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<!DOCTYPE html> <html lang="en" class="grade-c"> <head> <title>Plasmarons in high-temperature cuprate superconductors | Communications Physics</title> <link rel="alternate" type="application/rss+xml" href="https://www.nature.com/commsphys.rss"/> <link rel="preconnect" href="https://cmp.nature.com" crossorigin> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width,initial-scale=1.0,maximum-scale=5,user-scalable=yes"> <meta name="360-site-verification" content="5a2dc4ab3fcb9b0393241ffbbb490480" /> <script data-test="dataLayer"> window.dataLayer = [{"content":{"category":{"contentType":"article","legacy":{"webtrendsPrimaryArticleType":"research","webtrendsSubjectTerms":"electronic-properties-and-materials;superconducting-properties-and-materials","webtrendsContentCategory":null,"webtrendsContentCollection":null,"webtrendsContentGroup":"Communications 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This fundamental concept was reinforced also in high-temperature cuprate superconductors recently, although cuprates are not only layered systems but also strongly correlated electron systems. Here, we study how such ubiquitous plasmons leave their marks on the electron dispersion in cuprates. In contrast to phonons and magnetic fluctuations, plasmons do not yield a kink in the electron dispersion. Instead, we find that the optical plasmon accounts for an emergent band—plasmarons—in the one-particle excitation spectrum; acoustic-like plasmons typical to a layered system are far less effective. Because of strong electron correlations, the plasmarons are generated by bosonic fluctuations associated with the local constraint, not by the usual charge-density fluctuations. Apart from this physical mechanism, the plasmarons are similar to those discussed in alkali metals, Bi, graphene, monolayer transition-metal dichalcogenides, semiconductors, diamond, two-dimensional electron systems, and SrIrO3 films, establishing a concept of plasmarons in metallic systems in general. Plasmarons are realized below (above) the quasiparticle band in electron-doped (hole-doped) cuprates, including a region around (π, 0) and (0, π) where the superconducting gap and the pseudogap are most enhanced. Recent research has revealed plasmon excitations in high-Tc cuprates, raising an important question—how do they renormalize the electron property? 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This fundamental concept was reinforced also in high-temperature cuprate superconductors recently, although cuprates are not only layered systems but also strongly correlated electron systems. Here, we study how such ubiquitous plasmons leave their marks on the electron dispersion in cuprates. In contrast to phonons and magnetic fluctuations, plasmons do not yield a kink in the electron dispersion. Instead, we find that the optical plasmon accounts for an emergent band—plasmarons—in the one-particle excitation spectrum; acoustic-like plasmons typical to a layered system are far less effective. Because of strong electron correlations, the plasmarons are generated by bosonic fluctuations associated with the local constraint, not by the usual charge-density fluctuations. Apart from this physical mechanism, the plasmarons are similar to those discussed in alkali metals, Bi, graphene, monolayer transition-metal dichalcogenides, semiconductors, diamond, two-dimensional electron systems, and SrIrO3 films, establishing a concept of plasmarons in metallic systems in general. Plasmarons are realized below (above) the quasiparticle band in electron-doped (hole-doped) cuprates, including a region around (π, 0) and (0, π) where the superconducting gap and the pseudogap are most enhanced. Recent research has revealed plasmon excitations in high-Tc cuprates, raising an important question—how do they renormalize the electron property? 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Apart from this physical mechanism, the plasmarons are similar to those discussed in alkali metals, Bi, graphene, monolayer transition-metal dichalcogenides, semiconductors, diamond, two-dimensional electron systems, and SrIrO3 films, establishing a concept of plasmarons in metallic systems in general. Plasmarons are realized below (above) the quasiparticle band in electron-doped (hole-doped) cuprates, including a region around (π, 0) and (0, π) where the superconducting gap and the pseudogap are most enhanced. Recent research has revealed plasmon excitations in high-Tc cuprates, raising an important question—how do they renormalize the electron property? 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class="c-article-header"> <header> <ul class="c-article-identifiers" data-test="article-identifier"> <li class="c-article-identifiers__item" data-test="article-category">Article</li> <li class="c-article-identifiers__item"> <a href="https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research" data-track="click" data-track-action="open access" data-track-label="link" class="u-color-open-access" data-test="open-access">Open access</a> </li> <li class="c-article-identifiers__item">Published: <time datetime="2023-07-08">08 July 2023</time></li> </ul> <h1 class="c-article-title" data-test="article-title" data-article-title="">Plasmarons in high-temperature cuprate superconductors</h1> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Hiroyuki-Yamase-Aff1" data-author-popup="auth-Hiroyuki-Yamase-Aff1" data-author-search="Yamase, Hiroyuki" data-corresp-id="c1">Hiroyuki Yamase<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0003-0328-5657"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0003-0328-5657</a></span><sup class="u-js-hide"><a href="#Aff1">1</a></sup>, </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Mat_as-Bejas-Aff2" data-author-popup="auth-Mat_as-Bejas-Aff2" data-author-search="Bejas, Matías">Matías Bejas</a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0003-4254-0622"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0003-4254-0622</a></span><sup class="u-js-hide"><a href="#Aff2">2</a></sup> & </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Andr_s-Greco-Aff2" data-author-popup="auth-Andr_s-Greco-Aff2" data-author-search="Greco, Andrés">Andrés Greco</a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0001-5958-5080"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0001-5958-5080</a></span><sup class="u-js-hide"><a href="#Aff2">2</a></sup> </li></ul> <p class="c-article-info-details" data-container-section="info"> <a data-test="journal-link" href="/commsphys" data-track="click" data-track-action="journal homepage" data-track-category="article body" data-track-label="link"><i data-test="journal-title">Communications Physics</i></a> <b data-test="journal-volume"><span class="u-visually-hidden">volume</span> 6</b>, Article number: <span data-test="article-number">168</span> (<span data-test="article-publication-year">2023</span>) <a href="#citeas" class="c-article-info-details__cite-as u-hide-print" data-track="click" data-track-action="cite this article" data-track-label="link">Cite this article</a> </p> <div class="c-article-metrics-bar__wrapper u-clear-both"> <ul class="c-article-metrics-bar u-list-reset"> <li class=" c-article-metrics-bar__item" data-test="access-count"> <p class="c-article-metrics-bar__count">1393 <span class="c-article-metrics-bar__label">Accesses</span></p> </li> <li class="c-article-metrics-bar__item" data-test="altmetric-score"> <p class="c-article-metrics-bar__count">4 <span class="c-article-metrics-bar__label">Altmetric</span></p> </li> <li class="c-article-metrics-bar__item"> <p class="c-article-metrics-bar__details"><a href="/articles/s42005-023-01276-z/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Metrics <span class="u-visually-hidden">details</span></a></p> </li> </ul> </div> </header> <div class="u-js-hide" data-component="article-subject-links"> <h3 class="c-article__sub-heading">Subjects</h3> <ul class="c-article-subject-list"> <li class="c-article-subject-list__subject"><a href="/subjects/electronic-properties-and-materials" data-track="click" data-track-action="view subject" data-track-label="link">Electronic properties and materials</a></li><li class="c-article-subject-list__subject"><a href="/subjects/superconducting-properties-and-materials" data-track="click" data-track-action="view subject" data-track-label="link">Superconducting properties and materials</a></li> </ul> </div> </div> <div class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>Metallic systems exhibit plasmons as elementary charge excitations. This fundamental concept was reinforced also in high-temperature cuprate superconductors recently, although cuprates are not only layered systems but also strongly correlated electron systems. Here, we study how such ubiquitous plasmons leave their marks on the electron dispersion in cuprates. In contrast to phonons and magnetic fluctuations, plasmons do not yield a kink in the electron dispersion. Instead, we find that the optical plasmon accounts for an emergent band—plasmarons—in the one-particle excitation spectrum; acoustic-like plasmons typical to a layered system are far less effective. Because of strong electron correlations, the plasmarons are generated by bosonic fluctuations associated with the local constraint, not by the usual charge-density fluctuations. Apart from this physical mechanism, the plasmarons are similar to those discussed in alkali metals, Bi, graphene, monolayer transition-metal dichalcogenides, semiconductors, diamond, two-dimensional electron systems, and SrIrO<sub>3</sub> films, establishing a concept of plasmarons in metallic systems in general. Plasmarons are realized below (above) the quasiparticle band in electron-doped (hole-doped) cuprates, including a region around (<i>π</i>, 0) and (0, <i>π</i>) where the superconducting gap and the pseudogap are most enhanced.</p></div></div></section> <noscript> </noscript> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41467-023-38408-6/MediaObjects/41467_2023_38408_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41467-023-38408-6?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1038/s41467-023-38408-6">Traces of electron-phonon coupling in one-dimensional cuprates </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">30 May 2023</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs42005-023-01373-z/MediaObjects/42005_2023_1373_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s42005-023-01373-z?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1038/s42005-023-01373-z">Investigating the Cuprates as a platform for high-order Van Hove singularities and flat-band physics </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">13 October 2023</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41467-022-30918-z/MediaObjects/41467_2022_30918_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41467-022-30918-z?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1038/s41467-022-30918-z">Paramagnons and high-temperature superconductivity in a model family of cuprates </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">07 June 2022</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1732705975, embedded_user: 'null' } }); </script> <div class="main-content"> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1">Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>Superconductivity is driven by forming Cooper pairs of electrons<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Cooper, L. N. Bound electron pairs in a degenerate Fermi gas. Phys. Rev. 104, 1189–1190 (1956)." href="/articles/s42005-023-01276-z#ref-CR1" id="ref-link-section-d10251175e375">1</a></sup>. This is achieved even above the boiling point of liquid nitrogen at ambient pressure in cuprate superconductors<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature 518, 179 (2015)." href="/articles/s42005-023-01276-z#ref-CR2" id="ref-link-section-d10251175e379">2</a></sup>. The mechanism of such a remarkable phenomenon has been a central issue in condensed matter physics since their discovery<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Bednorz, J. G. & Müller, K. A. Possible high Tc superconductivity in the Ba–La–Cu–O system. Z. Phys. B: Condens. Matter 64, 189–193 (1986)." href="/articles/s42005-023-01276-z#ref-CR3" id="ref-link-section-d10251175e383">3</a></sup>. The one-particle property of electrons may possess an important hint to solve it. In fact, electrons are not independent inside a material, but acquire the self-energy through various interactions—it is such electrons which drive the superconductivity.</p><p>A well-known interaction is electron–phonon coupling, which renormalizes the electron band to yield a kink at the corresponding phonon energy<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Mahan, G. D. Many-Particle Physics 2nd edn (Plunum Press, 1990)." href="/articles/s42005-023-01276-z#ref-CR4" id="ref-link-section-d10251175e390">4</a></sup>. A kink was actually observed in not only conventional metals<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Hengsberger, M., Purdie, D., Segovia, P., Garnier, M. & Baer, Y. Photoemission study of a strongly coupled electron-phonon system. Phys. Rev. Lett. 83, 592–595 (1999)." href="/articles/s42005-023-01276-z#ref-CR5" id="ref-link-section-d10251175e394">5</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Valla, T., Fedorov, A. V., Johnson, P. D. & Hulbert, S. L. Many-body effects in angle-resolved photoemission: quasiparticle energy and lifetime of a Mo(110) surface state. Phys. Rev. Lett. 83, 2085–2088 (1999)." href="/articles/s42005-023-01276-z#ref-CR6" id="ref-link-section-d10251175e397">6</a></sup> but also high-temperature cuprates superconductors<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Lanzara, A. et al. Evidence for ubiquitous strong electron–phonon coupling in high-temperature superconductors. Nature 412, 510–514 (2001)." href="/articles/s42005-023-01276-z#ref-CR7" id="ref-link-section-d10251175e401">7</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Zhou, X. J. et al. Multiple bosonic mode coupling in the electron self-energy of (La2−xSrx)CuO4. Phys. Rev. Lett. 95, 117001 (2005)." href="/articles/s42005-023-01276-z#ref-CR8" id="ref-link-section-d10251175e404">8</a></sup>. Given that the electron–phonon coupling is the conventional mechanism of superconductivity<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Bardeen, J., Cooper, L. N. & Schrieffer, J. R. Theory of superconductivity. Phys. Rev. 108, 1175–1204 (1957)." href="/articles/s42005-023-01276-z#ref-CR9" id="ref-link-section-d10251175e408">9</a></sup>, its role in the high-<i>T</i><sub>c</sub> mechanism drew much attention. On the other hand, it is also well recognized that the formation of a kink is not a special feature of electron–phonon coupling, but rather a manifestation of coupling to bosonic fluctuations from a general point of view<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Carbotte, J. P., Timusk, T. & Hwang, J. Bosons in high-temperature superconductors: an experimental survey. Rep. Prog. Phys. 74, 066501 (2011)." href="/articles/s42005-023-01276-z#ref-CR10" id="ref-link-section-d10251175e417">10</a></sup>. Magnetic fluctuations are bosonic fluctuations and can in fact yield a kink in the electron dispersion at the energy of the magnetic resonance mode<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Kaminski, A. et al. Renormalization of spectral line shape and dispersion below Tc in Bi2Sr2CaCu2O8+δ. Phys. Rev. Lett. 86, 1070–1073 (2001)." href="#ref-CR11" id="ref-link-section-d10251175e421">11</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Johnson, P. D. et al. Doping and temperature dependence of the mass enhancement observed in the cuprate Bi2Sr2CaCu2O8+δ. Phys. Rev. Lett. 87, 177007 (2001)." href="#ref-CR12" id="ref-link-section-d10251175e421_1">12</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Gromko, A. D. et al. Mass-renormalized electronic excitations at (π, 0) in the superconducting state of Bi2Sr2CaCu2O8+δ. Phys. Rev. B 68, 174520 (2003)." href="#ref-CR13" id="ref-link-section-d10251175e421_2">13</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Mou, Y. & Feng, S. Doping dependence of charge order in electron-doped cuprate superconductors. Philos. Mag. 97, 3361–3380 (2017)." href="/articles/s42005-023-01276-z#ref-CR14" id="ref-link-section-d10251175e424">14</a></sup>.