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<span class="vector-toc-numb">2</span> <span>Similarities to Fermi gas</span> </div> </a> <ul id="toc-Similarities_to_Fermi_gas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differences_from_Fermi_gas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Differences_from_Fermi_gas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Differences from Fermi gas</span> </div> </a> <button aria-controls="toc-Differences_from_Fermi_gas-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Differences from Fermi gas subsection</span> </button> <ul id="toc-Differences_from_Fermi_gas-sublist" class="vector-toc-list"> <li id="toc-Energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Energy</span> </div> </a> <ul id="toc-Energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Specific_heat_and_compressibility" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Specific_heat_and_compressibility"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Specific heat and compressibility</span> </div> </a> <ul id="toc-Specific_heat_and_compressibility-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interactions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interactions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Interactions</span> </div> </a> <ul id="toc-Interactions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Structure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Structure"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Structure</span> </div> </a> <ul id="toc-Structure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Distribution</span> </div> </a> <ul id="toc-Distribution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electrical_resistivity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electrical_resistivity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Electrical resistivity</span> </div> </a> <ul id="toc-Electrical_resistivity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optical_response" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optical_response"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Optical response</span> </div> </a> <ul id="toc-Optical_response-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Instabilities" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Instabilities"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Instabilities</span> </div> </a> <ul id="toc-Instabilities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-Fermi_liquids" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Non-Fermi_liquids"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Non-Fermi liquids</span> </div> </a> <ul id="toc-Non-Fermi_liquids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</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">8</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" 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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Fermi liquid theory</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A4%D0%B5%D1%80%D0%BC%D0%B8-%D1%82%D0%B5%D1%87%D0%BD%D0%BE%D1%81%D1%82" title="Ферми-течност – Bulgarian" lang="bg" hreflang="bg" data-title="Ферми-течност" 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/Teoria_l%C3%ADquida_de_Fermi" title="Teoria líquida de Fermi – Catalan" lang="ca" hreflang="ca" data-title="Teoria líquida de Fermi" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Fermi-Fl%C3%BCssigkeits-Theorie" title="Fermi-Flüssigkeits-Theorie – German" lang="de" hreflang="de" data-title="Fermi-Flüssigkeits-Theorie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/L%C3%ADquido_de_Fermi" title="Líquido de Fermi – Spanish" lang="es" hreflang="es" data-title="Líquido de Fermi" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Liquide_de_Fermi" title="Liquide de Fermi – French" lang="fr" hreflang="fr" data-title="Liquide de Fermi" 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/Liquido_di_Fermi" title="Liquido di Fermi – Italian" lang="it" hreflang="it" data-title="Liquido di Fermi" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A4%D0%B5%D1%80%D0%BC%D0%B8_%D1%81%D2%B1%D0%B9%D1%8B%D2%9B" title="Ферми сұйық – Kazakh" lang="kk" hreflang="kk" data-title="Ферми сұйық" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A7%E3%83%AB%E3%83%9F%E6%B6%B2%E4%BD%93%E8%AB%96" title="フェルミ液体論 – Japanese" lang="ja" hreflang="ja" data-title="フェルミ液体論" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ciecz_Fermiego" title="Ciecz Fermiego – Polish" lang="pl" hreflang="pl" data-title="Ciecz Fermiego" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/L%C3%ADquido_de_Fermi" title="Líquido de Fermi – Portuguese" lang="pt" hreflang="pt" data-title="Líquido de Fermi" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D0%B5%D1%80%D0%BC%D0%B8-%D0%B6%D0%B8%D0%B4%D0%BA%D0%BE%D1%81%D1%82%D1%8C" title="Ферми-жидкость – Russian" lang="ru" hreflang="ru" data-title="Ферми-жидкость" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Fermi_s%C4%B1v%C4%B1lar_teorisi" title="Fermi sıvılar teorisi – Turkish" lang="tr" hreflang="tr" data-title="Fermi sıvılar teorisi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D0%B5%D1%80%D0%BC%D1%96-%D1%80%D1%96%D0%B4%D0%B8%D0%BD%D0%B0" title="Фермі-рідина – Ukrainian" lang="uk" hreflang="uk" data-title="Фермі-рідина" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%9C%97%E9%81%93-%E8%B4%B9%E7%B1%B3%E6%B6%B2%E4%BD%93%E7%90%86%E8%AE%BA" title="朗道-费米液体理论 – Chinese" lang="zh" hreflang="zh" data-title="朗道-费米液体理论" data-language-autonym="中文" data-language-local-name="Chinese" 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks hlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a></th></tr><tr><td class="sidebar-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:QuantumPhaseTransition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/220px-QuantumPhaseTransition.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/330px-QuantumPhaseTransition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/440px-QuantumPhaseTransition.svg.png 2x" data-file-width="512" data-file-height="369" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="hlist"><ul><li><a href="/wiki/Phase_(matter)" title="Phase (matter)">Phases</a></li><li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li><li><a href="/wiki/Quantum_critical_point" title="Quantum critical point">QCP</a></li></ul></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/State_of_matter" title="State of matter">States of matter</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Solid" title="Solid">Solid</a></li><li><a href="/wiki/Liquid" title="Liquid">Liquid</a></li><li><a href="/wiki/Gas" title="Gas">Gas</a></li><li><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></li><li><a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a></li><li><a href="/wiki/Bose_gas" title="Bose gas">Bose gas</a></li><li><a href="/wiki/Fermionic_condensate" title="Fermionic condensate">Fermionic condensate</a></li><li><a href="/wiki/Fermi_gas" title="Fermi gas">Fermi gas</a></li><li><a class="mw-selflink selflink">Fermi liquid</a></li><li><a href="/wiki/Supersolid" title="Supersolid">Supersolid</a></li><li><a href="/wiki/Superfluidity" title="Superfluidity">Superfluidity</a></li><li><a href="/wiki/Luttinger_liquid" title="Luttinger liquid">Luttinger liquid</a></li><li><a href="/wiki/Time_crystal" title="Time crystal">Time crystal</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Phase phenomena</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Order_parameter" class="mw-redirect" title="Order parameter">Order parameter</a></li><li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li><li><a href="/wiki/Quantum_critical_point" title="Quantum critical point">QCP</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Electronic phases</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Electronic_band_structure" title="Electronic band structure">Electronic band