled+band+case&amp;rft.btitle=Models+of+Quantum+Matter&amp;rft.pub=Oxford+University+Press&amp;rft.date=2019&amp;rft.isbn=9780199678839&amp;rft.aulast=Eckle&amp;rft.aufirst=Hans-Peter&amp;rft_id=http%3A%2F%2Fphysics.gu.se%2F~tfkhj%2FHubbardEckle.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIzyumovChashchin1998" class="citation journal cs1">Izyumov, Yu. A.; Chashchin, N. I. (1998). <a rel="nofollow" class="external text" href="https://doi.org/10.5488%2FCMP.1.1.41">"tJ -model in terms of equations with variational derivatives"</a>. <i><a href="/wiki/Condensed_Matter_Physics" class="mw-redirect" title="Condensed Matter Physics">Condensed Matter Physics</a></i>. <b>1</b> (1): 41–56. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998CMPh....1...41I">1998CMPh....1...41I</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.5488%2FCMP.1.1.41">10.5488/CMP.1.1.41</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11254082">11254082</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Condensed+Matter+Physics&amp;rft.atitle=tJ+-model+in+terms+of+equations+with+variational+derivatives&amp;rft.volume=1&amp;rft.issue=1&amp;rft.pages=41-56&amp;rft.date=1998&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11254082%23id-name%3DS2CID&amp;rft_id=info%3Adoi%2F10.5488%2FCMP.1.1.41&amp;rft_id=info%3Abibcode%2F1998CMPh....1...41I&amp;rft.aulast=Izyumov&amp;rft.aufirst=Yu.+A.&amp;rft.au=Chashchin%2C+N.+I.&amp;rft_id=https%3A%2F%2Fdoi.org%2F10.5488%252FCMP.1.1.41&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarchev1998" class="citation journal cs1">Karchev, Naoum (1998). "Generalized <i>CP</i><sup>1</sup> model from the <i>t</i><sub>1</sub>-<i>t</i><sub>2</sub>-<i>J</i> model". <i>Phys. Rev. B</i>. <b>57</b> (17): 10913. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/9706105">cond-mat/9706105</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998PhRvB..5710913K">1998PhRvB..5710913K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevB.57.10913">10.1103/PhysRevB.57.10913</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:12865671">12865671</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Phys.+Rev.+B&amp;rft.atitle=Generalized+CP%3Csup%3E1%3C%2Fsup%3E+model+from+the+t%3Csub%3E1%3C%2Fsub%3E-t%3Csub%3E2%3C%2Fsub%3E-J+model&amp;rft.volume=57&amp;rft.issue=17&amp;rft.pages=10913&amp;rft.date=1998&amp;rft_id=info%3Aarxiv%2Fcond-mat%2F9706105&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A12865671%23id-name%3DS2CID&amp;rft_id=info%3Adoi%2F10.1103%2FPhysRevB.57.10913&amp;rft_id=info%3Abibcode%2F1998PhRvB..5710913K&amp;rft.aulast=Karchev&amp;rft.aufirst=Naoum&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYanagisawa2008" class="citation journal cs1">Yanagisawa, Takashi (2008-02-12). <a rel="nofollow" class="external text" href="https://iopscience.iop.org/article/10.1088/1367-2630/10/2/023014">"Phase diagram of the<i>t</i>–<i>U</i><sup>2</sup>Hamiltonian of the weak coupling Hubbard model"</a>. <i>New Journal of Physics</i>. <b>10</b> (2): 023014. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0803.1739">0803.1739</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008NJPh...10b3014Y">2008NJPh...10b3014Y</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1367-2630%2F10%2F2%2F023014">10.1088/1367-2630/10/2/023014</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a>&#160;<a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1367-2630">1367-2630</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:55405863">55405863</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=New+Journal+of+Physics&amp;rft.atitle=Phase+diagram+of+thet%E2%80%93U%3Csup%3E2%3C%2Fsup%3EHamiltonian+of+the+weak+coupling+Hubbard+model&amp;rft.volume=10&amp;rft.issue=2&amp;rft.pages=023014&amp;rft.date=2008-02-12&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A55405863%23id-name%3DS2CID&amp;rft_id=info%3Abibcode%2F2008NJPh...10b3014Y&amp;rft_id=info%3Aarxiv%2F0803.1739&amp;rft.issn=1367-2630&amp;rft_id=info%3Adoi%2F10.1088%2F1367-2630%2F10%2F2%2F023014&amp;rft.aulast=Yanagisawa&amp;rft.aufirst=Takashi&amp;rft_id=https%3A%2F%2Fiopscience.iop.org%2Farticle%2F10.1088%2F1367-2630%2F10%2F2%2F023014&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T-J_model&amp;action=edit&amp;section=3" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFazekas1999" class="citation book cs1">Fazekas, Patrik (1999). <i>Lectures on Correlation and Magnetism</i>. Series in Modern Condensed Matter Physics: Volume 5. Vol.&#160;5. <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>. p.&#160;199. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F2945">10.1142/2945</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-981-4499-62-0" title="Special:BookSources/978-981-4499-62-0"><bdi>978-981-4499-62-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Lectures+on+Correlation+and+Magnetism&amp;rft.series=Series+in+Modern+Condensed+Matter+Physics%3A+Volume+5&amp;rft.pages=199&amp;rft.pub=World+Scientific&amp;rft.date=1999&amp;rft_id=info%3Adoi%2F10.1142%2F2945&amp;rft.isbn=978-981-4499-62-0&amp;rft.aulast=Fazekas&amp;rft.aufirst=Patrik&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpałek2007" class="citation journal cs1">Spałek, Józef (2007). "<i>t</i>-<i>J</i> model then and now: A personal perspective from the pioneering times". <i>Acta Phys. Pol. A</i>. <b>111</b> (4): 409–424. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0706.4236">0706.4236</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007AcPPA.111..409S">2007AcPPA.111..409S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.12693%2FAPhysPolA.111.409">10.12693/APhysPolA.111.409</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:53117123">53117123</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Acta+Phys.+Pol.+A&amp;rft.atitle=t-J+model+then+and+now%3A+A+personal+perspective+from+the+pioneering+times&amp;rft.volume=111&amp;rft.issue=4&amp;rft.pages=409-424&amp;rft.date=2007&amp;rft_id=info%3Aarxiv%2F0706.4236&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A53117123%23id-name%3DS2CID&amp;rft_id=info%3Adoi%2F10.12693%2FAPhysPolA.111.409&amp;rft_id=info%3Abibcode%2F2007AcPPA.111..409S&amp;rft.aulast=Spa%C5%82ek&amp;rft.aufirst=J%C3%B3zef&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDr_Mitchell" class="citation cs2">Dr Mitchell, <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=wtYQlELm3tk"><i>Electron interactions and the Hubbard model</i></a><span class="reference-accessdate">, retrieved <span class="nowrap">2022-08-29</span></span></cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Electron+interactions+and+the+Hubbard+model&amp;rft.au=Dr+Mitchell&amp;rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DwtYQlELm3tk&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AT-J+model" class="Z3988"></span></li></ul> <!-- NewPP limit report Parsed by mw‐web.eqiad.main‐6b6c9bdc8b‐r5vlk Cached time: 20241102164359 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1] CPU time usage: 0.213 seconds Real time usage: 0.384 seconds Preprocessor visited node count: 856/1000000 Post‐expand include size: 32592/2097152 bytes Template argument size: 123/2097152 bytes Highest expansion depth: 8/100 Expensive parser function count: 1/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 44996/5000000 bytes Lua time usage: 0.110/10.000 seconds Lua memory usage: 4249265/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 256.380 1 -total 78.74% 201.875 1 Template:Reflist 52.39% 134.322 9 Template:Cite_journal 7.48% 19.168 4 Template:Su 5.18% 13.293 1 Template:Sfrac 4.52% 11.585 2 Template:Cite_book 2.62% 6.713 1 Template:Citation 1.84% 4.721 2 Template:Underset 0.81% 2.084 1 Template:Main_other --> <!-- Saved in parser cache with key enwiki:pcache:idhash:10477190-0!canonical and timestamp 20241102164359 and revision id 1232260190. Rendering was triggered because: page-view --> </div><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=T-J_model&amp;oldid=1232260190">https://en.wikipedia.org/w/index.php?title=T-J_model&amp;oldid=1232260190</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Quantum_lattice_models" title="Category:Quantum lattice models">Quantum lattice models</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 2 July 2024, at 20:26<span class="anonymous-show">&#160;(UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=T-J_model&amp;mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-6df7948d6c-5k98v","wgBackendResponseTime":166,"wgPageParseReport":{"limitreport":{"cputime":"0.213","walltime":"0.384","ppvisitednodes":{"value":856,"limit":1000000},"postexpandincludesize":{"value":32592,"limit":2097152},"templateargumentsize":{"value":123,"limit":2097152},"expansiondepth":{"value":8,"limit":100},"expensivefunctioncount":{"value":1,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":44996,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 256.380 1 -total"," 78.74% 201.875 1 Template:Reflist"," 52.39% 134.322 9 Template:Cite_journal"," 7.48% 19.168 4 Template:Su"," 5.18% 13.293 1 Template:Sfrac"," 4.52% 11.585 2 Template:Cite_book"," 2.62% 6.713 1 Template:Citation"," 1.84% 4.721 2 Template:Underset"," 0.81% 2.084 1 Template:Main_other"]},"scribunto":{"limitreport-timeusage":{"value":"0.110","limit":"10.000"},"limitreport-memusage":{"value":4249265,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-6b6c9bdc8b-r5vlk","timestamp":"20241102164359","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"T-J model","url":"https:\/\/en.wikipedia.org\/wiki\/T-J_model","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1049381","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1049381","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2007-04-05T14:57:16Z","dateModified":"2024-07-02T20:26:32Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/a\/a8\/2D-Hubbard-model.png","headline":"model of electrical resistance"}</script> </body> </html>

Pages: 1 2 3 4 5 6 7 8 9 10