</p><p>How about charge fluctuations, which are also bosonic ones? While the understanding of the charge fluctuations is crucial to the high-<i>T</i><sub>c</sub> mechanism in cuprates, it is only recently when the charge dynamics was revealed in momentum-energy space by an advanced technique of resonance inelastic X-ray scattering (RIXS). In particular, low-energy collective charge excitations were revealed in electron-doped<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Hepting, M. et al. Three-dimensional collective charge excitations in electron-doped copper oxide superconductors. Nature 563, 374–378 (2018)." href="#ref-CR15" id="ref-link-section-d10251175e435">15</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lin, J. et al. Doping evolution of the charge excitations and electron correlations in electron-doped superconducting La2−xCexCuO4. npj Quantum Mater. 5, 4 (2020)." href="#ref-CR16" id="ref-link-section-d10251175e435_1">16</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Hepting, M. et al. Gapped collective charge excitations and interlayer hopping in cuprate superconductors. Phys. Rev. Lett. 129, 047001 (2022)." href="/articles/s42005-023-01276-z#ref-CR17" id="ref-link-section-d10251175e438">17</a></sup> and hole-doped cuprates<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Nag, A. et al. Detection of acoustic plasmons in hole-doped lanthanum and bismuth cuprate superconductors using resonant inelastic x-ray scattering. Phys. Rev. Lett. 125, 257002 (2020)." href="/articles/s42005-023-01276-z#ref-CR18" id="ref-link-section-d10251175e442">18</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Singh, A. et al. Acoustic plasmons and conducting carriers in hole-doped cuprate superconductors. Phys. Rev. B 105, 235105 (2022)." href="/articles/s42005-023-01276-z#ref-CR19" id="ref-link-section-d10251175e445">19</a></sup>. The characteristic in-plane and out-of-plane dependence allowed to identify these excitations as low-energy plasmons<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Greco, A., Yamase, H. & Bejas, M. Plasmon excitations in layered high-Tc cuprates. Phys. Rev. B 94, 075139 (2016)." href="#ref-CR20" id="ref-link-section-d10251175e449">20</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Greco, A., Yamase, H. & Bejas, M. Origin of high-energy charge excitations observed by resonant inelastic x-ray scattering in cuprate superconductors. Commun. Phys. 2, 3 (2019)." href="#ref-CR21" id="ref-link-section-d10251175e449_1">21</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Greco, A., Yamase, H. & Bejas, M. Close inspection of plasmon excitations in cuprate superconductors. Phys. Rev. B 102, 024509 (2020)." href="#ref-CR22" id="ref-link-section-d10251175e449_2">22</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Fidrysiak, M. & Spałek, J. Unified theory of spin and charge excitations in high-Tc cuprate superconductors: a quantitative comparison with experiment and interpretation. Phys. Rev. B 104, L020510 (2021)." href="/articles/s42005-023-01276-z#ref-CR23" id="ref-link-section-d10251175e452">23</a></sup>, which were also discussed for layered metallic systems in the 1970s<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Grecu, D. Plasma frequency of the electron gas in layered structures. Phys. Rev. B 8, 1958–1961 (1973)." href="#ref-CR24" id="ref-link-section-d10251175e456">24</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Fetter, A. L. Electrodynamics of a layered electron gas. II. Periodic array. Ann. Phys. 88, 1 (1974)." href="#ref-CR25" id="ref-link-section-d10251175e456_1">25</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title="Grecu, D. Self-consistent field approximation for the plasma frequencies of an electron gas in a layered thin film. J. Phys. C: Solid State Phys. 8, 2627–2641 (1975)." href="/articles/s42005-023-01276-z#ref-CR26" id="ref-link-section-d10251175e459">26</a></sup>. Dispersing charge modes were reported previously by other groups, too, but interpreted differently, not as plasmons<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Ishii, K. et al. Momentum dependence of charge excitations in the electron-doped superconductor Nd1.85Ce0.15CuO4: a resonant inelastic X-ray scattering study. Phys. Rev. Lett. 94, 207003 (2005)." href="#ref-CR27" id="ref-link-section-d10251175e464">27</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lee, W. S. et al. Asymmetry of collective excitations in electron- and hole-doped cuprate superconductors. Nat. Phys. 10, 883–889 (2014)." href="#ref-CR28" id="ref-link-section-d10251175e464_1">28</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Ishii, K. et al. High-energy spin and charge excitations in electron-doped copper oxide superconductors. Nat. Commun. 5, 3714 (2014)." href="#ref-CR29" id="ref-link-section-d10251175e464_2">29</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Ishii, K. et al. Observation of momentum-dependent charge excitations in hole-doped cuprates using resonant inelastic x-ray scattering at the oxygen K edge. Phys. Rev. B 96, 115148 (2017)." href="#ref-CR30" id="ref-link-section-d10251175e464_3">30</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Dellea, G. et al. Spin and charge excitations in artificial hole- and electron-doped infinite layer cuprate superconductors. Phys. Rev. B 96, 115117 (2017)." href="/articles/s42005-023-01276-z#ref-CR31" id="ref-link-section-d10251175e467">31</a></sup>. While the dispersion was presumed to be likely an acoustic mode<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Hepting, M. et al. Three-dimensional collective charge excitations in electron-doped copper oxide superconductors. Nature 563, 374–378 (2018)." href="/articles/s42005-023-01276-z#ref-CR15" id="ref-link-section-d10251175e471">15</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Lin, J. et al. Doping evolution of the charge excitations and electron correlations in electron-doped superconducting La2−xCexCuO4. npj Quantum Mater. 5, 4 (2020)." href="/articles/s42005-023-01276-z#ref-CR16" id="ref-link-section-d10251175e474">16</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Nag, A. et al. Detection of acoustic plasmons in hole-doped lanthanum and bismuth cuprate superconductors using resonant inelastic x-ray scattering. Phys. Rev. Lett. 125, 257002 (2020)." href="/articles/s42005-023-01276-z#ref-CR18" id="ref-link-section-d10251175e477">18</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Singh, A. et al. Acoustic plasmons and conducting carriers in hole-doped cuprate superconductors. Phys. Rev. B 105, 235105 (2022)." href="/articles/s42005-023-01276-z#ref-CR19" id="ref-link-section-d10251175e480">19</a></sup>, it was found very recently that the low-energy plasmons are gapped at the in-plane zone center for the infinite-layered electron-doped cuprate Sr<sub>0.9</sub>La<sub>0.1</sub>CuO<sub>2</sub><sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Hepting, M. et al. Gapped collective charge excitations and interlayer hopping in cuprate superconductors. Phys. Rev. Lett. 129, 047001 (2022)." href="/articles/s42005-023-01276-z#ref-CR17" id="ref-link-section-d10251175e489">17</a></sup>, in agreement with a theoretical study—the gap is predicted to be proportional to the interlayer hopping<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Greco, A., Yamase, H. & Bejas, M. Plasmon excitations in layered high-Tc cuprates. Phys. Rev. B 94, 075139 (2016)." href="/articles/s42005-023-01276-z#ref-CR20" id="ref-link-section-d10251175e494">20</a></sup>. These low-energy plasmons may be referred to as acoustic-like plasmons. While the optical plasmon itself was observed already around 1990 by electron energy-loss spectroscopy<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Nücker, N. et al. Plasmons and interband transitions in Bi2Sr2CaCu2O8. Phys. Rev. B 39, 12379–12382 (1989)." href="/articles/s42005-023-01276-z#ref-CR32" id="ref-link-section-d10251175e498">32</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="Romberg, H. et al. Dielectric function of YBa2Cu3O7−δ between 50 meV and 50 eV. Z. Phys. B Condens. Matter 78, 367–380 (1990)." href="/articles/s42005-023-01276-z#ref-CR33" id="ref-link-section-d10251175e501">33</a></sup> and optical spectroscopy<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title="Bozovic, I. Plasmons in cuprate superconductors. Phys. Rev. B 42, 1969–1984 (1990)." href="/articles/s42005-023-01276-z#ref-CR34" id="ref-link-section-d10251175e505">34</a></sup>, these recent advances to reveal the charge excitation spectrum in cuprates attract renewed interest and plasmons offer a hot topic in the research of cuprate superconductors.</p><p>Electron–plasmon coupling was studied mainly for weakly correlated materials: alkali metals such as Na and Al<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 35" title="Aryasetiawan, F., Hedin, L. & Karlsson, K. Multiple plasmon satellites in Na and Al spectral functions from ab initio cumulant expansion. Phys. Rev. Lett. 77, 2268–2271 (1996)." href="/articles/s42005-023-01276-z#ref-CR35" id="ref-link-section-d10251175e512">35</a></sup>, Bi<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 36" title="Tediosi, R., Armitage, N. P., Giannini, E. & van der Marel, D. Charge carrier interaction with a purely electronic collective mode: Plasmarons and the infrared response of elemental bismuth. Phys. Rev. Lett. 99, 016406 (2007)." href="/articles/s42005-023-01276-z#ref-CR36" id="ref-link-section-d10251175e516">36</a></sup>, graphene<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 37" title="Polini, M. et al. Plasmons and the spectral function of graphene. Phys. Rev. B 77, 081411 (2008)." href="/articles/s42005-023-01276-z#ref-CR37" id="ref-link-section-d10251175e520">37</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 38" title="Hwang, E. H. & Das Sarma, S. Quasiparticle spectral function in doped graphene: electron-electron interaction effects in ARPES. Phys. Rev. B 77, 081412 (2008)." href="/articles/s42005-023-01276-z#ref-CR38" id="ref-link-section-d10251175e523">38</a></sup>, monolayer transition-metal dichalcogenides (Mo, W)(S, Se)<sub>2</sub><sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 39" title="Caruso, F., Lambert, H. & Giustino, F. Band structures of plasmonic polarons. Phys. Rev. Lett. 114, 146404 (2015)." href="/articles/s42005-023-01276-z#ref-CR39" id="ref-link-section-d10251175e528">39</a></sup>, semiconductors<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Caruso, F., Lambert, H. & Giustino, F. Band structures of plasmonic polarons. Phys. Rev. Lett. 114, 146404 (2015)." href="#ref-CR39" id="ref-link-section-d10251175e532">39</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Kheifets, A. S., Sashin, V. A., Vos, M., Weigold, E. & Aryasetiawan, F. Spectral properties of quasiparticles in silicon: a test of many-body theory. Phys. Rev. B 68, 233205 (2003)." href="#ref-CR40" id="ref-link-section-d10251175e532_1">40</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title="Caruso, F. & Giustino, F. Spectral fingerprints of electron-plasmon coupling. Phys. Rev. B 92, 045123 (2015)." href="/articles/s42005-023-01276-z#ref-CR41" id="ref-link-section-d10251175e535">41</a></sup>, and diamond<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title="Caruso, F. & Giustino, F. Spectral fingerprints of electron-plasmon coupling. Phys. Rev. B 92, 045123 (2015)." href="/articles/s42005-023-01276-z#ref-CR41" id="ref-link-section-d10251175e540">41</a></sup>. Here the so-called replica bands are known to be generated by coupling to plasmons; they are also referred to as plasmon satellites<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Caruso, F. & Giustino, F. Spectral fingerprints of electron-plasmon coupling. Phys. Rev. B 92, 045123 (2015)." href="#ref-CR41" id="ref-link-section-d10251175e544">41</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Brar, V. W. et al. Observation of carrier-density-dependent many-body effects in graphene via tunneling spectroscopy. Phys. Rev. Lett. 104, 036805 (2010)." href="#ref-CR42" id="ref-link-section-d10251175e544_1">42</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 43" title="Guzzo, M. et al. Valence electron photoemission spectrum of semiconductors: ab initio description of multiple satellites. Phys. Rev. Lett. 107, 166401 (2011)." href="/articles/s42005-023-01276-z#ref-CR43" id="ref-link-section-d10251175e547">43</a></sup> especially when momentum is integrated. The spectrum of the replica band is usually very broad and has low spectral weight<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Hwang, E. H. & Das Sarma, S. Quasiparticle spectral function in doped graphene: electron-electron interaction effects in ARPES. Phys. Rev. B 77, 081412 (2008)." href="#ref-CR38" id="ref-link-section-d10251175e551">38</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Caruso, F., Lambert, H. & Giustino, F. Band structures of plasmonic polarons. Phys. Rev. Lett. 114, 146404 (2015)." href="#ref-CR39" id="ref-link-section-d10251175e551_1">39</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Kheifets, A. S., Sashin, V. A., Vos, M., Weigold, E. & Aryasetiawan, F. Spectral properties of quasiparticles in silicon: a test of many-body theory. Phys. Rev. B 68, 233205 (2003)." href="#ref-CR40" id="ref-link-section-d10251175e551_2">40</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title="Caruso, F. & Giustino, F. Spectral fingerprints of electron-plasmon coupling. Phys. Rev. B 92, 045123 (2015)." href="/articles/s42005-023-01276-z#ref-CR41" id="ref-link-section-d10251175e554">41</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Markiewicz, R. S. & Bansil, A. Dispersion anomalies induced by the low-energy plasmon in the cuprates. Phys. Rev. B 75, 020508(R) (2007)." href="#ref-CR44" id="ref-link-section-d10251175e557">44</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lischner, J., Vigil-Fowler, D. & Louie, S. G. Physical origin of satellites in photoemission of doped graphene: an ab initio GW plus cumulant study. Phys. Rev. Lett. 110, 146801 (2013)." href="#ref-CR45" id="ref-link-section-d10251175e557_1">45</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="Lischner, J. et al. Satellite band structure in silicon caused by