structure</a></li><li><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></li><li><a href="/wiki/Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li><li><a href="/wiki/Mott_insulator" title="Mott insulator">Mott insulator</a></li><li><a href="/wiki/Semiconductor" title="Semiconductor">Semiconductor</a></li><li><a href="/wiki/Semimetal" title="Semimetal">Semimetal</a></li><li><a href="/wiki/Electrical_conductor" title="Electrical conductor">Conductor</a></li><li><a href="/wiki/Superconductivity" title="Superconductivity">Superconductor</a></li><li><a href="/wiki/Thermoelectric_effect" title="Thermoelectric effect">Thermoelectric</a></li><li><a href="/wiki/Piezoelectricity" title="Piezoelectricity">Piezoelectric</a></li><li><a href="/wiki/Ferroelectricity" title="Ferroelectricity">Ferroelectric</a></li><li><a href="/wiki/Topological_insulator" title="Topological insulator">Topological insulator</a></li><li><a href="/wiki/Spin_gapless_semiconductor" title="Spin gapless semiconductor">Spin gapless semiconductor</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Electronic phenomena</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Quantum_Hall_effect" title="Quantum Hall effect">Quantum Hall effect</a></li><li><a href="/wiki/Spin_Hall_effect" title="Spin Hall effect">Spin Hall effect</a></li><li><a href="/wiki/Kondo_effect" title="Kondo effect">Kondo effect</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Magnetic phases</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Diamagnetism" title="Diamagnetism">Diamagnet</a></li><li><a href="/wiki/Superdiamagnetism" title="Superdiamagnetism">Superdiamagnet</a><br /> <a href="/wiki/Paramagnetism" title="Paramagnetism">Paramagnet</a></li><li><a href="/wiki/Superparamagnetism" title="Superparamagnetism">Superparamagnet</a><br /><a href="/wiki/Ferromagnetism" title="Ferromagnetism">Ferromagnet</a></li><li><a href="/wiki/Antiferromagnetism" title="Antiferromagnetism">Antiferromagnet</a><br /><a href="/wiki/Metamagnetism" title="Metamagnetism">Metamagnet</a></li><li><a href="/wiki/Spin_glass" title="Spin glass">Spin glass</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Quasiparticle" title="Quasiparticle">Quasiparticles</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Phonon" title="Phonon">Phonon</a></li><li><a href="/wiki/Exciton" title="Exciton">Exciton</a></li><li><a href="/wiki/Plasmon" title="Plasmon">Plasmon</a><br /><a href="/wiki/Polariton" title="Polariton">Polariton</a></li><li><a href="/wiki/Polaron" title="Polaron">Polaron</a></li><li><a href="/wiki/Magnon" title="Magnon">Magnon</a></li><li><a href="/wiki/Roton" title="Roton">Roton</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Soft_matter" title="Soft matter">Soft matter</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Amorphous_solid" title="Amorphous solid">Amorphous solid</a></li><li><a href="/wiki/Colloid" title="Colloid">Colloid</a></li><li><a href="/wiki/Granular_material" title="Granular material">Granular material</a></li><li><a href="/wiki/Liquid_crystal" title="Liquid crystal">Liquid crystal</a></li><li><a href="/wiki/Polymer" title="Polymer">Polymer</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">Van der Waals</a></li><li><a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li><li><a href="/wiki/Max_von_Laue" title="Max von Laue">von Laue</a></li><li><a href="/wiki/William_Henry_Bragg" title="William Henry Bragg">Bragg</a></li><li><a href="/wiki/Peter_Debye" title="Peter Debye">Debye</a></li><li><a href="/wiki/Felix_Bloch" title="Felix Bloch">Bloch</a></li><li><a href="/wiki/Lars_Onsager" title="Lars Onsager">Onsager</a></li><li><a href="/wiki/Neville_Mott" class="mw-redirect" title="Neville Mott">Mott</a></li><li><a href="/wiki/Rudolf_Peierls" title="Rudolf Peierls">Peierls</a></li><li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li><li><a href="/wiki/Luttinger" class="mw-redirect" title="Luttinger">Luttinger</a></li><li><a href="/wiki/Philip_Warren_Anderson" class="mw-redirect" title="Philip Warren Anderson">Anderson</a></li><li><a href="/wiki/John_Hasbrouck_Van_Vleck" title="John Hasbrouck Van Vleck">Van Vleck</a></li><li><a href="/wiki/John_Hubbard_(physicist)" title="John Hubbard (physicist)">Hubbard</a></li><li><a href="/wiki/William_Bradford_Shockley" class="mw-redirect" title="William Bradford Shockley">Shockley</a></li><li><a href="/wiki/John_Bardeen" title="John Bardeen">Bardeen</a></li><li><a href="/wiki/Leon_Cooper" title="Leon Cooper">Cooper</a></li><li><a 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href="/wiki/Template:Condensed_matter_physics" title="Template:Condensed matter physics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Condensed_matter_physics" title="Template talk:Condensed matter physics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Condensed_matter_physics" title="Special:EditPage/Template:Condensed matter physics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Fermi liquid theory</b> (also known as <b>Landau's Fermi-liquid theory</b>) is a theoretical model of interacting <a href="/wiki/Fermion" title="Fermion">fermions</a> that describes the normal state of the conduction electrons in most <a href="/wiki/Metal" title="Metal">metals</a> at sufficiently low temperatures.<sup id="cite_ref-phillips_1-0" class="reference"><a href="#cite_note-phillips-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The theory describes the behavior of <a href="/wiki/Many-body_problem" title="Many-body problem">many-body systems</a> of particles in which the interactions between particles may be strong. The <a href="/wiki/Phenomenological_model" title="Phenomenological model">phenomenological</a> theory of <b>Fermi liquids</b> was introduced by the Soviet physicist <a href="/wiki/Lev_Davidovich_Landau" class="mw-redirect" title="Lev Davidovich Landau">Lev Davidovich Landau</a> in 1956,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and later developed by <a href="/wiki/Alexei_Alexeyevich_Abrikosov" class="mw-redirect" title="Alexei Alexeyevich Abrikosov">Alexei Abrikosov</a> and <a href="/wiki/Isaak_Markovich_Khalatnikov" class="mw-redirect" title="Isaak Markovich Khalatnikov">Isaak Khalatnikov</a> using <a href="/wiki/Feynman_diagrams" class="mw-redirect" title="Feynman diagrams">diagrammatic</a> <a href="/wiki/Perturbation_theory" title="Perturbation theory">perturbation theory</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The theory explains why some of the properties of an interacting fermion system are very similar to those of the ideal <a href="/wiki/Fermi_gas" title="Fermi gas">Fermi gas</a> (collection of non-interacting fermions), and why other properties differ. </p><p>Fermi liquid theory applies most notably to conduction electrons in normal (non-<a href="/wiki/Superconductivity" title="Superconductivity">superconducting</a>) metals, and to <a href="/wiki/Liquid_helium" title="Liquid helium">liquid helium</a>-3.<sup id="cite_ref-schulz_4-0" class="reference"><a href="#cite_note-schulz-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Liquid <a href="/wiki/Helium-3" title="Helium-3">helium-3</a> is a Fermi liquid at low temperatures (but not low enough to be in its <a href="/wiki/Superfluid" class="mw-redirect" title="Superfluid">superfluid</a> <a href="/wiki/Phase_(matter)" title="Phase (matter)">phase</a>). An atom of helium-3 has two <a href="/wiki/Proton" title="Proton">protons</a>, one <a href="/wiki/Neutron" title="Neutron">neutron</a> and two <a href="/wiki/Electron" title="Electron">electrons</a>, giving an odd number of <a href="/wiki/Fermions" class="mw-redirect" title="Fermions">fermions</a>, so the atom itself is a fermion. Fermi liquid theory also describes the low-temperature behavior of electrons in <a href="/wiki/Heavy_fermion_material" title="Heavy fermion material">heavy fermion materials</a>, which are metallic rare-earth alloys having partially filled f orbitals. The <a href="/wiki/Effective_mass_(solid-state_physics)" title="Effective mass (solid-state physics)">effective mass</a> of electrons in these materials is much larger than the free-electron mass because of interactions with other electrons, so these systems are known as <i>heavy Fermi liquids</i>. <a href="/wiki/Distrontium_ruthenate" title="Distrontium ruthenate">Strontium ruthenate</a> displays some key properties of Fermi liquids, despite being a <a href="/wiki/Strongly_correlated_material" title="Strongly correlated material">strongly correlated material</a> that is similar to <a href="/wiki/High_temperature_superconductor" class="mw-redirect" title="High temperature superconductor">high temperature superconductors</a> such as the <a href="/wiki/Cuprate_superconductor" title="Cuprate superconductor">cuprates</a>.<sup id="cite_ref-wysokinski_5-0" class="reference"><a href="#cite_note-wysokinski-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The low-momentum interactions of <a href="/wiki/Nucleons" class="mw-redirect" title="Nucleons">nucleons</a> (protons and neutrons) in <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">atomic nuclei</a> are also described by Fermi liquid theory.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermi_liquid_theory&action=edit&section=1" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The key ideas behind Landau's theory are the notion of <a href="/wiki/Adiabatic_process" title="Adiabatic process"><i>adiabaticity</i></a> and the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a>.<sup id="cite_ref-coleman_7-0" class="reference"><a href="#cite_note-coleman-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Consider a non-interacting fermion system (a <a href="/wiki/Fermi_gas" title="Fermi gas">Fermi gas</a>), and suppose we "turn on" the interaction slowly. Landau argued that in this situation, the ground state of the Fermi gas would adiabatically transform into the ground state of the interacting system. </p><p>By Pauli's exclusion principle, the ground state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bbf983eb36150518e9c5a7f69bbc50374367cbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:2.509ex;" alt="{\displaystyle \Psi _{0}}"></span> of a Fermi gas consists of fermions occupying all momentum states corresponding to momentum <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 p<p_{\rm {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo><</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p<p_{\rm {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bcd774318688f5ea5964adecf0e770a381c5464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.832ex; height:2.176ex;" alt="{\displaystyle p<p_{\rm {F}}}"></span> with all higher momentum states unoccupied. As the interaction is turned on, the spin, charge and momentum of the fermions corresponding to the occupied states remain unchanged, while their dynamical properties, such as their mass, magnetic moment etc. are <i><a href="/wiki/Renormalization" title="Renormalization">renormalized</a></i> to new values.<sup id="cite_ref-coleman_7-1" class="reference"><a href="#cite_note-coleman-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Thus, there is a one-to-one correspondence between the elementary excitations of a Fermi gas system and a Fermi liquid system. In the context of Fermi liquids, these excitations are called "<a href="/wiki/Quasiparticle" title="Quasiparticle">quasiparticles</a>".<sup id="cite_ref-phillips_1-1" class="reference"><a href="#cite_note-phillips-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Landau quasiparticles are long-lived excitations with a lifetime <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> that satisfies <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 {\hbar }/{\tau }\ll \varepsilon _{\rm {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant">ℏ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mo>≪</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hbar }/{\tau }\ll \varepsilon _{\rm {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f47a49785a5ba7eb869a48253a75973a4dc67114" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.515ex; height:3.009ex;" alt="{\displaystyle {\hbar }/{\tau }\ll \varepsilon _{\rm {p}}}"></span> where <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 \varepsilon _{\rm {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\rm {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49e885818c44b447ff969fd4d3e2ad7490a99a75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.23ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{\rm {p}}}"></span> is the quasiparticle energy (measured from the <a href="/wiki/Fermi_energy" title="Fermi energy">Fermi energy</a>). At finite temperature, <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 \varepsilon _{\rm {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\rm {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49e885818c44b447ff969fd4d3e2ad7490a99a75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.23ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{\rm {p}}}"></span> is on the order of the thermal energy <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 k_{\rm {B}}T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{\rm {B}}T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81730a1844016dda66a50d36d65a9ffe418d8ac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.243ex; height:2.509ex;" alt="{\displaystyle k_{\rm {B}}T}"></span>, and the condition for Landau quasiparticles can be reformulated as <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 {\hbar }/{\tau }\ll k_{\rm {B}}T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant">ℏ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mo>≪</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hbar }/{\tau }\ll k_{\rm {B}}T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61b48beadd9e322374f0adeff8bce23ac4ec8055" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.528ex; height:2.843ex;" alt="{\displaystyle {\hbar }/{\tau }\ll k_{\rm {B}}T}"></span>. </p><p>For this system, the <a href="/wiki/Green%27s_function_(many-body_theory)" title="Green's function (many-body theory)">many-body Green's function</a> can be written<sup id="cite_ref-landau_8-0" class="reference"><a href="#cite_note-landau-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> (near its poles) in the form </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 G(\omega ,\mathbf {p} )\approx {\frac {Z}{\omega +\mu -\varepsilon (\mathbf {p} )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Z</mi> <mrow> <mi>ω</mi> <mo>+</mo> <mi>μ</mi> <mo>−</mo> <mi>ε</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(\omega ,\mathbf {p} )\approx {\frac {Z}{\omega +\mu -\varepsilon (\mathbf {p} )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ee0c60a6da2e34f8f5d3345b52a50be56ada4f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.442ex; height:6.009ex;" alt="{\displaystyle G(\omega ,\mathbf {p} )\approx {\frac {Z}{\omega +\mu -\varepsilon (\mathbf {p} )}}}"></span></dd></dl> <p>where <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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> is the <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</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 \varepsilon (\mathbf {p} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon (\mathbf {p} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e4d5bcfd43aaab77e3aa8619b70e12694bcdb70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.378ex; height:2.843ex;" alt="{\displaystyle \varepsilon (\mathbf {p} )}"></span> is the energy corresponding to the given momentum state and <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 Z>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4d608d0346c00e51273f906f08e2273fb9acd85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.941ex; height:2.176ex;" alt="{\displaystyle Z>0}"></span> is called the <i>quasiparticle residue</i> or <i>renormalisation constant</i> which is