electron-plasmon coupling. Phys. Rev. B 91, 205113 (2015)." href="/articles/s42005-023-01276-z#ref-CR46" id="ref-link-section-d10251175e560">46</a></sup>, making it difficult to confirm it in experiments. Recently, however, the replica band was successfully resolved in various systems—graphene<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title="Brar, V. W. et al. Observation of carrier-density-dependent many-body effects in graphene via tunneling spectroscopy. Phys. Rev. Lett. 104, 036805 (2010)." href="/articles/s42005-023-01276-z#ref-CR42" id="ref-link-section-d10251175e564">42</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Bostwick, A. et al. Observation of plasmarons in quasi-freestanding doped graphene. Science 328, 999–1002 (2010)." href="/articles/s42005-023-01276-z#ref-CR47" id="ref-link-section-d10251175e567">47</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 48" title="Walter, A. L. et al. Effective screening and the plasmaron bands in graphene. Phys. Rev. B 84, 085410 (2011)." href="/articles/s42005-023-01276-z#ref-CR48" id="ref-link-section-d10251175e570">48</a></sup>, two-dimensional electron systems<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 49" title="Dial, O. E., Ashoori, R. C., Pfeiffer, L. N. & West, K. W. Observations of plasmarons in a two-dimensional system: tunneling measurements using time-domain capacitance spectroscopy. Phys. Rev. B 85, 081306 (2012)." href="/articles/s42005-023-01276-z#ref-CR49" id="ref-link-section-d10251175e574">49</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 50" title="Jang, J. et al. Full momentum- and energy-resolved spectral function of a 2D electronic system. Science 358, 901–906 (2017)." href="/articles/s42005-023-01276-z#ref-CR50" id="ref-link-section-d10251175e577">50</a></sup>, and SrIrO<sub>3</sub> films<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 51" title="Liu, Z. et al. Electron-plasmon interaction induced plasmonic-polaron band replication in epitaxial perovskite SrIrO3 films. Sci. Bull. 66, 433–440 (2021)." href="/articles/s42005-023-01276-z#ref-CR51" id="ref-link-section-d10251175e584">51</a></sup>. The presence of the replica band implies that the one-particle Green’s function has poles. That is, the replica band corresponds to the dispersion relation of quasiparticles dubbed as plasmarons<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Hedin, L., Lundqvist, B. & Lundqvist, S. New structure in the single-particle spectrum of an electron gas. Solid State Commun. 5, 237–239 (1967)." href="#ref-CR52" id="ref-link-section-d10251175e588">52</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lundqvist, B. I. Single-particle spectrum of the degenerate electron gas I. The structure of the spectral weight function. Phys. Kondens. Mater. 6, 193–205 (1967)." href="#ref-CR53" id="ref-link-section-d10251175e588_1">53</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lundqvist, B. I. Single-particle spectrum of the degenerate electron gas II. Numerical results for electrons coupled to plasmons. Phys. Kondens. Mater. 6, 206–217 (1967)." href="#ref-CR54" id="ref-link-section-d10251175e588_2">54</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 55" title="Lundqvist, B. I. Single-particle spectrum of the degenerate electron gas III. Numerical results in the random phase approximation. Phys. Kondens. Mater. 7, 117–123 (1968)." href="/articles/s42005-023-01276-z#ref-CR55" id="ref-link-section-d10251175e591">55</a></sup>.</p><p>Can we expect plasmarons in high-temperature cuprate superconductors? This is a far from obvious issue. First of all, cuprates are strongly correlated electron systems and thus it is reasonable to distinguish cuprates from weakly correlated systems as the ones discussed in the previous paragraph—a direct analogy between them is not trivial. Second, cuprates are layered systems, where not only the conventional optical plasmon, but also many acoustic-like plasmons are present<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Greco, A., Yamase, H. & Bejas, M. Plasmon excitations in layered high-Tc cuprates. Phys. Rev. B 94, 075139 (2016)." href="/articles/s42005-023-01276-z#ref-CR20" id="ref-link-section-d10251175e599">20</a></sup>. This is also a situation different from previous studies of plasmarons<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Aryasetiawan, F., Hedin, L. & Karlsson, K. Multiple plasmon satellites in Na and Al spectral functions from ab initio cumulant expansion. Phys. Rev. Lett. 77, 2268–2271 (1996)." href="#ref-CR35" id="ref-link-section-d10251175e603">35</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Tediosi, R., Armitage, N. P., Giannini, E. & van der Marel, D. Charge carrier interaction with a purely electronic collective mode: Plasmarons and the infrared response of elemental bismuth. Phys. Rev. Lett. 99, 016406 (2007)." href="#ref-CR36" id="ref-link-section-d10251175e603_1">36</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Polini, M. et al. Plasmons and the spectral function of graphene. Phys. Rev. B 77, 081411 (2008)." href="#ref-CR37" id="ref-link-section-d10251175e603_2">37</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Hwang, E. H. & Das Sarma, S. Quasiparticle spectral function in doped graphene: electron-electron interaction effects in ARPES. Phys. Rev. B 77, 081412 (2008)." href="#ref-CR38" id="ref-link-section-d10251175e603_3">38</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Caruso, F., Lambert, H. & Giustino, F. Band structures of plasmonic polarons. Phys. Rev. Lett. 114, 146404 (2015)." href="#ref-CR39" id="ref-link-section-d10251175e603_4">39</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Kheifets, A. S., Sashin, V. A., Vos, M., Weigold, E. & Aryasetiawan, F. Spectral properties of quasiparticles in silicon: a test of many-body theory. Phys. Rev. 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Lett. 107, 166401 (2011)." href="#ref-CR43" id="ref-link-section-d10251175e603_8">43</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Markiewicz, R. S. & Bansil, A. Dispersion anomalies induced by the low-energy plasmon in the cuprates. Phys. Rev. B 75, 020508(R) (2007)." href="#ref-CR44" id="ref-link-section-d10251175e603_9">44</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lischner, J., Vigil-Fowler, D. & Louie, S. G. Physical origin of satellites in photoemission of doped graphene: an ab initio GW plus cumulant study. Phys. Rev. Lett. 110, 146801 (2013)." href="#ref-CR45" id="ref-link-section-d10251175e603_10">45</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lischner, J. et al. Satellite band structure in silicon caused by electron-plasmon coupling. Phys. Rev. B 91, 205113 (2015)." href="#ref-CR46" id="ref-link-section-d10251175e603_11">46</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Bostwick, A. et al. Observation of plasmarons in quasi-freestanding doped graphene. Science 328, 999–1002 (2010)." href="#ref-CR47" id="ref-link-section-d10251175e603_12">47</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Walter, A. L. et al. Effective screening and the plasmaron bands in graphene. Phys. Rev. B 84, 085410 (2011)." href="#ref-CR48" id="ref-link-section-d10251175e603_13">48</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Dial, O. E., Ashoori, R. C., Pfeiffer, L. N. & West, K. W. Observations of plasmarons in a two-dimensional system: tunneling measurements using time-domain capacitance spectroscopy. Phys. Rev. B 85, 081306 (2012)." href="#ref-CR49" id="ref-link-section-d10251175e603_14">49</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Jang, J. et al. Full momentum- and energy-resolved spectral function of a 2D electronic system. Science 358, 901–906 (2017)." href="#ref-CR50" id="ref-link-section-d10251175e603_15">50</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 51" title="Liu, Z. et al. Electron-plasmon interaction induced plasmonic-polaron band replication in epitaxial perovskite SrIrO3 films. Sci. Bull. 66, 433–440 (2021)." href="/articles/s42005-023-01276-z#ref-CR51" id="ref-link-section-d10251175e606">51</a></sup>.</p><p>In this paper, we show that instead of yielding a kink, plasmons in cuprates lead to plasmarons—similar to weakly correlated systems. A common feature lies in the singularity of the long-range Coulomb interaction in the limit of long wavelength. However, the underlying physics is different. Instead of usual charge density-density correlations, fluctuations associated with the local constraint—non-double occupancy of electrons at any site—are responsible for the emergence of plasmarons. We find that cuprates can host plasmarons near the optical plasmon energy below (above) the quasiparticle band in electron-doped (hole-doped) cuprates.</p></div></div></section><section data-title="Results"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2">Results</h2><div class="c-article-section__content" id="Sec2-content"><h3 class="c-article__sub-heading" id="Sec3">Analytical scheme</h3><p>Cuprate superconductors are doped Mott insulators—strong correlations of electrons are believed to be crucial<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 56" title="Anderson, P. W. The resonating valence bond state in La2CuO4 and superconductivity. Science 235, 1196–1198 (1987)." href="/articles/s42005-023-01276-z#ref-CR56" id="ref-link-section-d10251175e625">56</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 57" title="Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006)." href="/articles/s42005-023-01276-z#ref-CR57" id="ref-link-section-d10251175e628">57</a></sup>. The <i>t</i>–<i>J</i> model is a microscopic model of cuprates superconductors and is derived from the three-band<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 58" title="Zhang, F. C. & Rice, T. M. Effective Hamiltonian for the superconducting Cu oxides. Phys. Rev. B 37, 3759–3761 (1988)." href="/articles/s42005-023-01276-z#ref-CR58" id="ref-link-section-d10251175e638">58</a></sup> and one-band<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 59" title="Chao, K. A., Spalek, J. & Oleś, A. M. Kinetic exchange interaction in a narrow s-band. J. Phys. C: Solid State Phys. 10, L271 (1977)." href="/articles/s42005-023-01276-z#ref-CR59" id="ref-link-section-d10251175e642">59</a></sup> Hubbard model. It reads</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$H=-\mathop{\sum}\limits_{i,j,\sigma }{t}_{ij}{\tilde{c}}_{i\sigma }^{{{{\dagger}}} }{\tilde{c}}_{j\sigma }+\mathop{\sum}\limits_{\langle i,j\rangle }{J}_{ij}\left({\overrightarrow{S}}_{i}\cdot {\overrightarrow{S}}_{j}-\frac{1}{4}{n}_{i}{n}_{j}\right)+\frac{1}{2}\mathop{\sum}\limits_{i\ne j}{V}_{ij}{n}_{i}{n}_{j}\,,$$</span></div><div class="c-article-equation__number"> (1) </div></div><p>where <span class="mathjax-tex">${\tilde{c}}_{i\sigma }^{{{{\dagger}}} }$</span> (<span class="mathjax-tex">${\tilde{c}}_{i\sigma }$</span>) are the creation (annihilation) operators of electrons with spin <i>σ</i>(=<i>↑</i>, <i>↓</i>) in the Fock space without double occupancy at any site—strong correlation effects, <span class="mathjax-tex">${n}_{i}={\sum }_{\sigma }{\tilde{c}}_{i\sigma }^{{{{\dagger}}} }{\tilde{c}}_{i\sigma }$</span> is the electron density operator, and <span class="mathjax-tex">${\overrightarrow{S}}_{i}$</span> is the spin operator. While cuprates are frequently modeled on a square lattice, we take the layered structure of cuprates into account and consider a three-dimensional lattice to describe plasmons correctly. The sites <i>i</i> and <i>j</i> run over such a three-dimensional lattice. The hopping <i>t</i><sub><i>i</i><i>j</i></sub> takes the value <i>t</i><span class="mathjax-tex">$({t}^{{\prime} })$</span> between the first (second) nearest-neighbor sites in the plane and is scaled by <i>t</i><sub><i>z</i></sub> between the planes. The exchange interaction <i>J</i><sub><i>i</i><i>j</i></sub> = <i>J</i> is considered only for the nearest-neighbor sites inside the plane as denoted by 〈<i>i</i>, <i>j</i>〉—the exchange term between the planes is much smaller than <i>J</i> (Thio et al.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 60" title="Thio, T. et al. Antisymmetric exchange and its influence on the magnetic structure and conductivity of La2CuO4. Phys. Rev. B 38, 905–908 (1988)." href="/articles/s42005-023-01276-z#ref-CR60" id="ref-link-section-d10251175e1294">60</a></sup>). <i>V</i><sub><i>i</i><i>j</i></sub> is the long-range Coulomb interaction.</p><p>It is highly nontrivial to analyze the strong correlation effects systematically. While a variational approach is powerful<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 61" title="Spałek, J., Fidrysiak, M., Zegrodnik, M. & Biborski, A. Superconductivity in high-Tc and related strongly correlated systems from variational perspective: beyond mean field theory. Phys. Rep. 959, 1–117 (2022)." href="/articles/s42005-023-01276-z#ref-CR61" id="ref-link-section-d10251175e1309">61</a></sup>, here we employ a large-<i>N</i> technique in a path integral representation in terms of the Hubbard operators<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 62" title="Foussats, A. & Greco, A. Large-N expansion based on the Hubbard operator path integral representation and its application to the t-J model. Phys. Rev. B 65, 195107 (2002)." href="/articles/s42005-023-01276-z#ref-CR62" id="ref-link-section-d10251175e1316">62</a></sup>. In the large-<i>N</i> scheme, the number of spin components is extended from 2 to <i>N</i> and physical quantities are computed by counting the power of 1/<i>N</i> systematically. One of the advantages of this method is that it treats all possible charge excitations on an equal footing<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 63" title="Bejas, M., Greco, A. & Yamase, H. Possible charge instabilities in two-dimensional doped Mott insulators. Phys. Rev. B 86, 224509 (2012)." href="/articles/s42005-023-01276-z#ref-CR63" id="ref-link-section-d10251175e1330">63</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 64" title="Bejas, M., Greco, A. & Yamase, H. Strong particle-hole asymmetry of charge instabilities in doped Mott insulators. New J. Phys. 16, 123002 (2014)." href="/articles/s42005-023-01276-z#ref-CR64" id="ref-link-section-d10251175e1333">64</a></sup>. There are two different charge fluctuations: on-site charge fluctuations describing usual charge-density-wave and plasmons, and bond-charge fluctuations describing charge-density-waves with an internal structure such as <i>d</i>-wave and <i>s</i>-wave symmetry, including the flux phase. Explicit calculations clarified that those two fluctuations are essentially decoupled to each other<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 65" title="Bejas, M., Yamase, H. & Greco, A. Dual structure in the charge excitation spectrum of electron-doped cuprates. Phys. Rev. B 96, 214513 (2017)." href="/articles/s42005-023-01276-z#ref-CR65" id="ref-link-section-d10251175e1343">65</a></sup>. Since we are interested in plasmons, we focus on the former fluctuations.</p><p>Because of the local constraint that double occupancy of electrons is prohibited at any lattice site, the charge fluctuations are described by a 2 × 2 matrix <i>D</i><sub><i>a</i><i>b</i></sub>(<b>q</b>, i<i>ν</i><sub><i>n</i></sub>) with <i>a</i>, <i>b</i> = 1, 2; <b>q</b> is the momentum of the charge fluctuations and <i>ν</i><sub><i>n</i></sub> a bosonic Matsubara frequency. While <i>D</i><sub>11</sub> is the usual density-density correlation function, <i>D</i><sub>22</sub> is a special feature of strong correlation effects—it describes fluctuations associated with the local constraint. As we shall clarify, this <i>D</i><sub>22</sub> plays the central role in the formation of plasmarons. Naturally there is also the off-diagonal component <i>D</i><sub>12</sub>(=<i>D</i><sub>21</sub>).</p><p>In the large-<i>N</i> theory, <i>D</i><sub><i>a</i><i>b</i></sub>(<b>q</b>, i<i>ν</i><sub><i>n</i></sub>) is renormalized already at leading order. After the analytical continuation i<i>ν</i><sub><i>n</i></sub> → <i>ν</i> + iΓ<sub>ch</sub>, where Γ<sub>ch</sub>(>0) is infinitesimally small, the full charge excitation spectrum is described by <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{D}_{ab}({{{{{{{\bf{q}}}}}}}},\nu )$</span>—Bejas et al.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 65" title="Bejas, M., Yamase, H. & Greco, A. Dual structure in the charge excitation spectrum of electron-doped cuprates. Phys. Rev. B 96, 214513 (2017)." href="/articles/s42005-023-01276-z#ref-CR65" id="ref-link-section-d10251175e1492">65</a></sup> reported a comprehensive analysis of <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{D}_{ab}({{{{{{{\bf{q}}}}}}}},\nu )$</span>. In particular, <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{D}_{11}({{{{{{{\bf{q}}}}}}}},\nu )$</span> predicted acoustic-like plasmon excitations with a gap at <b>q</b> = (0, 0, <i>q</i><sub><i>z</i></sub>) for <i>q</i><sub><i>z</i></sub> ≠ 0 as well as the well-known optical plasmon at <b>q</b> = (0, 0, 0)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Greco, A., Yamase, H. & Bejas, M. Plasmon excitations in layered high-Tc cuprates. Phys. Rev. B 94, 075139 (2016)." href="/articles/s42005-023-01276-z#ref-CR20" id="ref-link-section-d10251175e1613">20</a></sup>. Close inspections revealed that the predicted plasmon excitations explain semiquantitatively charge excitation spectra reported by RIXS for both hole-doped<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Nag, A. et al. Detection of acoustic plasmons in hole-doped lanthanum and bismuth cuprate superconductors using resonant inelastic x-ray scattering. Phys. Rev. Lett. 125, 257002 (2020)." href="/articles/s42005-023-01276-z#ref-CR18" id="ref-link-section-d10251175e1617">18</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Greco, A., Yamase, H. & Bejas, M. Origin of high-energy charge excitations observed by resonant inelastic x-ray scattering in cuprate superconductors. Commun. Phys. 2, 3 (2019)." href="/articles/s42005-023-01276-z#ref-CR21" id="ref-link-section-d10251175e1620">21</a></sup> and electron-doped<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Hepting, M. et al. Gapped collective charge excitations and interlayer hopping in cuprate superconductors. Phys. Rev. Lett. 129, 047001 (2022)." href="/articles/s42005-023-01276-z#ref-CR17" id="ref-link-section-d10251175e1624">17</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Greco, A., Yamase, H. & Bejas, M. Origin of high-energy charge excitations observed by resonant inelastic x-ray scattering in cuprate superconductors. Commun. Phys. 2, 3 (2019)." href="/articles/s42005-023-01276-z#ref-CR21" id="ref-link-section-d10251175e1627">21</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Greco, A., Yamase, H. & Bejas, M. Close inspection of plasmon excitations in cuprate superconductors. Phys. Rev. B 102, 024509 (2020)." href="/articles/s42005-023-01276-z#ref-CR22" id="ref-link-section-d10251175e1630">22</a></sup> cuprates.</p><p>Charge fluctuations renormalize the one-particle property of electrons, which can be analyzed by computing the electron self-energy. This requires involved calculations in the large-<i>N</i> theory because one needs to go beyond leading order theory. At order of 1/<i>N</i>, the imaginary part of the self-energy is obtained as<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 66" title="Yamase, H., Bejas, M. & Greco, A. Electron self-energy from quantum charge fluctuations in the layered t-J model with long-range coulomb interaction. Phys. Rev. B 104, 045141 (2021)." href="/articles/s42005-023-01276-z#ref-CR66" id="ref-link-section-d10251175e1644">66</a></sup></p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${{{{{{{\rm{Im}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )=\mathop{\sum}\limits_{a,b=1,2}{{{{{{{\rm{Im}}}}}}}}{\Sigma }_{ab}({{{{{{{\bf{k}}}}}}}},\omega )\,,$$</span></div><div class="c-article-equation__number"> (2) </div></div><p>where</p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${{{{{\rm{Im}}}}}}{\Sigma }_{ab}({{{{{\bf{k}}}}}},\omega )= \frac{-1}{{N}_{s}{N}_{z}}\mathop{\sum}\limits_{{{{{{\bf{q}}}}}}}{{{{{\rm{Im}}}}}}{D}_{ab}({{{{{\bf{q}}}}}},\nu ){h}_{a}({{{{{\bf{k}}}}}},{{{{{\bf{q}}}}}},\nu ){h}_{b}({{{{{\bf{k}}}}}},{{{{{\bf{q}}}}}},\nu )\\ \times \left[{n}_{{{{{{\rm{F}}}}}}}\left(-{\varepsilon }_{{{{{{\bf{k}}}}}}-{{{{{\bf{q}}}}}}}\right)+{n}_{{{{{{\rm{B}}}}}}}\left(\nu \right)\right].$$</span></div><div class="c-article-equation__number"> (3) </div></div><p>Here <i>ν</i> = <i>ω</i> − <i>ε</i><sub><b>k</b>−<b>q</b></sub>, <i>ε</i><sub><b>k</b></sub> is the electron dispersion obtained at leading order, <i>h</i><sub><i>a</i></sub>(<b>k</b>, <b>q</b>, <i>ν</i>) a vertex describing the coupling between electrons and charge excitations, <i>n</i><sub>F</sub> and <i>n</i><sub>B</sub> the Fermi and Bose distribution functions, respectively, <i>N</i><sub><i>s</i></sub> the total number of lattice sites in each layer, and <i>N</i><sub><i>z</i></sub> the number of layers; see “Methods” for the explicit forms of <i>D</i><sub><i>a</i><i>b</i></sub>(<b>q</b>, <i>ν</i>), <i>ε</i><sub><b>k</b></sub>, and <i>h</i><sub><i>a</i></sub>(<b>k</b>, <b>q</b>, <i>ν</i>). The real part of Σ(<b>k</b>, <i>ω</i>) is computed by the Kramers-Kronig relations. Since the electron Green’s function <i>G</i>(<b>k</b>, <i>ω</i>) is written as <i>G</i><sup>−1</sup>(<b>k</b>, <i>ω</i>) = <i>ω</i> + iΓ<sub>sf</sub> − <i>ε</i><sub><b>k</b></sub> − Σ(<b>k</b>, <i>ω</i>), we obtain the one-particle spectral function <span class="mathjax-tex">$A({{{{{{{\bf{k}}}}}}}},\omega )=-\frac{1}{\pi }{{{{{{{\rm{Im}}}}}}}}G({{{{{{{\bf{k}}}}}}}},\omega )$</span>:</p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$A({{{{{{{\bf{k}}}}}}}},\omega )=-\frac{1}{\pi }\frac{{{{{{{{\rm{Im}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )-{\Gamma }_{{{{{{{{\rm{sf}}}}}}}}}}{{[\omega -{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}-{{{{{{{\rm{Re}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )]}^{2}+{[{{{{{{{\rm{Im}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )-{\Gamma }_{{{{{{{{\rm{sf}}}}}}}}}]}^{2}}\,,$$</span></div><div class="c-article-equation__number"> (4) </div></div><p>where Γ<sub>sf</sub>(>0) originates from the analytical continuation in the electron Green’s function.</p><p>The quasiparticle dispersion appears as poles of <i>A</i>(<b>k</b>, <i>ω</i>), i.e., a sharp peak structure of <i>A</i>(<b>k</b>, <i>ω</i>), and crosses the Fermi energy. On top of that, <i>A</i>(<b>k</b>, <i>ω</i>) can exhibit other sharp features. If plasmons themselves are responsible for yielding additional poles of <i>A</i>(<b>k</b>, <i>ω</i>), namely fulfilling the condition of <span class="mathjax-tex">$\omega -{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}-{{{{{{{\rm{Re}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )=0$</span> with a relatively small value of <span class="mathjax-tex">$| {{{{{{{\rm{Im}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )|$</span>, <i>A</i>(<b>k</b>, <i>ω</i>) exhibits a peak describing electrons coupling to plasmons, namely plasmarons<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Hedin, L., Lundqvist, B. & Lundqvist, S. New structure in the single-particle spectrum of an electron gas. Solid State Commun. 5, 237–239 (1967)." href="#ref-CR52" id="ref-link-section-d10251175e2670">52</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lundqvist, B. I. Single-particle spectrum of the degenerate electron gas I. The structure of the spectral weight function. Phys. Kondens. Mater. 6, 193–205 (1967)." href="#ref-CR53" id="ref-link-section-d10251175e2670_1">53</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lundqvist, B. I. Single-particle spectrum of the degenerate electron gas II. Numerical results for electrons coupled to plasmons. Phys. Kondens. Mater. 6, 206–217 (1967)." href="#ref-CR54" id="ref-link-section-d10251175e2670_2">54</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 55" title="Lundqvist, B. I. Single-particle spectrum of the degenerate electron gas III. Numerical results in the random phase approximation. Phys. Kondens. Mater. 7, 117–123 (1968)." href="/articles/s42005-023-01276-z#ref-CR55" id="ref-link-section-d10251175e2673">55</a></sup>, with a damping controlled by <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )$</span>. Note that the charge excitation spectrum also contains usual particle-hole excitations, the so-called continuum spectrum, which can also lead to poles in <i>A</i>(<b>k</b>, <i>ω</i>). Thus additional poles of <i>A</i>(<b>k</b>, <i>ω</i>) do not necessarily signal the emergence of plasmarons.</p><p>While the <i>t</i>–<i>J</i> model in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ1">1</a>) contains spin fluctuations, they appear at order of <i>O</i>(1/<i>N</i>) in the present theory whereas charge fluctuations appear at <i>O</i>(1). Hence when we compute the electron self-energy at order of 1/<i>N</i>, only charge fluctuations enter Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ3">3</a>), which is suitable to study the role of plasmons exclusively in the one-particle spectral function.</p><h3 class="c-article__sub-heading" id="Sec4">One-particle spectral function</h3><p>A choice of model parameters is not crucial to our major conclusions. Here we present results which can be applied directly to electron-doped cuprates, especially La<sub>1.825</sub>Ce<sub>0.175</sub>CuO<sub>4</sub> (LCCO); details of model parameters are given in “Methods” and Supplementary Note <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s42005-023-01276-z#MOESM2">3</a>.</p><p>Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig1">1</a> shows the one-particle spectral function <i>A</i>(<b>k</b>, <i>ω</i>) along the direction (<i>π</i>, <i>π</i>)–(0, 0)–(<i>π</i>, 0)–(<i>π</i>, <i>π</i>); <i>k</i><sub><i>z</i></sub> dependence is weak and thus <i>k</i><sub><i>z</i></sub> = <i>π</i> is taken throughout the present paper—the value of <i>k</i><sub><i>z</i></sub> shall be omitted for simplicity. The spectrum around <i>ω</i> = 0 is the quasiparticle dispersion renormalized by charge fluctuations. In contrast to the case of electron–phonon coupling<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 67" title="Zeyher, R. & Greco, A. Low-energy renormalization of the electron dispersion of high-Tc superconductors. Phys. Rev. B 64, 140510(R) (2001)." href="/articles/s42005-023-01276-z#ref-CR67" id="ref-link-section-d10251175e2835">67</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 68" title="Li, Z., Wu, M., Chan, Y.-H. & Louie, S. G. Unmasking the origin of kinks in the photoemission spectra of cuprate superconductors. Phys. Rev. Lett. 126, 146401 (2021)." href="/articles/s42005-023-01276-z#ref-CR68" id="ref-link-section-d10251175e2838">68</a></sup> and magnetic fluctuations<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 69" title="Eschrig, M. & Norman, M. R. Neutron resonance: modeling photoemission and tunneling data in the superconducting state of $${{{{{{{{{\rm{Bi}}}}}}}}}_{2}{{{{{{{{\rm{Sr}}}}}}}}}_{2}{{{{{{{{\rm{CaCu}}}}}}}}}_{2}O}_{8+{\delta }}$$ Bi 2 Sr 2 CaCu 2 O 8 + δ . Phys. Rev. Lett. 85, 3261–3264 (2000)." href="/articles/s42005-023-01276-z#ref-CR69" id="ref-link-section-d10251175e2843">69</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 70" title="Markiewicz, R. S., Sahrakorpi, S. & Bansil, A. Paramagnon-induced dispersion anomalies in the cuprates. Phys. Rev. B 76, 174514 (2007)." href="/articles/s42005-023-01276-z#ref-CR70" id="ref-link-section-d10251175e2846">70</a></sup>, it does not exhibit a kink structure. Rather it is described by the dispersion <i>ε</i><sub><b>k</b></sub> (solid curve) multiplied by some constant <i>Z</i>(=0.29) as shown by a dotted curve in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig1">1</a>. This implies that the renormalization factor depends weakly on <b>k</b> and the quasiparticle spectral weight is reduced down to 0.29 by charge fluctuations.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-1" data-title="Intensity map of the one-particle spectral function A(k, ω)."><figure><figcaption><b id="Fig1" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 1: Intensity map of the one-particle spectral function <b><i>A</i></b>(<b>k</b>, <i>ω</i>).</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s42005-023-01276-z/figures/1" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs42005-023-01276-z/MediaObjects/42005_2023_1276_Fig1_HTML.png?as=webp"><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs42005-023-01276-z/MediaObjects/42005_2023_1276_Fig1_HTML.png" alt="figure 1" loading="lazy" width="685" height="645"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p>The map is focused on a low-energy region along the direction (<i>π</i>, <i>π</i>)–(0, 0)–(<i>π</i>, 0)–(<i>π</i>, <i>π</i>) with <i>k</i><sub><i>z</i></sub> = <i>π</i>. For comparison, the quasiparticle dispersion obtained at leading order (solid curve) and a renormalized dispersion multiplied by <i>Z</i> = 0.29 (dotted curve) are superimposed. The inset shows <i>A</i>(<b>k</b>,<i>ω</i>) in a wider energy window. The additional band in <i>ω</i> < 0 corresponds to plasmarons, which we shall clarify in the present work.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s42005-023-01276-z/figures/1" data-track-dest="link:Figure1 Full size image" aria-label="Full size image figure 1" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>Charge fluctuations also generate additional bands as shown in the inset in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig1">1</a>. There are two major bands: a low-energy incoherent band near <i>ω</i> ≈ −1.5<i>t</i> and a high-energy side band with a large dispersion in 4 < <i>ω</i>/<i>t</i> < 7—the spectral weight of the former is about 10% and that of the latter is about 60%. The reason to call a side band instead of an incoherent one for the high-energy feature lies in that <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )$</span> almost vanishes in such a high-energy region, leading to a coherent feature.</p><p>Since the low-energy incoherent band disappears when the long-range Coulomb interaction is replaced by a short-range Coulomb interaction—the high-energy one still remains<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 66" title="Yamase, H., Bejas, M. & Greco, A. Electron self-energy from quantum charge fluctuations in the layered t-J model with long-range coulomb interaction. Phys. Rev. B 104, 045141 (2021)." href="/articles/s42005-023-01276-z#ref-CR66" id="ref-link-section-d10251175e2997">66</a></sup>, a coupling to plasmons is crucial to the low-energy feature. The major point of the present work is to elucidate that the low-energy one corresponds to plasmarons<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Hedin, L., Lundqvist, B. & Lundqvist, S. New structure in the single-particle spectrum of an electron gas. Solid State Commun. 