very characteristic of Fermi liquid theory. The spectral function for the system can be directly observed via <a href="/wiki/Angle-resolved_photoemission_spectroscopy" title="Angle-resolved photoemission spectroscopy">angle-resolved photoemission spectroscopy</a> (ARPES), and can be written (in the limit of low-lying excitations) in the form: </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 A(\mathbf {k} ,\omega )=Z\delta (\omega -v_{\rm {F}}k_{\|})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Z</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(\mathbf {k} ,\omega )=Z\delta (\omega -v_{\rm {F}}k_{\|})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/736d68e05cd35c7cdbbb0532e757800d3f486fd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.065ex; height:3.176ex;" alt="{\displaystyle A(\mathbf {k} ,\omega )=Z\delta (\omega -v_{\rm {F}}k_{\|})}"></span></dd></dl> <p>where <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 v_{\rm {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\rm {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c052762468b05e25bcadeaba5b40a1775d341d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.433ex; height:2.009ex;" alt="{\displaystyle v_{\rm {F}}}"></span> is the Fermi velocity.<sup id="cite_ref-senthil_9-0" class="reference"><a href="#cite_note-senthil-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Physically, we can say that a propagating fermion interacts with its surrounding in such a way that the net effect of the interactions is to make the fermion behave as a "dressed" fermion, altering its effective mass and other dynamical properties. These "dressed" fermions are what we think of as "quasiparticles".<sup id="cite_ref-caltech_10-0" class="reference"><a href="#cite_note-caltech-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Another important property of Fermi liquids is related to the scattering cross section for electrons. Suppose we have an electron with energy <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 \varepsilon _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e900f9bee793f99d10877ef108da074cbca60ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{1}}"></span> above the Fermi surface, and suppose it scatters with a particle in the <a href="/wiki/Composite_fermion#Fermi_sea" title="Composite fermion">Fermi sea</a> with energy <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 \varepsilon _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/453175f483471f03e90376edf8b31faeccd83e28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{2}}"></span>. By Pauli's exclusion principle, both the particles after scattering have to lie above the Fermi surface, with energies <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 \varepsilon _{3},\varepsilon _{4}>\varepsilon _{\rm {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>></mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{3},\varepsilon _{4}>\varepsilon _{\rm {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d85fec4ed2bff255ff9392b6c45d5310f0ca02f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.797ex; height:2.176ex;" alt="{\displaystyle \varepsilon _{3},\varepsilon _{4}>\varepsilon _{\rm {F}}}"></span>. Now, suppose the initial electron has energy very close to the Fermi surface <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 \varepsilon \approx \varepsilon _{\rm {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>≈</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon \approx \varepsilon _{\rm {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c84d2e0a6a5df5b433c9eb165f41237032d3390d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.571ex; height:2.009ex;" alt="{\displaystyle \varepsilon \approx \varepsilon _{\rm {F}}}"></span> Then, we have that <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 \varepsilon _{2},\varepsilon _{3},\varepsilon _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{2},\varepsilon _{3},\varepsilon _{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a84e99a78f829b27d2a5e838f21d8b724ea50cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.481ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{2},\varepsilon _{3},\varepsilon _{4}}"></span> also have to be very close to the Fermi surface. This reduces the <a href="/wiki/Phase_space" title="Phase space">phase space</a> volume of the possible states after scattering, and hence, by <a href="/wiki/Fermi%27s_golden_rule" title="Fermi's golden rule">Fermi's golden rule</a>, the <a href="/wiki/Scattering_cross_section" class="mw-redirect" title="Scattering cross section">scattering cross section</a> goes to zero. Thus we can say that the lifetime of particles at the Fermi surface goes to infinity.<sup id="cite_ref-phillips_1-2" class="reference"><a href="#cite_note-phillips-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Similarities_to_Fermi_gas">Similarities to Fermi gas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermi_liquid_theory&action=edit&section=2" title="Edit section: Similarities to Fermi gas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Fermi liquid is qualitatively analogous to the non-interacting <a href="/wiki/Fermi_gas" title="Fermi gas">Fermi gas</a>, in the following sense: The system's dynamics and thermodynamics at low excitation energies and temperatures may be described by substituting the non-interacting fermions with interacting <a href="/wiki/Quasiparticle" title="Quasiparticle">quasiparticles</a>, each of which carries the same <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>, <a href="/wiki/Electric_charge" title="Electric charge">charge</a> and <a href="/wiki/Momentum" title="Momentum">momentum</a> as the original particles. Physically these may be thought of as being particles whose motion is disturbed by the surrounding particles and which themselves perturb the particles in their vicinity. Each many-particle excited state of the interacting system may be described by listing all occupied momentum states, just as in the non-interacting system. As a consequence, quantities such as the heat capacity of the Fermi liquid behave qualitatively in the same way as in the Fermi gas (e.g. the heat capacity rises linearly with temperature). </p> <div class="mw-heading mw-heading2"><h2 id="Differences_from_Fermi_gas">Differences from Fermi gas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermi_liquid_theory&action=edit&section=3" title="Edit section: Differences from Fermi gas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following differences to the non-interacting Fermi gas arise: </p> <div class="mw-heading mw-heading3"><h3 id="Energy">Energy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermi_liquid_theory&action=edit&section=4" title="Edit section: Energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Energy" title="Energy">energy</a> of a many-particle state is not simply a sum of the single-particle energies of all occupied states. Instead, the change in energy for a given change <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 \delta n_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta n_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/039d28602b4eec226cb7b329896c149d0de2a655" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.532ex; height:2.676ex;" alt="{\displaystyle \delta n_{k}}"></span> in occupation of states <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> contains terms both linear and quadratic in <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 \delta n_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta n_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/039d28602b4eec226cb7b329896c149d0de2a655" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.532ex; height:2.676ex;" alt="{\displaystyle \delta n_{k}}"></span> (for the Fermi gas, it would only be linear, <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 \delta n_{k}\varepsilon _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta n_{k}\varepsilon _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d7a022e558e57d6868be4f6d0198c2a15f51a04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.704ex; height:2.676ex;" alt="{\displaystyle \delta n_{k}\varepsilon _{k}}"></span>, where <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 \varepsilon _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae7c5753b846b2bdee985d0816274ee2b83a1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.172ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{k}}"></span> denotes the single-particle energies). The linear contribution corresponds to renormalized single-particle energies, which involve, e.g., a change in the effective mass of particles. The quadratic terms correspond to a sort of "mean-field" interaction between quasiparticles, which is parametrized by so-called Landau Fermi liquid parameters and determines the behaviour of density oscillations (and spin-density oscillations) in the Fermi liquid. Still, these mean-field interactions do not lead to a scattering of quasi-particles with a transfer of particles between different momentum states. </p><p>The renormalization of the mass of a fluid of interacting fermions can be calculated from first principles using many-body computational techniques. For the two-dimensional <a href="/wiki/Homogeneous_electron_gas" class="mw-redirect" title="Homogeneous electron gas">homogeneous electron gas</a>, <a href="/wiki/GW_approximation" title="GW approximation">GW calculations</a><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Quantum_Monte_Carlo" title="Quantum Monte Carlo">quantum Monte Carlo</a> methods<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> have been used to calculate renormalized quasiparticle effective masses. </p> <div class="mw-heading mw-heading3"><h3 id="Specific_heat_and_compressibility">Specific heat and compressibility</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermi_liquid_theory&action=edit&section=5" title="Edit section: Specific heat and compressibility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Specific_heat" class="mw-redirect" title="Specific heat">Specific heat</a>, <a href="/wiki/Compressibility" title="Compressibility">compressibility</a>, <a href="/w/index.php?title=Spin-susceptibility&action=edit&redlink=1" class="new" title="Spin-susceptibility (page does not exist)">spin-susceptibility</a> and other quantities show the same qualitative behaviour (e.g. dependence on temperature) as in the Fermi gas, but the magnitude is (sometimes strongly) changed. </p> <div class="mw-heading mw-heading3"><h3 id="Interactions">Interactions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermi_liquid_theory&action=edit&section=6" title="Edit section: Interactions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In addition to the mean-field interactions, some weak interactions between quasiparticles remain, which lead to scattering of quasiparticles off each other. Therefore, quasiparticles acquire a finite lifetime. However, at low enough energies above the Fermi surface, this lifetime becomes very long, such that the product of excitation energy (expressed in frequency) and lifetime is much larger than one. In this sense, the quasiparticle energy is still well-defined (in the opposite limit, <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a>'s <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty relation</a> would prevent an accurate definition of the energy). </p> <div class="mw-heading mw-heading3"><h3 id="Structure">Structure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermi_liquid_theory&action=edit&section=7" title="Edit section: Structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The structure of the "bare" particles (as opposed to quasiparticle) <a href="/wiki/Green%27s_function_(many-body_theory)" title="Green's function (many-body theory)">many-body Green's function</a> is similar to that in the Fermi gas (where, for a given momentum, the Green's function in frequency space is a delta peak at the respective single-particle energy). The delta peak in the density-of-states is broadened (with a width given by the quasiparticle lifetime). In addition (and in contrast to the quasiparticle Green's function), its weight (integral over frequency) is suppressed by a quasiparticle weight factor <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 0<Z<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>Z</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<Z<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8999f9344f832ed29178b2196824f8e786e1977e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.202ex; height:2.176ex;" alt="{\displaystyle 0<Z<1}"></span>. The remainder of the total weight is in a broad "incoherent background", corresponding to the strong effects of interactions on the fermions at short time scales. </p> <div class="mw-heading mw-heading3"><h3 id="Distribution">Distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermi_liquid_theory&action=edit&section=8" title="Edit section: Distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The distribution of particles (as opposed to quasiparticles) over momentum states at zero temperature still shows a discontinuous jump at the Fermi surface (as in the Fermi gas), but it does not drop from 1 to 0: the step is only of size <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 Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Electrical_resistivity">Electrical resistivity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermi_liquid_theory&action=edit&section=9" title="Edit section: Electrical resistivity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a metal the resistivity at low temperatures is dominated by electron–electron scattering in combination with <a href="/wiki/Umklapp_scattering" title="Umklapp scattering">umklapp scattering</a>. For a Fermi liquid, the resistivity from this mechanism varies as <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 T^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f632936d25e5207ab3ad0e6182882ea3c6a9c7ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.774ex; height:2.676ex;" alt="{\displaystyle T^{2}}"></span>, which is often taken as an experimental check for Fermi liquid behaviour (in addition to the linear temperature-dependence of the specific heat), although it only arises in combination with the lattice. In certain cases, umklapp scattering is not required. For example, the resistivity of compensated <a href="/wiki/Semimetal" title="Semimetal">semimetals</a> scales as <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 T^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f632936d25e5207ab3ad0e6182882ea3c6a9c7ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.774ex; height:2.676ex;" alt="{\displaystyle T^{2}}"></span> because of mutual scattering of electron and hole. This is known as the Baber mechanism.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Optical_response">Optical response</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermi_liquid_theory&action=edit&section=10" title="Edit section: Optical response"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fermi liquid theory predicts that the scattering rate, which governs the optical response of metals, not only depends quadratically on temperature (thus causing 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 T^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f632936d25e5207ab3ad0e6182882ea3c6a9c7ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.774ex; height:2.676ex;" alt="{\displaystyle T^{2}}"></span> dependence of the DC resistance), but it also depends quadratically on frequency.