5, 237–239 (1967)." href="#ref-CR52" id="ref-link-section-d10251175e3001">52</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lundqvist, B. I. Single-particle spectrum of the degenerate electron gas I. The structure of the spectral weight function. Phys. Kondens. 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Mater. 7, 117–123 (1968)." href="/articles/s42005-023-01276-z#ref-CR55" id="ref-link-section-d10251175e3004">55</a></sup> and is essentially the same as the so-called replica band discussed in weakly correlated electron systems<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Aryasetiawan, F., Hedin, L. & Karlsson, K. Multiple plasmon satellites in Na and Al spectral functions from ab initio cumulant expansion. Phys. Rev. Lett. 77, 2268–2271 (1996)." href="#ref-CR35" id="ref-link-section-d10251175e3008">35</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Tediosi, R., Armitage, N. P., Giannini, E. & van der Marel, D. Charge carrier interaction with a purely electronic collective mode: Plasmarons and the infrared response of elemental bismuth. Phys. Rev. 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Lett. 104, 036805 (2010)." href="#ref-CR42" id="ref-link-section-d10251175e3008_7">42</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Guzzo, M. et al. Valence electron photoemission spectrum of semiconductors: ab initio description of multiple satellites. Phys. Rev. Lett. 107, 166401 (2011)." href="#ref-CR43" id="ref-link-section-d10251175e3008_8">43</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Markiewicz, R. S. & Bansil, A. Dispersion anomalies induced by the low-energy plasmon in the cuprates. Phys. Rev. B 75, 020508(R) (2007)." href="#ref-CR44" id="ref-link-section-d10251175e3008_9">44</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lischner, J., Vigil-Fowler, D. & Louie, S. G. Physical origin of satellites in photoemission of doped graphene: an ab initio GW plus cumulant study. Phys. Rev. Lett. 110, 146801 (2013)." href="#ref-CR45" id="ref-link-section-d10251175e3008_10">45</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lischner, J. et al. Satellite band structure in silicon caused by electron-plasmon coupling. Phys. Rev. B 91, 205113 (2015)." href="#ref-CR46" id="ref-link-section-d10251175e3008_11">46</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Bostwick, A. et al. Observation of plasmarons in quasi-freestanding doped graphene. Science 328, 999–1002 (2010)." href="#ref-CR47" id="ref-link-section-d10251175e3008_12">47</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Walter, A. L. et al. Effective screening and the plasmaron bands in graphene. Phys. Rev. B 84, 085410 (2011)." href="#ref-CR48" id="ref-link-section-d10251175e3008_13">48</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Dial, O. E., Ashoori, R. C., Pfeiffer, L. N. & West, K. W. Observations of plasmarons in a two-dimensional system: tunneling measurements using time-domain capacitance spectroscopy. Phys. Rev. B 85, 081306 (2012)." href="#ref-CR49" id="ref-link-section-d10251175e3008_14">49</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Jang, J. et al. Full momentum- and energy-resolved spectral function of a 2D electronic system. Science 358, 901–906 (2017)." href="#ref-CR50" id="ref-link-section-d10251175e3008_15">50</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 51" title="Liu, Z. et al. Electron-plasmon interaction induced plasmonic-polaron band replication in epitaxial perovskite SrIrO3 films. Sci. Bull. 66, 433–440 (2021)." href="/articles/s42005-023-01276-z#ref-CR51" id="ref-link-section-d10251175e3011">51</a></sup>. In the following we focus on an energy window −2 ≤ <i>ω</i>/<i>t</i> ≤ −1.</p><h3 class="c-article__sub-heading" id="Sec5">Relevant contributions to the formation of plasmarons</h3><p>As seen in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ2">2</a>), ImΣ(<b>k</b>, <i>ω</i>) is given by the sum of four components. To elucidate the relevant contribution to forming plasmarons, we may introduce an auxiliary parameter <i>r</i>(≥0) as</p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${{{{{{{\rm{Im}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega ;r)={{{{{{{\rm{Im}}}}}}}}{\Sigma }_{11}+r\times \left({{{{{{{\rm{Im}}}}}}}}{\Sigma }_{22}+2\,{{{{{{{\rm{Im}}}}}}}}{\Sigma }_{12}\right)\,,$$</span></div><div class="c-article-equation__number"> (5) </div></div><p>where the arguments on the right hand side are omitted for simplicity and the fact that ImΣ<sub>12</sub> is equal to ImΣ<sub>21</sub> was used. The case of <i>r</i> = 1 corresponds to the physical situation seen in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ2">2</a>). We then compute <i>A</i>(<b>k</b>, <i>ω</i>) for several choices of <i>r</i> in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig2">2</a>a–d, where a different color scale is used to highlight the weak feature in <i>r</i> < 1; see Supplementary Note <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s42005-023-01276-z#MOESM2">1</a> for a wider energy window. Upon decreasing <i>r</i>, the incoherent band loses intensity substantially, fades away, and finally becomes invisible in <i>r</i> <span class="stix">≲</span> 0.4. This clearly indicates that the incoherent band is driven by components involving <i>a</i>, <i>b</i> = 2, namely fluctuations associated with the local constraint—a direct consequence of the strong correlation effect.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-2" data-title="Analysis of plasmarons in terms of Eq. (5)."><figure><figcaption><b id="Fig2" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 2: Analysis of plasmarons in terms of Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ5">5</a>).</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s42005-023-01276-z/figures/2" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs42005-023-01276-z/MediaObjects/42005_2023_1276_Fig2_HTML.png?as=webp"><img aria-describedby="Fig2" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs42005-023-01276-z/MediaObjects/42005_2023_1276_Fig2_HTML.png" alt="figure 2" loading="lazy" width="685" height="530"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-2-desc"><p><b>a</b>–<b>d</b> Intensity map of the spectral function <i>A</i>(<b>k</b>, <i>ω</i>) computed with ImΣ(<b>k</b>, <i>ω</i>; <i>r</i>) [Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ5">5</a>)] in −2 ≤ <i>ω</i>/<i>t</i> ≤ −1 for several choices of <i>r</i> along the direction (<i>π</i>, <i>π</i>)–(0, 0)–(<i>π</i>, 0)–(<i>π</i>, <i>π</i>) with <i>k</i><sub><i>z</i></sub> = <i>π</i>. The plasmaron band in (<b>a</b>) fades away upon decreasing <i>r</i>, indicating that fluctuations associated with the local constraint are crucially important to the plasmarons. Note a different color scale in each panel. <b>e</b>–<b>h</b> Imaginary and real parts of Σ(<b>k</b>, <i>ω</i>) as a function of <i>ω</i> at <b>k</b> = (0, 0) and (<i>π</i>, 0) for several choices of <i>r</i>. The peak of ImΣ(<b>k</b>, <i>ω</i>) in (<b>e</b>) and (<b>g</b>) is determined by the optical plasmon. The line of <i>ω</i> − <i>ε</i><sub><b>k</b></sub> is also shown in (<b>f</b>) and (<b>h</b>). The plasmaron energy is determined by its crossing point of ReΣ(<b>k</b>, <i>ω</i>) on the lower energy side when <i>r</i> is close to 1.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s42005-023-01276-z/figures/2" data-track-dest="link:Figure2 Full size image" aria-label="Full size image figure 2" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>The corresponding <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )$</span> and <span class="mathjax-tex">${{{{{{{\rm{Re}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )$</span> are also shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig2">2</a>e–h for two choices of <b>k</b>. <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )$</span> exhibits a sharp peak at <i>ω</i> ≈ −1.5<i>t</i> and − 1.3<i>t</i> for <b>k</b> = (0, 0) and (<i>π</i>, 0), respectively, and the peak is suppressed and broadened with decreasing <i>r</i>. The peak structure of <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )$</span> yields a large dip structure in <span class="mathjax-tex">${{{{{{{\rm{Re}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )$</span> at slightly lower energy than the peak energy of <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )$</span> via the Kramers-Kronig relations. Consequently, the term <span class="mathjax-tex">$\omega -{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}-{{{{{{{\rm{Re}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )$</span> in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ4">4</a>) can vanish at two energies when <i>r</i> is close to 1: one is very close to the peak energy of ImΣ(<b>k</b>, <i>ω</i>) and the other corresponds to the tail of the dip structure of ReΣ(<b>k</b>, <i>ω</i>). Since ImΣ(<b>k</b>, <i>ω</i>) becomes small at the latter energy, <i>A</i>(<b>k</b>, <i>ω</i>) forms a peak there with a damping controlled by ImΣ(<b>k</b>, <i>ω</i>). Hence this peak is incoherent, but is a resonance in the sense that <span class="mathjax-tex">$\omega -{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}-{{{{{{{\rm{Re}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )=0$</span> is fulfilled.</p><p>Which one is more crucial to the incoherent band, ImΣ<sub>12</sub> or ImΣ<sub>22</sub>? To answer this, we have studied each component of ImΣ<sub><i>a</i><i>b</i></sub>(<b>k</b>, <i>ω</i>) and computed the spectral function for each of them. We can check that ImΣ<sub>22</sub> is responsible for the formation of the incoherent band. The component of ImΣ<sub>12</sub> works to sharpen the incoherent band by reducing the absolute value of the imaginary part of the self-energy (see Supplementary Note <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s42005-023-01276-z#MOESM2">2</a> for details).</p><h3 class="c-article__sub-heading" id="Sec6">Role of plasmons</h3><p>What is then the role of plasmons? Plasmons are described in terms of the charge-charge correlation function, namely poles of <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{D}_{11}$</span> in the present theory. Because of the matrix structure of <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{D}_{ab}$</span>, the poles are determined by the zeros of its determinant. Thus all components of <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{D}_{ab}$</span> contain the same poles as those in <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{D}_{11}$</span> and thus describe the same plasmons equally (see Figs. 1 and 8 in Bejas et al.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 65" title="Bejas, M., Yamase, H. & Greco, A. Dual structure in the charge excitation spectrum of electron-doped cuprates. Phys. Rev. B 96, 214513 (2017)." href="/articles/s42005-023-01276-z#ref-CR65" id="ref-link-section-d10251175e3957">65</a></sup> for explicit calculations). Because of the layered structure of cuprates, plasmons have various branches depending on the value of <i>q</i><sub><i>z</i></sub>: the usual optical plasmon corresponds to <i>q</i><sub><i>z</i></sub> = 0 and the acoustic-like branches to <i>q</i><sub><i>z</i></sub> ≠ 0<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Greco, A., Yamase, H. & Bejas, M. Plasmon excitations in layered high-Tc cuprates. Phys. Rev. B 94, 075139 (2016)." href="/articles/s42005-023-01276-z#ref-CR20" id="ref-link-section-d10251175e3980">20</a></sup>. Their energy varies in 0.07 ≤ <i>ν</i>/<i>t</i> ≤ 1.15 around <b>q</b> = (0, 0, <i>q</i><sub><i>z</i></sub>) for the present parameters. The peak of <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )$</span> at <i>ω</i> = −1.5<i>t</i> (−1.3<i>t</i>) in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig2">2</a>e [Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig2">2</a>g] is determined by the optical plasmon at <i>ν</i> = 1.15<i>t</i>—the energy difference is easily read off from Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ3">3</a>): the energy of <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{D}_{ab}$</span> is given by <i>ν</i> = <i>ω</i> − <i>ε</i><sub><b>k</b>−<b>q</b></sub>, the optical plasmon is realized at <b>q</b> = <b>0</b>, Im<i>D</i><sub><i>a</i><i>b</i></sub> is odd with respect to <i>ν</i>, and thus the peak of ImΣ(<b>k</b>, <i>ω</i>) is shifted by <i>ε</i><sub><b>k</b>−<b>q</b></sub> with <b>q</b> = <b>0</b>.</p><p>The crucial role of the optical plasmon is also confirmed numerically. In Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig3">3</a>, we compute the self-energy ImΣ(<b>k</b>, <i>ω</i>) by removing a region <span class="stix">∣</span><i>q</i><sub><i>z</i></sub><span class="stix">∣</span> ≤ 2<i>π</i>/10 in the <i>q</i><sub><i>z</i></sub> summation in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ3">3</a>) so that the contribution from the optical plasmon becomes zero. We then observe that the peak structure in ImΣ(<b>k</b>, <i>ω</i>) completely disappears, not shifts to another energy window. The resulting <i>A</i>(<b>k</b>, <i>ω</i>) no longer forms any structure there.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-3" data-title="Role of the optical plasmon for plasmarons."><figure><figcaption><b id="Fig3" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 3: Role of the optical plasmon for plasmarons.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s42005-023-01276-z/figures/3" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs42005-023-01276-z/MediaObjects/42005_2023_1276_Fig3_HTML.png?as=webp"><img aria-describedby="Fig3" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs42005-023-01276-z/MediaObjects/42005_2023_1276_Fig3_HTML.png" alt="figure 3" loading="lazy" width="685" height="420"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-3-desc"><p>The self-energy and the spectral function are computed at <b>k</b> = (<i>π</i>, 0) by removing a region <span class="stix">∣</span><i>q</i><sub><i>z</i></sub><span class="stix">∣</span> ≤ 2<i>π</i>/10 in the <i>q</i><sub><i>z</i></sub> summation in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ3">3</a>), namely without contributions from the optical plasmon. They do not form any structure. Dotted curves are the corresponding results with the full <i>q</i><sub><i>z</i></sub> summation. The line of <i>ω</i> − <i>ε</i><sub><b>k</b></sub> is also given. The contrast between the solid and dotted curves demonstrates the importance of the optical plasmon to forming the plasmarons.