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> This is in contrast to the <a href="/wiki/Drude_model" title="Drude model">Drude prediction</a> for non-interacting metallic electrons, where the scattering rate is a constant as a function of frequency. One material in which optical Fermi liquid behavior was experimentally observed is the low-temperature metallic phase of <a href="/wiki/Sr2RuO4" class="mw-redirect" title="Sr2RuO4">Sr<sub>2</sub>RuO<sub>4</sub></a>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Instabilities">Instabilities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermi_liquid_theory&action=edit&section=11" title="Edit section: Instabilities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The experimental observation of exotic phases in strongly correlated systems has triggered an enormous effort from the theoretical community to try to understand their microscopic origin. One possible route to detect instabilities of a Fermi liquid is precisely the analysis done by <a href="/wiki/Isaak_Pomeranchuk" title="Isaak Pomeranchuk">Isaak Pomeranchuk</a>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Due to that, the <a href="/wiki/Pomeranchuk_instability" title="Pomeranchuk instability">Pomeranchuk instability</a> has been studied by several authors <sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> with different techniques in the last few years and in particular, the instability of the Fermi liquid towards the nematic phase was investigated for several models. </p> <div class="mw-heading mw-heading2"><h2 id="Non-Fermi_liquids">Non-Fermi liquids</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermi_liquid_theory&action=edit&section=12" title="Edit section: Non-Fermi liquids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Non-Fermi liquids</b> are systems in which the Fermi-liquid behaviour breaks down. The simplest example is a system of interacting fermions in one dimension, called the <a href="/wiki/Luttinger_liquid" title="Luttinger liquid">Luttinger liquid</a>.<sup id="cite_ref-schulz_4-1" class="reference"><a href="#cite_note-schulz-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Although Luttinger liquids are physically similar to Fermi liquids, the restriction to one dimension gives rise to several qualitative differences such as the absence of a <i>quasiparticle peak</i> in the momentum dependent spectral function, and the presence of <a href="/wiki/Spin-charge_separation" class="mw-redirect" title="Spin-charge separation">spin-charge separation</a> and of <a href="/wiki/Spin-density_wave" class="mw-redirect" title="Spin-density wave">spin-density waves</a>. One cannot ignore the existence of interactions in one dimension and has to describe the problem with a non-Fermi theory, where Luttinger liquid is one of them. At small finite spin temperatures in one dimension the ground state of the system is described by spin-incoherent Luttinger liquid (SILL).<sup id="cite_ref-soltanieh-ha_22-0" class="reference"><a href="#cite_note-soltanieh-ha-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>Another example of non-Fermi-liquid behaviour is observed at <a href="/wiki/Quantum_critical_point" title="Quantum critical point">quantum critical points</a> of certain second-order <a href="/wiki/Phase_transition" title="Phase transition">phase transitions</a>, such as <a href="/wiki/Heavy_fermion" class="mw-redirect" title="Heavy fermion">heavy fermion</a> criticality, <a href="/wiki/Mott_insulator" title="Mott insulator">Mott criticality</a> and high-<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 T_{\rm {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\rm {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f94e364a8c6313a30beb7688803bf3996ace1c94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.32ex; height:2.509ex;" alt="{\displaystyle T_{\rm {c}}}"></span> <a href="/wiki/High-temperature_superconductivity#Cuprates" title="High-temperature superconductivity">cuprate</a> phase transitions.<sup id="cite_ref-senthil_9-1" class="reference"><a href="#cite_note-senthil-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The ground state of such transitions is characterized by the presence of a sharp Fermi surface, although there may not be well-defined quasiparticles. That is, on approaching the critical point, it is observed that the quasiparticle residue <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 Z\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bfbe5f7e028c165befb0461d6f4b037dccbf1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.457ex; height:2.176ex;" alt="{\displaystyle Z\to 0}"></span>. </p><p>In optimally doped cuprates and iron-based superconductors, the normal state above the critical temperature shows signs of non-Fermi liquid behaviour, and is often called a <b>strange metal</b>. In this region of phase diagram, resistivity increases linearly in temperature and the Hall coefficient is found to depend on temperature.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>Understanding the behaviour of non-Fermi liquids is an important problem in condensed matter physics. Approaches towards explaining these phenomena include the treatment of <i>marginal Fermi liquids</i>; attempts to understand critical points and derive <a href="/wiki/Critical_scaling" class="mw-redirect" title="Critical scaling">scaling relations</a>; and descriptions using <i>emergent</i> <a href="/wiki/Gauge_theory" title="Gauge theory">gauge theories</a> with techniques of <a href="/wiki/Holographic_principle" title="Holographic principle">holographic</a> gauge/gravity duality.<sup id="cite_ref-polchinsky_25-0" class="reference"><a href="#cite_note-polchinsky-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermi_liquid_theory&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Classical_fluid" title="Classical fluid">Classical fluid</a></li> <li><a href="/wiki/Fermionic_condensate" title="Fermionic condensate">Fermionic condensate</a></li> <li><a href="/wiki/Luttinger_liquid" title="Luttinger liquid">Luttinger liquid</a></li> <li><a href="/wiki/Luttinger%27s_theorem" title="Luttinger's theorem">Luttinger's theorem</a></li> <li><a href="/wiki/Strongly_correlated_quantum_spin_liquid" class="mw-redirect" title="Strongly correlated quantum spin liquid">Strongly correlated quantum spin liquid</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermi_liquid_theory&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-phillips-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-phillips_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-phillips_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-phillips_1-2"><sup><i><b>c</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="CITEREFPhillips2008" class="citation book cs1">Phillips, Philip (2008). <i>Advanced Solid State Physics</i>. Perseus Books. p. 224. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-89938-16-1" title="Special:BookSources/978-81-89938-16-1"><bdi>978-81-89938-16-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Solid+State+Physics&rft.pages=224&rft.pub=Perseus+Books&rft.date=2008&rft.isbn=978-81-89938-16-1&rft.aulast=Phillips&rft.aufirst=Philip&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermi+liquid+theory" 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="CITEREFLandau1957" class="citation journal cs1">Landau, L. D. (1957). <a rel="nofollow" class="external text" href="http://www.jetp.ras.ru/cgi-bin/dn/e_003_06_0920.pdf">"The Theory of a Fermi Liquid"</a> <span class="cs1-format">(PDF)</span>. <i>Sov. Phys. JETP</i>. <b>3</b> (6): <span class="nowrap">920–</span>925.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sov.+Phys.+JETP&rft.atitle=The+Theory+of+a+Fermi+Liquid&rft.volume=3&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E920-%3C%2Fspan%3E925&rft.date=1957&rft.aulast=Landau&rft.aufirst=L.+D.&rft_id=http%3A%2F%2Fwww.jetp.ras.ru%2Fcgi-bin%2Fdn%2Fe_003_06_0920.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermi+liquid+theory" 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="CITEREFAbrikosov,_A.A.Khalatnikov,_I.M.1959" class="citation journal cs1">Abrikosov, A.A.; Khalatnikov, I.M. (1959). 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(2020-07-07). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/RevModPhys.92.031001">"Colloquium : Linear in temperature resistivity and associated mysteries including high temperature superconductivity"</a>. <i>Reviews of Modern Physics</i>. <b>92</b> (3): 031001. <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/1908.05686">1908.05686</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/2020RvMP...92c1001V">2020RvMP...92c1001V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.92.031001">10.1103/RevModPhys.92.031001</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0034-6861">0034-6861</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Colloquium+%3A+Linear+in+temperature+resistivity+and+associated+mysteries+including+high+temperature+superconductivity&rft.volume=92&rft.issue=3&rft.pages=031001&rft.date=2020-07-07&rft_id=info%3Aarxiv%2F1908.05686&rft.issn=0034-6861&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.92.031001&rft_id=info%3Abibcode%2F2020RvMP...92c1001V&rft.aulast=Varma&rft.aufirst=Chandra+M.&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FRevModPhys.92.031001&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermi+liquid+theory" class="Z3988"></span></span> </li> <li id="cite_note-polchinsky-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-polchinsky_25-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFaulknerPolchinski,_Joseph2010" class="citation journal cs1">Faulkner, Thomas; Polchinski, Joseph (2010). "Semi-Holographic Fermi Liquids". <i>Journal of High Energy Physics</i>. <b>2011</b> (6): 12. <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/1001.5049">1001.5049</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/2011JHEP...06..012F">2011JHEP...06..012F</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.755.3304">10.1.1.755.3304</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FJHEP06%282011%29012">10.1007/JHEP06(2011)012</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119243857">119243857</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+High+Energy+Physics&rft.atitle=Semi-Holographic+Fermi+Liquids&rft.volume=2011&rft.issue=6&rft.pages=12&rft.date=2010&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119243857%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2011JHEP...06..012F&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.755.3304%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1007%2FJHEP06%282011%29012&rft_id=info%3Aarxiv%2F1001.5049&rft.aulast=Faulkner&rft.aufirst=Thomas&rft.au=Polchinski%2C+Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermi+liquid+theory" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuoGuSachdev2020" class="citation journal cs1">Guo, Haoyu; Gu, Yingfei; Sachdev, Subir (2020). "Linear in temperature resistivity in the limit of zero temperature from the time reparameterization soft mode". <i>Annals of Physics</i>. <b>418</b>: 168202. <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/2004.05182">2004.05182</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/2020AnPhy.41868202G">2020AnPhy.41868202G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.aop.2020.168202">10.1016/j.aop.2020.168202</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Physics&rft.atitle=Linear+in+temperature+resistivity+in+the+limit+of+zero+temperature+from+the+time+reparameterization+soft+mode&rft.volume=418&rft.pages=168202&rft.date=2020&rft_id=info%3Aarxiv%2F2004.05182&rft_id=info%3Adoi%2F10.1016%2Fj.aop.2020.168202&rft_id=info%3Abibcode%2F2020AnPhy.41868202G&rft.aulast=Guo&rft.aufirst=Haoyu&rft.au=Gu%2C+Yingfei&rft.au=Sachdev%2C+Subir&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermi+liquid+theory" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeiSedrakyan2023" class="citation journal cs1">Wei, Chenan; Sedrakyan, Tigran A. (2023-08-02). "Strange metal phase of disordered magic-angle twisted bilayer graphene at low temperatures: From flat bands to weakly coupled Sachdev-Ye-Kitaev bundles". <i>Physical Review B</i>. <b>108</b> (6): 064202. <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/2205.09766">2205.09766</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/2023PhRvB.108f4202W">2023PhRvB.108f4202W</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.108.064202">10.1103/PhysRevB.108.064202</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2469-9950">2469-9950</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+B&rft.atitle=Strange+metal+phase+of+disordered+magic-angle+twisted+bilayer+graphene+at+low+temperatures%3A+From+flat+bands+to+weakly+coupled+Sachdev-Ye-Kitaev+bundles&rft.volume=108&rft.issue=6&rft.pages=064202&rft.date=2023-08-02&rft_id=info%3Aarxiv%2F2205.09766&rft.issn=2469-9950&rft_id=info%3Adoi%2F10.1103%2FPhysRevB.108.064202&rft_id=info%3Abibcode%2F2023PhRvB.108f4202W&rft.aulast=Wei&rft.aufirst=Chenan&rft.au=Sedrakyan%2C+Tigran+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermi+liquid+theory" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fermi_liquid_theory&action=edit&section=15" 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="CITEREFBaymPethick1991" class="citation book cs1">Baym, Gordon; Pethick, Christopher (1991). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/book/10.1002/9783527617159"><i>Landau Fermi-Liquid Theory: Concepts and Applications</i></a> (1 ed.). Wiley. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F9783527617159">10.1002/9783527617159</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-82418-3" title="Special:BookSources/978-0-471-82418-3"><bdi>978-0-471-82418-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Landau+Fermi-Liquid+Theory%3A+Concepts+and+Applications&rft.edition=1&rft.pub=Wiley&rft.date=1991&rft_id=info%3Adoi%2F10.1002%2F9783527617159&rft.isbn=978-0-471-82418-3&rft.aulast=Baym&rft.aufirst=Gordon&rft.au=Pethick%2C+Christopher&rft_id=https%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2Fbook%2F10.1002%2F9783527617159&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermi+liquid+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFColeman2015" class="citation book cs1">Coleman, Piers (2015). "Landau Fermi-liquid theory". <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/books/introduction-to-manybody-physics/B7598FC1FCEE0285F5EC767E835854C8"><i>Introduction to Many-Body Physics</i></a>. Cambridge, U.K.: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521864886" title="Special:BookSources/9780521864886"><bdi>9780521864886</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Landau+Fermi-liquid+theory&rft.btitle=Introduction+to+Many-Body+Physics&rft.place=Cambridge%2C+U.K.&rft.pub=Cambridge+University+Press&rft.date=2015&rft.isbn=9780521864886&rft.aulast=Coleman&rft.aufirst=Piers&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fbooks%2Fintroduction-to-manybody-physics%2FB7598FC1FCEE0285F5EC767E835854C8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermi+liquid+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPinesNozières1989" class="citation book cs1">Pines, David; Nozières, Philippe (1989). <a rel="nofollow" class="external text" href="https://www.taylorfrancis.com/books/9780429961212"><i>The Theory of Quantum Liquids: Normal Fermi Liquids</i></a>. CRC Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4324%2F9780429492662">10.4324/9780429492662</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-429-49266-2" title="Special:BookSources/978-0-429-49266-2"><bdi>978-0-429-49266-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Quantum+Liquids%3A+Normal+Fermi+Liquids&rft.pub=CRC+Press&rft.date=1989&rft_id=info%3Adoi%2F10.4324%2F9780429492662&rft.isbn=978-0-429-49266-2&rft.aulast=Pines&rft.aufirst=David&rft.au=Nozi%C3%A8res%2C+Philippe&rft_id=https%3A%2F%2Fwww.taylorfrancis.com%2Fbooks%2F9780429961212&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermi+liquid+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVignale2022" class="citation book cs1">Vignale, Giovanni (2022). <a rel="nofollow" class="external text" href="https://www.cond-mat.de/events/correl22/manuscripts/vignale.pdf">"Fermi Liquids"</a> <span class="cs1-format">(PDF)</span>. In Pavarini, Eva; Koch, Erik; Lichtenstein, Alexander; Vollhardt, Dieter (eds.). <i>Dynamical Mean-Field Theory of Correlated Electrons</i>. Verlag des Forschungszentrum Jülich. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-95806-619-9" title="Special:BookSources/978-3-95806-619-9"><bdi>978-3-95806-619-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Fermi+Liquids&rft.btitle=Dynamical+Mean-Field+Theory+of+Correlated+Electrons&rft.pub=Verlag+des+Forschungszentrum+J%C3%BClich&rft.date=2022&rft.isbn=978-3-95806-619-9&rft.aulast=Vignale&rft.aufirst=Giovanni&rft_id=https%3A%2F%2Fwww.cond-mat.de%2Fevents%2Fcorrel22%2Fmanuscripts%2Fvignale.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFermi+liquid+theory" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid 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condensate</a></li> <li><a href="/wiki/Fermionic_condensate" title="Fermionic condensate">Fermionic condensate</a></li> <li><a href="/wiki/Fermi_gas" title="Fermi gas">Fermi gas</a></li> <li><a href="/wiki/Supersolid" title="Supersolid">Supersolid</a></li> <li><a href="/wiki/Superfluidity" title="Superfluidity">Superfluid</a></li> <li><a href="/wiki/Luttinger_liquid" title="Luttinger liquid">Luttinger liquid</a></li> <li><a href="/wiki/Time_crystal" title="Time crystal">Time crystal</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:QuantumPhaseTransition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/220px-QuantumPhaseTransition.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/330px-QuantumPhaseTransition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/QuantumPhaseTransition.svg/440px-QuantumPhaseTransition.svg.png 2x" data-file-width="512" data-file-height="369" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Phase phenomena</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Order_parameter" class="mw-redirect" title="Order parameter">Order parameter</a></li> <li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li> <li><a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li> <li><a href="/wiki/Critical_phenomena" title="Critical phenomena">Critical phenomena</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Electrons in solids</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Phenomena</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hall_effect" title="Hall effect">Hall effect</a></li> <li><a href="/wiki/Quantum_Hall_effect" title="Quantum Hall effect">Quantum Hall effect</a></li> <li><a href="/wiki/Spin_Hall_effect" title="Spin Hall effect">Spin Hall effect</a></li> <li><a href="/wiki/Quantum_spin_Hall_effect" title="Quantum spin Hall effect">Quantum spin Hall effect</a></li> <li><a href="/wiki/Berry_phase" class="mw-redirect" title="Berry phase">Berry phase</a></li> <li><a href="/wiki/Aharonov%E2%80%93Bohm_effect" title="Aharonov–Bohm effect">Aharonov–Bohm effect</a></li> <li><a href="/wiki/Josephson_effect" title="Josephson effect">Josephson effect</a></li> <li><a href="/wiki/Kondo_effect" title="Kondo effect">Kondo effect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Drude_model" title="Drude model">Drude model</a></li> <li><a href="/wiki/Free_electron_model" title="Free electron model">Free electron model</a></li> <li><a href="/wiki/Nearly_free_electron_model" title="Nearly free electron model">Nearly free electron model</a></li> <li><a href="/wiki/Bloch%27s_theorem" title="Bloch's theorem">Bloch's theorem</a></li> <li><a class="mw-selflink selflink">Fermi liquid theory</a></li> <li><a href="/wiki/Electronic_band_structure" title="Electronic band structure">electronic band structure</a></li> <li><a href="/wiki/Anderson_localization" title="Anderson localization">Anderson localization</a></li> <li><a href="/wiki/BCS_theory" title="BCS theory">BCS theory</a></li> <li><a href="/wiki/Tight_binding_model" class="mw-redirect" title="Tight binding model">tight binding model</a></li> <li><a href="/wiki/Hubbard_model" title="Hubbard model">Hubbard model</a></li> <li><a href="/wiki/Density_functional_theory" title="Density functional theory">Density functional theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Conduction</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li> <li><a href="/wiki/Mott_insulator" title="Mott insulator">Mott insulator</a></li> <li><a href="/wiki/Semiconductor" title="Semiconductor">Semiconductor</a></li> <li><a href="/wiki/Semimetal" title="Semimetal">Semimetal</a></li> <li><a href="/wiki/Electrical_conductor" title="Electrical conductor">Conductor</a></li> <li><a href="/wiki/Superconductivity" title="Superconductivity">Superconductor</a></li> <li><a href="/wiki/Topological_insulator" title="Topological insulator">Topological insulator</a></li> <li><a href="/wiki/Spin_gapless_semiconductor" title="Spin gapless semiconductor">Spin gapless semiconductor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Couplings</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Thermoelectric_effect" title="Thermoelectric effect">Thermoelectricity</a></li> <li><a href="/wiki/Piezoelectricity" title="Piezoelectricity">Piezoelectricity</a></li> <li><a href="/wiki/Ferroelectricity" title="Ferroelectricity">Ferroelectricity</a></li> <li><a href="/wiki/Flexoelectricity" title="Flexoelectricity">Flexoelectricity</a></li> <li><a href="/wiki/Electrostriction" title="Electrostriction">Electrostriction</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Magnetic phases</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amorphous_magnet" title="Amorphous magnet">Amorphous magnet</a></li> <li><a href="/wiki/Diamagnetism" title="Diamagnetism">Diamagnet</a></li> <li><a href="/wiki/Superdiamagnetism" title="Superdiamagnetism">Superdiamagnet</a></li> <li><a href="/wiki/Paramagnetism" title="Paramagnetism">Paramagnet</a></li> <li><a href="/wiki/Superparamagnetism" title="Superparamagnetism">Superparamagnet</a></li> <li><a href="/wiki/Ferromagnetism" title="Ferromagnetism">Ferromagnet</a></li> <li><a href="/wiki/Antiferromagnetism" title="Antiferromagnetism">Antiferromagnet</a></li> <li><a href="/wiki/Metamagnetism" title="Metamagnetism">Metamagnet</a></li> <li><a href="/wiki/Spin_glass" title="Spin glass">Spin glass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quasiparticle" title="Quasiparticle">Quasiparticles</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anyon" title="Anyon">Anyon</a></li> <li><a href="/wiki/Bogoliubov_quasiparticle" title="Bogoliubov quasiparticle">Bogoliubov quasiparticle</a></li> <li><a href="/wiki/Electron_hole" title="Electron hole">Hole</a></li> <li><a href="/wiki/Exciton" title="Exciton">Exciton</a></li> <li><a href="/wiki/Magnon" title="Magnon">Magnon</a></li> <li><a href="/wiki/Phonon" title="Phonon">Phonon</a></li> <li><a href="/wiki/Pines%27_demon" title="Pines' demon">Pines' demon</a></li> <li><a href="/wiki/Plasmon" title="Plasmon">Plasmon</a></li> <li><a href="/wiki/Polariton" title="Polariton">Polariton</a> <ul><li><a href="/wiki/Exciton-polariton" title="Exciton-polariton">Exciton-polariton</a></li> <li><a href="/wiki/Phonon_polariton" title="Phonon polariton">Phonon polariton</a></li></ul></li> <li><a href="/wiki/Polaron" title="Polaron">Polaron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Soft_matter" title="Soft matter">Soft matter</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amorphous_solid" title="Amorphous solid">Amorphous solid</a></li> <li><a href="/wiki/Granular_material" title="Granular material">Granular matter</a></li> <li><a href="/wiki/Liquid_crystal" title="Liquid crystal">Liquid crystal</a></li> <li><a href="/wiki/Polymer" title="Polymer">Polymer</a></li> <li><a href="/wiki/Interface_and_colloid_science" title="Interface and colloid science">Colloids</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" 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