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s42005-023-01276-z/figures/3" data-track-dest="link:Figure3 Full size image" aria-label="Full size image figure 3" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>It might be puzzling—on one hand, the optical plasmon is responsible for the formation of the plasmarons (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig3">3</a>), but on the other hand, the same plasmons in <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{D}_{11}$</span> and <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{D}_{12}$</span> do not generate them (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig2">2</a>). The vertex function <i>h</i><sub><i>a</i></sub>(<b>k</b>, <b>q</b>, <i>ν</i>) in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ3">3</a>) was checked not to be important. The key lies in the role of the long-range Coulomb interaction <i>V</i><sub><b>q</b></sub>, which diverges as <i>q</i><sup>−2</sup> in the limit of <b>q</b> → <b>0</b>. As is well known, this is the very reason why the plasmons are realized<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Mahan, G. D. Many-Particle Physics 2nd edn (Plunum Press, 1990)." href="/articles/s42005-023-01276-z#ref-CR4" id="ref-link-section-d10251175e4366">4</a></sup>—the inverse of the charge response function or the determinant of <i>D</i><sub><i>a</i><i>b</i></sub> vanishes along the plasmon dispersion. The crucial difference among <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{\Sigma }_{ab}$</span> appears in the numerator. To see this we study the explicit form of <i>D</i><sub><i>a</i><i>b</i></sub>, which is given by</p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${D}_{ab}({{{{{{{\bf{q}}}}}}}},\nu )=\frac{1}{{{{{{{{\mathcal{D}}}}}}}}}\left(\begin{array}{cc}-{{{\Pi }}}_{22}({{{{{{{\bf{q}}}}}}}},\nu )&{{{\Pi }}}_{12}({{{{{{{\bf{q}}}}}}}},\nu )-\frac{N\delta }{2}\\ {{{\Pi }}}_{12}({{{{{{{\bf{q}}}}}}}},\nu )-\frac{N\delta }{2}\quad &-{{{\Pi }}}_{11}({{{{{{{\bf{q}}}}}}}},\nu )+\frac{N{\delta }^{2}}{2}\left({V}_{{{{{{{{\bf{q}}}}}}}}}-{J}_{{{{{{{{\bf{q}}}}}}}}}\right)\end{array}\right)\,.$$</span></div><div class="c-article-equation__number"> (6) </div></div><p>Here <span class="mathjax-tex">${{{{{{{\mathcal{D}}}}}}}}$</span> is the determinant of the matrix <span class="mathjax-tex">${[{D}_{ab}({{{{{{{\bf{q}}}}}}}},\nu )]}^{-1}$</span>, <i>δ</i> the doping rate, <i>J</i><sub><b>q</b></sub> the superexchange interaction in momentum space, Π<sub><i>a</i><i>b</i></sub> a bubble describing particle-hole excitations with appropriate vertex functions <i>h</i><sub><i>a</i></sub>(<b>k</b>, <b>q</b>, <i>ν</i>), and <i>N</i> the number of spin components (<i>N</i> = 2 corresponds to the physical situations); a complete expression of each quantity is given in “Methods”. In the limit of <b>q</b> → <b>0</b>, we can obtain by virtue of <i>V</i><sub><b>q</b></sub> ~ <i>q</i><sup>−2</sup></p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${{{{{{{\rm{Im}}}}}}}}{D}_{22}({{{{{{{\bf{q}}}}}}}},\nu ) \sim \frac{{V}_{{{{{{{{\bf{q}}}}}}}}}^{2}{{{{{{{\rm{Im}}}}}}}}{{{\Pi }}}_{22}}{{({{{{{{{\rm{Re}}}}}}}}{{{{{{{\mathcal{D}}}}}}}})}^{2}+{({{{{{{{\rm{Im}}}}}}}}{{{{{{{\mathcal{D}}}}}}}})}^{2}}\,.$$</span></div><div class="c-article-equation__number"> (7) </div></div><p>The other components of <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{D}_{11}$</span> and <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{D}_{12}$</span> become smaller by order of <span class="mathjax-tex">${V}_{{{{{{{{\bf{q}}}}}}}}}^{-2}$</span> and <span class="mathjax-tex">${V}_{{{{{{{{\bf{q}}}}}}}}}^{-1}$</span>, respectively. Hence, <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{D}_{22}$</span> becomes dominant over the other components in the limit of <b>q</b> → <b>0</b> and thus <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{\Sigma }_{22}$</span> has a sizable contribution compared with the other components. This explains the reason why <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{\Sigma }_{22}$</span> is responsible for the plasmarons, although all components of <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{D}_{ab}$</span> equally describe the same plasmons.</p></div></div></section><section data-title="Discussion"><div class="c-article-section" id="Sec7-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec7">Discussion</h2><div class="c-article-section__content" id="Sec7-content"><p>From Eqs. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ3">3</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ7">7</a>), one may recognize that the mathematical structure of <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{\Sigma }_{22}$</span> in the limit of <b>q</b> → <b>0</b> is the same as the well-known expression of the self-energy in weak coupling theory,</p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${{{{{{{\rm{Im}}}}}}}}{\Sigma }^{{{{{{{{\rm{RPA}}}}}}}}}({{{{{{{\bf{k}}}}}}}},\omega )=\frac{-1}{{N}_{z}{N}_{s}}\mathop{\sum}\limits_{{{{{{{{\bf{q}}}}}}}}}{{{{{{{\rm{Im}}}}}}}}{D}^{{{{{{{{\rm{RPA}}}}}}}}}({{{{{{{\bf{q}}}}}}}},\nu )\left[{n}_{{{{{{{{\rm{F}}}}}}}}}\left(-{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}-{{{{{{{\bf{q}}}}}}}}}^{{{{{{{{\rm{RPA}}}}}}}}}\right)+{n}_{{{{{{{{\rm{B}}}}}}}}}(\nu )\right]\,,$$</span></div><div class="c-article-equation__number"> (8) </div></div><p>where <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{D}^{{{{{{{{\rm{RPA}}}}}}}}}({{{{{{{\bf{q}}}}}}}},\nu )={V}_{{{{{{{{\bf{q}}}}}}}}}^{2}{{{{{{{\rm{Im}}}}}}}}{{{\Pi }}}^{{{{{{{{\rm{RPA}}}}}}}}}({{{{{{{\bf{q}}}}}}}},\nu )$</span> is the imaginary part of the screened Coulomb interaction computed in the random phase approximation (RPA); <span class="mathjax-tex">$\nu =\omega -{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}-{{{{{{{\bf{q}}}}}}}}}^{{{{{{{{\rm{RPA}}}}}}}}}$</span>. Therefore weakly correlated electron systems in general can also host plasmarons in principle<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Aryasetiawan, F., Hedin, L. & Karlsson, K. Multiple plasmon satellites in Na and Al spectral functions from ab initio cumulant expansion. Phys. Rev. Lett. 77, 2268–2271 (1996)." href="#ref-CR35" id="ref-link-section-d10251175e5760">35</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Tediosi, R., Armitage, N. P., Giannini, E. & van der Marel, D. Charge carrier interaction with a purely electronic collective mode: Plasmarons and the infrared response of elemental bismuth. Phys. Rev. Lett. 99, 016406 (2007)." href="#ref-CR36" id="ref-link-section-d10251175e5760_1">36</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Polini, M. et al. Plasmons and the spectral function of graphene. Phys. Rev. B 77, 081411 (2008)." href="#ref-CR37" id="ref-link-section-d10251175e5760_2">37</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Hwang, E. H. & Das Sarma, S. Quasiparticle spectral function in doped graphene: electron-electron interaction effects in ARPES. Phys. Rev. B 77, 081412 (2008)." href="#ref-CR38" id="ref-link-section-d10251175e5760_3">38</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Caruso, F., Lambert, H. & Giustino, F. Band structures of plasmonic polarons. Phys. Rev. Lett. 114, 146404 (2015)." href="#ref-CR39" id="ref-link-section-d10251175e5760_4">39</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Kheifets, A. S., Sashin, V. A., Vos, M., Weigold, E. & Aryasetiawan, F. Spectral properties of quasiparticles in silicon: a test of many-body theory. Phys. Rev. B 68, 233205 (2003)." href="#ref-CR40" id="ref-link-section-d10251175e5760_5">40</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Caruso, F. & Giustino, F. Spectral fingerprints of electron-plasmon coupling. Phys. Rev. B 92, 045123 (2015)." href="#ref-CR41" id="ref-link-section-d10251175e5760_6">41</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Brar, V. W. et al. Observation of carrier-density-dependent many-body effects in graphene via tunneling spectroscopy. Phys. Rev. Lett. 104, 036805 (2010)." href="#ref-CR42" id="ref-link-section-d10251175e5760_7">42</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Guzzo, M. et al. Valence electron photoemission spectrum of semiconductors: ab initio description of multiple satellites. Phys. Rev. Lett. 107, 166401 (2011)." href="#ref-CR43" id="ref-link-section-d10251175e5760_8">43</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Markiewicz, R. S. & Bansil, A. Dispersion anomalies induced by the low-energy plasmon in the cuprates. Phys. Rev. B 75, 020508(R) (2007)." href="#ref-CR44" id="ref-link-section-d10251175e5760_9">44</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lischner, J., Vigil-Fowler, D. & Louie, S. G. Physical origin of satellites in photoemission of doped graphene: an ab initio GW plus cumulant study. Phys. Rev. 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B 84, 085410 (2011)." href="#ref-CR48" id="ref-link-section-d10251175e5760_13">48</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Dial, O. E., Ashoori, R. C., Pfeiffer, L. N. & West, K. W. Observations of plasmarons in a two-dimensional system: tunneling measurements using time-domain capacitance spectroscopy. Phys. Rev. B 85, 081306 (2012)." href="#ref-CR49" id="ref-link-section-d10251175e5760_14">49</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Jang, J. et al. Full momentum- and energy-resolved spectral function of a 2D electronic system. Science 358, 901–906 (2017)." href="#ref-CR50" id="ref-link-section-d10251175e5760_15">50</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 51" title="Liu, Z. et al. Electron-plasmon interaction induced plasmonic-polaron band replication in epitaxial perovskite SrIrO3 films. Sci. Bull. 66, 433–440 (2021)." href="/articles/s42005-023-01276-z#ref-CR51" id="ref-link-section-d10251175e5763">51</a></sup>. However, plasmarons are overdamped in many cases and leave faint spectral weight<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Hwang, E. H. & Das Sarma, S. Quasiparticle spectral function in doped graphene: electron-electron interaction effects in ARPES. Phys. Rev. B 77, 081412 (2008)." href="#ref-CR38" id="ref-link-section-d10251175e5767">38</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Caruso, F., Lambert, H. & Giustino, F. Band structures of plasmonic polarons. Phys. Rev. Lett. 114, 146404 (2015)." href="#ref-CR39" id="ref-link-section-d10251175e5767_1">39</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Kheifets, A. S., Sashin, V. A., Vos, M., Weigold, E. & Aryasetiawan, F. Spectral properties of quasiparticles in silicon: a test of many-body theory. Phys. Rev. B 68, 233205 (2003)." href="#ref-CR40" id="ref-link-section-d10251175e5767_2">40</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title="Caruso, F. & Giustino, F. Spectral fingerprints of electron-plasmon coupling. Phys. Rev. B 92, 045123 (2015)." href="/articles/s42005-023-01276-z#ref-CR41" id="ref-link-section-d10251175e5770">41</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Markiewicz, R. S. & Bansil, A. Dispersion anomalies induced by the low-energy plasmon in the cuprates. Phys. Rev. B 75, 020508(R) (2007)." href="#ref-CR44" id="ref-link-section-d10251175e5773">44</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lischner, J., Vigil-Fowler, D. & Louie, S. G. Physical origin of satellites in photoemission of doped graphene: an ab initio GW plus cumulant study. Phys. Rev. Lett. 110, 146801 (2013)." href="#ref-CR45" id="ref-link-section-d10251175e5773_1">45</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="Lischner, J. et al. Satellite band structure in silicon caused by electron-plasmon coupling. Phys. Rev. B 91, 205113 (2015)." href="/articles/s42005-023-01276-z#ref-CR46" id="ref-link-section-d10251175e5776">46</a></sup>. This unfavorable situation is soften when the system has a relatively small band width so that the correlation effect becomes relatively large. In fact, the importance of the small band width to plasmarons was discussed in SrIrO<sub>3</sub><sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 51" title="Liu, Z. et al. Electron-plasmon interaction induced plasmonic-polaron band replication in epitaxial perovskite SrIrO3 films. Sci. Bull. 66, 433–440 (2021)." href="/articles/s42005-023-01276-z#ref-CR51" id="ref-link-section-d10251175e5782">51</a></sup>. See Supplementary Note <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s42005-023-01276-z#MOESM2">5</a> for explicit results.</p><p>In the present <i>t</i>–<i>J</i> model, the band width is very small at order of <i>t</i><i>δ</i>/2 and <i>δ</i> ≈ 0.1–0.2. Furthermore, the magnitude of all the components of <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{\Sigma }_{ab}$</span> is comparable to each other, but the sign of <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{\Sigma }_{12}$</span> is the opposite to those of <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{\Sigma }_{11}$</span> and <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}{\Sigma }_{22}$</span> in <i>ω</i> < 0 (see Supplementary Note <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s42005-023-01276-z#MOESM2">2)</a>. Hence after the summation in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ2">2</a>), <span class="mathjax-tex">${{{{{{{\rm{Im}}}}}}}}\Sigma$</span> is substantially reduced in <i>ω</i> < 0. Nonetheless, a small band width allows to fulfill the resonance condition <span class="mathjax-tex">$\omega -{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}-{{{{{{{\rm{Re}}}}}}}}\Sigma ({{{{{{{\bf{k}}}}}}}},\omega )=0$</span> as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig2">2</a>f and h for <i>r</i> = 1. These features work constructively to host plasmarons in a strongly correlated electron system more than a weakly correlated one.</p><p>The dispersion of plasmarons exhibits a dispersive feature similar to the quasiparticle dispersion as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig4">4</a>—the plasmaron dispersion follows 0.98<i>ε</i><sub><b>k</b></sub> − 1.33<i>t</i>. The value of −1.33<i>t</i> is related to, but not exactly equal to, the optical plasmon energy <i>ν</i> = 1.15<i>t</i> and the factor 0.98 is a renormalization. The plasmaron dispersion can also be fitted to <i>ε</i><sub><b>k</b></sub> − 1.33<i>t</i> approximately. This feature is easily understood intuitively. The energy of the one-particle excitation <i>ω</i> is related to the charge fluctuation energy <i>ν</i> via <i>ω</i> = <i>ν</i> + <i>ε</i><sub><b>k</b>−<b>q</b></sub> in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ3">3</a>). Since it is the optical plasmon which generates the plasmarons, <i>ν</i> is estimated by its energy and <b>q</b> may be put to zero; recall that Im<i>D</i><sub><i>a</i><i>b</i></sub>(<b>q</b>, <i>ν</i>) is an odd function with respect to <i>ν</i>. Consequently, the dispersion of plasmarons essentially follows the (bare) quasiparticle dispersion <i>ε</i><sub><b>k</b></sub>. In this sense a term of “replica band” used in weakly correlated electron systems can be inherited even in strongly correlated electron systems.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-4" data-title="Dispersion of plasmarons."><figure><figcaption><b id="Fig4" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 4: Dispersion of plasmarons.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s42005-023-01276-z/figures/4" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs42005-023-01276-z/MediaObjects/42005_2023_1276_Fig4_HTML.png?as=webp"><img aria-describedby="Fig4" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs42005-023-01276-z/MediaObjects/42005_2023_1276_Fig4_HTML.png" alt="figure 4" loading="lazy" width="685" height="395"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-4-desc"><p>It follows 0.98<i>ε</i><sub><b>k</b></sub> − 1.33<i>t</i> (dashed curve) and in this sense, it is a replica band of <i>ε</i><sub><b>k</b></sub>.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s42005-023-01276-z/figures/4" data-track-dest="link:Figure4 Full size image" aria-label="Full size image figure 4" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>Figures <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig1">1</a>, <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig2">2</a>a, and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig4">4</a> can be applied directly to electron-doped cuprates, especially LCCO. The energy of plasmarons is controlled by the optical plasmon energy, which can be determined precisely by electron energy-loss spectroscopy and optical spectroscopy. Given that the typical energy scale of the optical plasmon in cuprates is around 1 eV, the plasmarons can be tested by angle-resolved photoemission spectroscopy (ARPES) by searching the energy region typically around 1 eV below the electron dispersion especially along the direction (0, 0)–(<i>π</i>, 0) [see the inset of Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig1">1</a> and Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig4">4</a>]. This energy window has not been studied in detail in ARPES<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 71" title="Armitage, N. P., Fournier, P. & Greene, R. L. Progress and perspectives on electron-doped cuprates. Rev. Mod. Phys. 82, 2421–2487 (2010)." href="/articles/s42005-023-01276-z#ref-CR71" id="ref-link-section-d10251175e6191">71</a></sup>. Recalling that the plasmarons were detected even in weakly correlated systems such as graphene<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title="Brar, V. W. et al. Observation of carrier-density-dependent many-body effects in graphene via tunneling spectroscopy. Phys. Rev. Lett. 104, 036805 (2010)." href="/articles/s42005-023-01276-z#ref-CR42" id="ref-link-section-d10251175e6195">42</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Bostwick, A. et al. Observation of plasmarons in quasi-freestanding doped graphene. Science 328, 999–1002 (2010)." href="/articles/s42005-023-01276-z#ref-CR47" id="ref-link-section-d10251175e6198">47</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 48" title="Walter, A. L. et al. Effective screening and the plasmaron bands in graphene. Phys. Rev. B 84, 085410 (2011)." href="/articles/s42005-023-01276-z#ref-CR48" id="ref-link-section-d10251175e6201">48</a></sup>, two-dimensional electron systems<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 49" title="Dial, O. E., Ashoori, R. C., Pfeiffer, L. N. & West, K. W. Observations of plasmarons in a two-dimensional system: tunneling measurements using time-domain capacitance spectroscopy. Phys. Rev. B 85, 081306 (2012)." href="/articles/s42005-023-01276-z#ref-CR49" id="ref-link-section-d10251175e6205">49</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 50" title="Jang, J. et al. Full momentum- and energy-resolved spectral function of a 2D electronic system. Science 358, 901–906 (2017)." href="/articles/s42005-023-01276-z#ref-CR50" id="ref-link-section-d10251175e6208">50</a></sup>, and SrIrO<sub>3</sub> films<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 51" title="Liu, Z. et al. Electron-plasmon interaction induced plasmonic-polaron band replication in epitaxial perovskite SrIrO3 films. Sci. Bull. 66, 433–440 (2021)." href="/articles/s42005-023-01276-z#ref-CR51" id="ref-link-section-d10251175e6214">51</a></sup>, there seems a good chance to reveal them also in cuprates.</p><p>In experiments<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 72" title="Uchida, S. et al. Optical spectra of La2−xSrxCuO4: effect of carrier doping on the electronic structure of the CuO2 plane. Phys. Rev. B 43, 7942–7954 (1991)." href="/articles/s42005-023-01276-z#ref-CR72" id="ref-link-section-d10251175e6222">72</a></sup>, the optical plasmon energy increases with carrier doping up to 20% doping. Therefore, the energy of plasmarons follows the same tendency. This feature can also be utilized to confirm plasmarons in cuprates. While ARPES is an ideal tool to test plasmarons, X-ray photoemission spectroscopy<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 43" title="Guzzo, M. et al. Valence electron photoemission spectrum of semiconductors: ab initio description of multiple satellites. Phys. Rev. Lett. 107, 166401 (2011)." href="/articles/s42005-023-01276-z#ref-CR43" id="ref-link-section-d10251175e6226">43</a></sup> and tunneling spectroscopy<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title="Brar, V. W. et al. Observation of carrier-density-dependent many-body effects in graphene via tunneling spectroscopy. Phys. Rev. Lett. 104, 036805 (2010)." href="/articles/s42005-023-01276-z#ref-CR42" id="ref-link-section-d10251175e6230">42</a></sup> can also be exploited to detect plasmarons as an emergent satellite peak.</p><p>What happens for hole-doped cuprates, where the optical plasmon<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Nücker, N. et al. Plasmons and interband transitions in Bi2Sr2CaCu2O8. Phys. Rev. B 39, 12379–12382 (1989)." href="#ref-CR32" id="ref-link-section-d10251175e6237">32</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Romberg, H. et al. Dielectric function of YBa2Cu3O7−δ between 50 meV and 50 eV. Z. Phys. B Condens. Matter 78, 367–380 (1990)." href="#ref-CR33" id="ref-link-section-d10251175e6237_1">33</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title="Bozovic, I. Plasmons in cuprate superconductors. Phys. Rev. B 42, 1969–1984 (1990)." href="/articles/s42005-023-01276-z#ref-CR34" id="ref-link-section-d10251175e6240">34</a></sup> as well as acoustic-like plasmons<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Hepting, M. et al. Gapped collective charge excitations and interlayer hopping in cuprate superconductors. Phys. Rev. Lett. 129, 047001 (2022)." href="#ref-CR17" id="ref-link-section-d10251175e6244">17</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Nag, A. et al. Detection of acoustic plasmons in hole-doped lanthanum and bismuth cuprate superconductors using resonant inelastic x-ray scattering. Phys. Rev. Lett. 125, 257002 (2020)." href="#ref-CR18" id="ref-link-section-d10251175e6244_1">18</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Singh, A. et al. Acoustic plasmons and conducting carriers in hole-doped cuprate superconductors. Phys. Rev. B 105, 235105 (2022)." href="/articles/s42005-023-01276-z#ref-CR19" id="ref-link-section-d10251175e6247">19</a></sup> typical to layered materials were actually observed? Performing the same analysis as that for the electron-doped cuprates, we predict plasmarons also in hole-doped cuprates. In contrast to the electron-doped case, however, they are realized along the direction (<i>π</i>, 0)–(<i>π</i>, <i>π</i>)–(<i>π</i>/2, <i>π</i>/2) in <i>ω</i> > 0, requiring inverse ARPES to test the plasmaron dispersion; See Supplementary Note <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s42005-023-01276-z#MOESM2">4</a> for details.</p><p>For cuprates, it has been discussed that coupling to bosonic fluctuations yields a kink in the electron dispersion. While plasmons are also bosonic fluctuations, their role in cuprates should be sharply distinguished from phonons<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Lanzara, A. et al. Evidence for ubiquitous strong electron–phonon coupling in high-temperature superconductors. Nature 412, 510–514 (2001)." href="/articles/s42005-023-01276-z#ref-CR7" id="ref-link-section-d10251175e6276">7</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Zhou, X. J. et al. Multiple bosonic mode coupling in the electron self-energy of (La2−xSrx)CuO4. Phys. Rev. Lett. 95, 117001 (2005)." href="/articles/s42005-023-01276-z#ref-CR8" id="ref-link-section-d10251175e6279">8</a></sup> and magnetic fluctuations<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Kaminski, A. et al. Renormalization of spectral line shape and dispersion below Tc in Bi2Sr2CaCu2O8+δ. Phys. Rev. Lett. 86, 1070–1073 (2001)." href="#ref-CR11" id="ref-link-section-d10251175e6283">11</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Johnson, P. D. et al. Doping and temperature dependence of the mass enhancement observed in the cuprate Bi2Sr2CaCu2O8+δ. Phys. Rev. Lett. 87, 177007 (2001)." href="#ref-CR12" id="ref-link-section-d10251175e6283_1">12</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Gromko, A. D. et al. Mass-renormalized electronic excitations at (π, 0) in the superconducting state of Bi2Sr2CaCu2O8+δ. Phys. Rev. B 68, 174520 (2003)." href="#ref-CR13" id="ref-link-section-d10251175e6283_2">13</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Mou, Y. & Feng, S. Doping dependence of charge order in electron-doped cuprate superconductors. Philos. Mag. 97, 3361–3380 (2017)." href="/articles/s42005-023-01276-z#ref-CR14" id="ref-link-section-d10251175e6286">14</a></sup>. Plasmons do not yield a kink (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig1">1</a>), but instead generate plasmarons as an emergent incoherent band (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s42005-023-01276-z#Fig4">4</a>).</p><p>The present calculations have been performed in a layered <i>t</i>–<i>J</i> model. If one employs a two-dimensional model, the plasmon dispersion in cuprates cannot be captured especially for the optical plasmon. In this sense, the inclusion of the three-dimensionality of the long-range Coulomb interaction is crucially important to discuss plasmarons in cuprates, although we checked that the interlayer hopping integral <i>t</i><sub><i>z</i></sub> is not relevant to plasmarons.</p><p>A replica band is also discussed in the polar electron–phonon coupling mechanism in TiO<sub>2</sub><sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Moser, S. et al. Tunable polaronic conduction in anatase TiO2. Phys. Rev. Lett. 110, 196403 (2013)." href="#ref-CR73" id="ref-link-section-d10251175e6315">73</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Verdi, C., Caruso, F. & Giustino, F. Origin of the crossover from polarons to Fermi liquids in transition metal oxides. Nat. Commun. 8, 15769 (2017)." href="#ref-CR74" id="ref-link-section-d10251175e6315_1">74</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 75" title="Caruso, F., Verdi, C., Poncé, S. & Giustino, F. Electron-plasmon and electron-phonon satellites in the angle-resolved photoelectron spectra of n-doped anatase TiO2. Phys. Rev. B 97, 165113 (2018)." href="/articles/s42005-023-01276-z#ref-CR75" id="ref-link-section-d10251175e6318">75</a></sup> and the interplay between the electron–phonon and electron–plasmon couplings was studied<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 76" title="Jalabert, R. & Das Sarma, S. Quasiparticle properties of a coupled two-dimensional electron-phonon system. Phys. Rev. B 40, 9723–9737 (1989)." href="/articles/s42005-023-01276-z#ref-CR76" id="ref-link-section-d10251175e6322">76</a></sup>. A clear distinction between those two couplings is made by studying the carrier density dependence of the replica band<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 51" title="Liu, Z. et al. Electron-plasmon interaction induced plasmonic-polaron band replication in epitaxial perovskite SrIrO3 films. Sci. Bull. 66, 433–440 (2021)." href="/articles/s42005-023-01276-z#ref-CR51" id="ref-link-section-d10251175e6326">51</a></sup>. In the present study, however, the electron–phonon coupling is irrelevant because phonon energy is limited below 100 meV in cuprates whereas our relevant energy scale is about 1 eV.</p></div></div></section><section data-title="Conclusions"><div class="c-article-section" id="Sec8-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec8">Conclusions</h2><div class="c-article-section__content" id="Sec8-content"><p>The present large-<i>N</i> theory captures the plasmon excitations observed in both electron- and hole-doped high-temperature cuprate superconductors with a good accuracy so that detailed comparisons with experimental data were made<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Hepting, M. et al. Gapped collective charge excitations and interlayer hopping in cuprate superconductors. Phys. Rev. Lett. 129, 047001 (2022)." href="/articles/s42005-023-01276-z#ref-CR17" id="ref-link-section-d10251175e6341">17</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Nag, A. et al. Detection of acoustic plasmons in hole-doped lanthanum and bismuth cuprate superconductors using resonant inelastic x-ray scattering. Phys. Rev. Lett. 125, 257002 (2020)." href="/articles/s42005-023-01276-z#ref-CR18" id="ref-link-section-d10251175e6344">18</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Greco, A., Yamase, H. & Bejas, M. Origin of high-energy charge excitations observed by resonant inelastic x-ray scattering in cuprate superconductors. Commun. Phys. 2, 3 (2019)." href="/articles/s42005-023-01276-z#ref-CR21" id="ref-link-section-d10251175e6347">21</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Greco, A., Yamase, H. & Bejas, M. Close inspection of plasmon excitations in cuprate superconductors. Phys. Rev. B 102, 024509 (2020)." href="/articles/s42005-023-01276-z#ref-CR22" id="ref-link-section-d10251175e6350">22</a></sup>. We have computed the electron self-energy in the same theoretical framework, but by going beyond leading order theory.</p><p>Our major point lies in the indication that cuprates can host plasmarons—quasiparticles coupling to plasmons—near the optical plasmon energy below (above) the quasiparticle dispersion in electron-doped (hole-doped) cuprates; plasmons do not yield a kink in the quasiparticle dispersion, in stark contrast to phonons and magnetic fluctuations. Since plasmarons are found clearly close to momentum (<i>π</i>, 0), where the superconducting gap as well as the pseudogap is enhanced, it is very interesting to explore further the role of plasmarons in the formation of the superconducting gap and the pseudogap in cuprate superconductors.</p><p>Our second major point lies in elucidating the mechanism of plasmarons: they are driven by the strong correlation effect—fluctuations associated with the local constraint that imposes no double occupancy of electrons at any site. The underlying physics to generate plasmarons in cuprates is thus different from that in weakly correlated electron systems<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Aryasetiawan, F., Hedin, L. & Karlsson, K. Multiple plasmon satellites in Na and Al spectral functions from ab initio cumulant expansion. Phys. Rev. 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Quasiparticle spectral function in doped graphene: electron-electron interaction effects in ARPES. Phys. Rev. B 77, 081412 (2008)." href="#ref-CR38" id="ref-link-section-d10251175e6363_3">38</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Caruso, F., Lambert, H. & Giustino, F. Band structures of plasmonic polarons. Phys. Rev. Lett. 114, 146404 (2015)." href="#ref-CR39" id="ref-link-section-d10251175e6363_4">39</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Kheifets, A. S., Sashin, V. A., Vos, M., Weigold, E. & Aryasetiawan, F. Spectral properties of quasiparticles in silicon: a test of many-body theory. Phys. Rev. B 68, 233205 (2003)." href="#ref-CR40" id="ref-link-section-d10251175e6363_5">40</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Caruso, F. & Giustino, F. 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Observation of plasmarons in quasi-freestanding doped graphene. Science 328, 999–1002 (2010)." href="#ref-CR47" id="ref-link-section-d10251175e6363_12">47</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Walter, A. L. et al. Effective screening and the plasmaron bands in graphene. Phys. Rev. B 84, 085410 (2011)." href="#ref-CR48" id="ref-link-section-d10251175e6363_13">48</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Dial, O. E., Ashoori, R. C., Pfeiffer, L. N. & West, K. W. Observations of plasmarons in a two-dimensional system: tunneling measurements using time-domain capacitance spectroscopy. Phys. Rev. B 85, 081306 (2012)." href="#ref-CR49" id="ref-link-section-d10251175e6363_14">49</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Jang, J. et al. Full momentum- and energy-resolved spectral function of a 2D electronic system. Science 358, 901–906 (2017)." href="#ref-CR50" id="ref-link-section-d10251175e6363_15">50</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 51" title="Liu, Z. et al. Electron-plasmon interaction induced plasmonic-polaron band replication in epitaxial perovskite SrIrO3 films. Sci. Bull. 66, 433–440 (2021)." href="/articles/s42005-023-01276-z#ref-CR51" id="ref-link-section-d10251175e6366">51</a></sup>. However, both have a common mathematical structure to yield plasmarons, establishing a general concept of plasmarons in metals. Plasmarons tend to be well-defined for a system with a smaller band width. This condition is usually fulfilled in cuprates because of strong correlations, but also in a weakly correlated system such as SrIrO<sub>3</sub> films<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 51" title="Liu, Z. et al. Electron-plasmon interaction induced plasmonic-polaron band replication in epitaxial perovskite SrIrO3 films. Sci. Bull. 66, 433–440 (2021)." href="/articles/s42005-023-01276-z#ref-CR51" id="ref-link-section-d10251175e6372">51</a></sup>.</p></div></div></section><section data-title="Methods"><div class="c-article-section" id="Sec9-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec9">Methods</h2><div class="c-article-section__content" id="Sec9-content"><p>We present a minimal description of the large-<i>N</i> theory of the layered <i>t</i>–<i>J</i> model with the long-range Coulomb interaction—a complete formalism is given in Yamase et al.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 66" title="Yamase, H., Bejas, M. & Greco, A. Electron self-energy from quantum charge fluctuations in the layered t-J model with long-range coulomb interaction. Phys. Rev. B 104, 045141 (2021)." href="/articles/s42005-023-01276-z#ref-CR66" id="ref-link-section-d10251175e6393">66</a></sup>.</p><p>The electron dispersion <i>ε</i><sub><b>k</b></sub> consists of the in-plane dispersion <span class="mathjax-tex">${\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}^{\parallel }$</span> and the out-of-plane dispersion <span class="mathjax-tex">${\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}^{\perp }$</span>,</p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}={\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}^{\parallel }+{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}^{\perp }\,.$$</span></div><div class="c-article-equation__number"> (9) </div></div><p>At leading order, they are calculated as</p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}^{\parallel }=-2\left(t\frac{\delta }{2}+\Delta \right)\left(\cos {k}_{x}+\cos {k}_{y}\right)-4{t}^{{\prime} }\frac{\delta }{2}\cos {k}_{x}\cos {k}_{y}-\mu \,,$$</span></div><div class="c-article-equation__number"> (10) </div></div><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}^{\perp }=-2{t}_{z}\frac{\delta }{2}{\left(\cos {k}_{x}-\cos {k}_{y}\right)}^{2}\cos {k}_{z}\,,$$</span></div><div class="c-article-equation__number"> (11) </div></div><p>where Δ is the mean value of the bond field, <i>δ</i> the doping rate, and <i>μ</i> the chemical potential. For a given <i>δ</i>, Δ and <i>μ</i> are determined self-consistently by solving the following coupled equations:</p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\Delta =\frac{J}{4{N}_{s}{N}_{z}}\mathop{\sum}\limits_{{{{{{{{\bf{k}}}}}}}}}\left(\cos {k}_{x}+\cos {k}_{y}\right){n}_{{{{{{{{\rm{F}}}}}}}}}\left({\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}\right)\,,$$</span></div><div class="c-article-equation__number"> (12) </div></div><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$(1-\delta )=\frac{2}{{N}_{s}{N}_{z}}\mathop{\sum}\limits_{{{{{{{{\bf{k}}}}}}}}}{n}_{{{{{{{{\rm{F}}}}}}}}}({\varepsilon }_{{{{{{{{\bf{k}}}}}}}}})\,.$$</span></div><div class="c-article-equation__number"> (13) </div></div><p>As already mentioned in the Analytical Scheme subsection, charge fluctuations in the <i>t</i>–<i>J</i> model are composed of on-site charge and bond-charge fluctuations. They are, however, essentially decoupled to each other<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 65" title="Bejas, M., Yamase, H. & Greco, A. Dual structure in the charge excitation spectrum of electron-doped cuprates. Phys. Rev. B 96, 214513 (2017)." href="/articles/s42005-023-01276-z#ref-CR65" id="ref-link-section-d10251175e7185">65</a></sup>. Since the former is relevant to the present work, we focus on that. In this case, the bosonic propagator of charge fluctuations, namely <i>D</i><sub><i>a</i><i>b</i></sub>(<b>q</b>, i<i>ν</i><sub><i>n</i></sub>), is described by a 2 × 2 matrix with <i>a</i>, <i>b</i> = 1, 2:</p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${\left[{D}_{ab}\left({{{{{{{\bf{q}}}}}}}},{{{{{{{\rm{i}}}}}}}}{\nu }_{n}\right)\right]}^{-1}={\left[{D}_{ab}^{(0)}\left({{{{{{{\bf{q}}}}}}}},{{{{{{{\rm{i}}}}}}}}{\nu }_{n}\right)\right]}^{-1}-{{{\Pi }}}_{ab}\left({{{{{{{\bf{q}}}}}}}},{{{{{{{\rm{i}}}}}}}}{\nu }_{n}\right)\,,$$</span></div><div class="c-article-equation__number"> (14) </div></div><p>where <span class="mathjax-tex">${D}_{ab}^{(0)}({{{{{{{\bf{q}}}}}}}},{{{{{{{\rm{i}}}}}}}}{\nu }_{n})$</span> is the bare bosonic propagator,</p><div id="Equ15" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${\left[{D}_{ab}^{(0)}\left({{{{{{{\bf{q}}}}}}}},{{{{{{{\rm{i}}}}}}}}{\nu }_{n}\right)\right]}^{-1}=N\left(\begin{array}{cc}\frac{{\delta }^{2}}{2}\left({V}_{{{{{{{{\bf{q}}}}}}}}}-{J}_{{{{{{{{\bf{q}}}}}}}}}\right)&\frac{\delta }{2}\\ \frac{\delta }{2}&0\end{array}\right)\,.$$</span></div><div class="c-article-equation__number"> (15) </div></div><p>Here <span class="mathjax-tex">${J}_{{{{{{{{\bf{q}}}}}}}}}=\frac{J}{2}(\cos {q}_{x}+\cos {q}_{y})$</span> is the superexchange interaction in momentum space and <i>V</i><sub><b>q</b></sub> is the long-range Coulomb interaction for a layered system<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 77" title="Becca, F., Tarquini, M., Grilli, M. & Di Castro, C. Charge-density waves and superconductivity as an alternative to phase separation in the infinite-U Hubbard-Holstein model. Phys. Rev. B 54, 12443–12457 (1996)." href="/articles/s42005-023-01276-z#ref-CR77" id="ref-link-section-d10251175e7811">77</a></sup>:</p><div id="Equ16" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${V}_{{{{{{{{\bf{q}}}}}}}}}=\frac{{V}_{{{{{{{{\rm{c}}}}}}}}}}{A({q}_{x},{q}_{y})-\cos {q}_{z}}\,,$$</span></div><div class="c-article-equation__number"> (16) </div></div><p>where <span class="mathjax-tex">${V}_{{{{{{{{\rm{c}}}}}}}}}={e}^{2}d{(2{\epsilon }_{\perp }{a}^{2})}^{-1}$</span> and <span class="mathjax-tex">$A({q}_{x},{q}_{y})=\alpha (2-\cos {q}_{x}-\cos {q}_{y})+1$</span>; <i>e</i> is the electric charge of electrons, <i>a</i> the unit length of the square lattice, <i>d</i> the distance between the layers, <i>α</i> describes the anisotropy between the in-plane and out-of-plane interaction and is given by <span class="mathjax-tex">$\alpha =\frac{\tilde{\epsilon }}{{(a/d)}^{2}}$</span> with <span class="mathjax-tex">$\tilde{\epsilon }={\epsilon }_{\parallel }/{\epsilon }_{\perp }$</span>, where <i>ϵ</i><sub><span class="stix">∥</span></sub> and <i>ϵ</i><sub><span class="stix">⊥</span></sub> are the dielectric constants parallel and perpendicular to the planes, respectively. The 2 × 2 matrix Π<sub><i>a</i><i>b</i></sub> is the bosonic self-energy at leading order</p><div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${{{\Pi }}}_{ab}({{{{{{{\bf{q}}}}}}}},{{{{{{{\rm{i}}}}}}}}{\nu }_{n})= -\frac{N}{{N}_{s}{N}_{z}}\mathop{\sum}\limits_{{{{{{{{\bf{k}}}}}}}}}{h}_{a}\left({{{{{{{\bf{k}}}}}}}},{{{{{{{\bf{q}}}}}}}},{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}-{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}-{{{{{{{\bf{q}}}}}}}}}\right)\frac{{n}_{{{{{{{{\rm{F}}}}}}}}}\left({\varepsilon }_{{{{{{{{\bf{k}}}}}}}}-{{{{{{{\bf{q}}}}}}}}}\right)-{n}_{{{{{{{{\rm{F}}}}}}}}}\left({\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}\right)}{{{{{{{{\rm{i}}}}}}}}{\nu }_{n}-{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}+{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}-{{{{{{{\bf{q}}}}}}}}}}{h}_{b}({{{{{{{\bf{k}}}}}}}},{{{{{{{\bf{q}}}}}}}},{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}-{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}-{{{{{{{\bf{q}}}}}}}}})\\ -{\delta }_{a1}{\delta }_{b1}\frac{N}{{N}_{s}{N}_{z}}\mathop{\sum}\limits_{{{{{{{{\bf{k}}}}}}}}}\frac{{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}-{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}-{{{{{{{\bf{q}}}}}}}}}}{2}{n}_{{{{{{{{\rm{F}}}}}}}}}\left({\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}\right),$$</span></div><div class="c-article-equation__number"> (17) </div></div><p>and the 2-component vertex is given by</p><div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${h}_{a}({{{{{{{\bf{k}}}}}}}},{{{{{{{\bf{q}}}}}}}},\nu )=\left(\frac{2{\varepsilon }_{{{{{{{{\bf{k}}}}}}}}-{{{{{{{\bf{q}}}}}}}}}+\nu +2\mu }{2}+2\Delta \left[\cos \left({k}_{x}-\frac{{q}_{x}}{2}\right)\cos \left(\frac{{q}_{x}}{2}\right) +\cos \left({k}_{y}-\frac{{q}_{y}}{2}\right)\cos \left(\frac{{q}_{y}}{2}\right)\right],1\right)\,.$$</span></div><div class="c-article-equation__number"> (18) </div></div><p>We then compute the electron self-energy from charge fluctuations described by Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ14">14</a>) at order of 1/<i>N</i>. This yields Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s42005-023-01276-z#Equ3">3</a>) in the main text—its derivation is elaborated in Yamase et al.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 66" title="Yamase, H., Bejas, M. & Greco, A. Electron self-energy from quantum charge fluctuations in the layered t-J model with long-range coulomb interaction. Phys. Rev. B 104, 045141 (2021)." href="/articles/s42005-023-01276-z#ref-CR66" id="ref-link-section-d10251175e9167">66</a></sup>.</p><p>Fixing temperature to zero, we choose parameters <i>J</i>/<i>t</i> = 0.3, <span class="mathjax-tex">${t}^{{\prime} }/t=0.3$</span>, <i>t</i><sub><i>z</i></sub>/<i>t</i> = 0.03, <i>α</i> = 2.9, <i>V</i><sub>c</sub>/<i>t</i> = 18, <i>δ</i> = 0.175, <i>N</i><sub><i>z</i></sub> = 10, Γ<sub>ch</sub>/<i>t</i> = 0.03, Γ<sub>sf</sub>/<i>t</i> = 0.03, and <i>t</i>/2 = 0.5 eV, which reproduce semiquantitatively the plasmon excitations observed in RIXS for one of the typical electron-doped cuprates LCCO<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Hepting, M. et al. Gapped collective charge excitations and interlayer hopping in cuprate superconductors. Phys. Rev. Lett. 129, 047001 (2022)." href="/articles/s42005-023-01276-z#ref-CR17" id="ref-link-section-d10251175e9263">17</a></sup>. The factor of 1/2 in <i>t</i>/2 comes from a large-<i>N</i> formalism where <i>t</i> is scaled by 1/<i>N</i>. We assume <i>N</i> = 2 in comparison with experiments. These parameters were obtained to achieve the best fit to the experimental data under an additional conditions that they should be realistic and do not contradict with the existing knowledge. While Γ<sub>ch</sub> and Γ<sub>sf</sub> are positive infinitesimals from the analytical point of view, we employ small, but finite values in actual numerical calculations. This may mimics broadening of the spectrum due to electron correlations at higher orders as well as instrumental resolution. In the figures we presented, all quantities with the dimension of energy are measured in units of <i>t</i>.</p></div></div></section> </div> <div> <section data-title="Data availability"><div class="c-article-section" id="data-availability-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="data-availability">Data availability</h2><div class="c-article-section__content" id="data-availability-content"> <p>The data that support the finding of this study are available from H.Y. and M.B. upon reasonable request.</p> </div></div></section><div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1">Cooper, L. 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P. Armitage, M. Hepting, and A. M. Oleś for valuable discussions. A part of the results presented in this work was obtained by using the facilities of the CCT-Rosario Computational Center, member of the High Performance Computing National System (SNCAD, MincyT-Argentina). A.G. is indebted to warm hospitality of Max-Planck-Institute for Solid State Research. H.Y. was supported by JSPS KAKENHI Grant No. JP20H01856.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">International Center of Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, 305-0047, Japan</p><p class="c-article-author-affiliation__authors-list">Hiroyuki Yamase</p></li><li id="Aff2"><p class="c-article-author-affiliation__address">Facultad de Ciencias Exactas, Ingeniería y Agrimensura and Instituto de Física Rosario (UNR-CONICET), Avenida Pellegrini 250, 2000, Rosario, Argentina</p><p class="c-article-author-affiliation__authors-list">Matías Bejas & Andrés Greco</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-Hiroyuki-Yamase-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">Hiroyuki Yamase</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?author=Hiroyuki%20Yamase" class="c-article-button" data-track="click" data-track-action="author link - 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M.B. performed numerical calculations with the help of A.G. and all authors analyzed the data together. H.Y. wrote the manuscript with input from M.B. and A.G.</p><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding author</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:yamase.hiroyuki@nims.go.jp">Hiroyuki Yamase</a>.</p></div></div></section><section data-title="Ethics declarations"><div class="c-article-section" id="ethics-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="ethics">Ethics declarations</h2><div class="c-article-section__content" id="ethics-content"> <h3 class="c-article__sub-heading" id="FPar2">Competing interests</h3> <p>The authors declare no competing interests.</p> </div></div></section><section data-title="Peer review"><div class="c-article-section" id="peer-review-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="peer-review">Peer review</h2><div class="c-article-section__content" id="peer-review-content"> <h3 class="c-article__sub-heading" id="FPar1">Peer review information</h3> <p><i>Communications Physics</i> thanks Andrzej Oleś and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. 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