Emergent Properties of the Periodic Anderson Model: A High-Resolution, Real-Frequency Study of Heavy-Fermion Quantum Criticality

IF 11.6 1区 物理与天体物理 Q1 PHYSICS, MULTIDISCIPLINARY Physical Review X Pub Date : 2024-11-07 DOI:10.1103/physrevx.14.041036
Andreas Gleis, Seung-Sup B. Lee, Gabriel Kotliar, Jan von Delft
{"title":"Emergent Properties of the Periodic Anderson Model: A High-Resolution, Real-Frequency Study of Heavy-Fermion Quantum Criticality","authors":"Andreas Gleis, Seung-Sup B. Lee, Gabriel Kotliar, Jan von Delft","doi":"10.1103/physrevx.14.041036","DOIUrl":null,"url":null,"abstract":"We study paramagnetic quantum criticality in the periodic Anderson model (PAM) using cellular dynamical mean-field theory (CDMFT), with the numerical renormalization group (NRG) as a cluster impurity solver. The PAM describes itinerant <mjx-container ctxtmenu_counter=\"273\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"c\" data-semantic-type=\"identifier\"><mjx-c>𝑐</mjx-c></mjx-mi></mjx-math></mjx-container> electrons hybridizing with a lattice of localized <mjx-container ctxtmenu_counter=\"274\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"f\" data-semantic-type=\"identifier\"><mjx-c>𝑓</mjx-c></mjx-mi></mjx-math></mjx-container> electrons. At zero temperature, it exhibits a much-studied quantum phase transition from a Kondo phase to a Ruderman-Kittel-Kasuya-Yosida (RKKY) phase when the hybridization is decreased through a so-called Kondo breakdown quantum critical point (KB QCP). There, Kondo screening of <mjx-container ctxtmenu_counter=\"275\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"f\" data-semantic-type=\"identifier\"><mjx-c>𝑓</mjx-c></mjx-mi></mjx-math></mjx-container> spins by <mjx-container ctxtmenu_counter=\"276\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"c\" data-semantic-type=\"identifier\"><mjx-c>𝑐</mjx-c></mjx-mi></mjx-math></mjx-container> electrons breaks down, so that <mjx-container ctxtmenu_counter=\"277\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"f\" data-semantic-type=\"identifier\"><mjx-c>𝑓</mjx-c></mjx-mi></mjx-math></mjx-container> excitations change their character from somewhat itinerant to mainly localized, while <mjx-container ctxtmenu_counter=\"278\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"c\" data-semantic-type=\"identifier\"><mjx-c>𝑐</mjx-c></mjx-mi></mjx-math></mjx-container> excitations remain itinerant. Building on <span>Phys. Rev. Lett.</span> <b>101</b>, 256404 (2008), which interpreted the KB transition as an orbital-selective Mott transition, we here elucidate its nature in great detail by performing a high-resolution, real-frequency study of various dynamical quantities (susceptibilities, self-energies, and spectral functions). NRG allows us to study the quantum critical regime governed by the QCP and located between two temperature scales, <mjx-container ctxtmenu_counter=\"279\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math breakable=\"true\" data-semantic-children=\"2,6\" data-semantic-content=\"3\" data-semantic- data-semantic-owns=\"2 3 6\" data-semantic-role=\"inequality\" data-semantic-speech=\"upper T Subscript upper F upper L Baseline less than upper T Subscript upper N upper F upper L\" data-semantic-structure=\"(7 (2 0 1) 3 (6 4 5))\" data-semantic-type=\"relseq\"><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-owns=\"0 1\" data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑇</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.048em;\"><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"unknown\" data-semantic-type=\"identifier\" size=\"s\"><mjx-c noic=\"true\" style=\"padding-top: 0.657em;\">F</mjx-c><mjx-c style=\"padding-top: 0.657em;\">L</mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-break size=\"4\"></mjx-break><mjx-mo data-semantic- data-semantic-operator=\"relseq,&lt;\" data-semantic-parent=\"7\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\"><mjx-c>&lt;</mjx-c></mjx-mo><mjx-msub data-semantic-children=\"4,5\" data-semantic- data-semantic-owns=\"4 5\" data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\" space=\"4\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑇</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.048em;\"><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"unknown\" data-semantic-type=\"identifier\" size=\"s\"><mjx-c noic=\"true\" style=\"padding-top: 0.657em;\">N</mjx-c><mjx-c noic=\"true\" style=\"padding-top: 0.657em;\">F</mjx-c><mjx-c style=\"padding-top: 0.657em;\">L</mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math></mjx-container>. In this regime, we find fingerprints of non-Fermi-liquid (NFL) behavior in several dynamical susceptibilities. Surprisingly, CDMFT self-consistency is essential to stabilize the QCP and the NFL regime. The Fermi-liquid (FL) scale <mjx-container ctxtmenu_counter=\"280\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"(2 0 1)\"><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-owns=\"0 1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper T Subscript upper F upper L\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑇</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.048em;\"><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"unknown\" data-semantic-type=\"identifier\" size=\"s\"><mjx-c noic=\"true\" style=\"padding-top: 0.657em;\">F</mjx-c><mjx-c style=\"padding-top: 0.657em;\">L</mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math></mjx-container> decreases toward and vanishes at the KB QCP; at temperatures below <mjx-container ctxtmenu_counter=\"281\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"(2 0 1)\"><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-owns=\"0 1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper T Subscript upper F upper L\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑇</mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.048em;\"><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"unknown\" data-semantic-type=\"identifier\" size=\"s\"><mjx-c noic=\"true\" style=\"padding-top: 0.657em;\">F</mjx-c><mjx-c style=\"padding-top: 0.657em;\">L</mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math></mjx-container>, FL behavior emerges. At <mjx-container ctxtmenu_counter=\"282\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math breakable=\"true\" data-semantic-children=\"0,2\" data-semantic-content=\"1\" data-semantic- data-semantic-owns=\"0 1 2\" data-semantic-role=\"equality\" data-semantic-speech=\"upper T equals 0\" data-semantic-structure=\"(3 0 1 2)\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c>𝑇</mjx-c></mjx-mi><mjx-break size=\"4\"></mjx-break><mjx-mo data-semantic- data-semantic-operator=\"relseq,=\" data-semantic-parent=\"3\" data-semantic-role=\"equality\" data-semantic-type=\"relation\"><mjx-c>=</mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"integer\" data-semantic-type=\"number\" space=\"4\"><mjx-c>0</mjx-c></mjx-mn></mjx-math></mjx-container>, we find the following properties. The KB transition is continuous. The <mjx-container ctxtmenu_counter=\"283\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"f\" data-semantic-type=\"identifier\"><mjx-c>𝑓</mjx-c></mjx-mi></mjx-math></mjx-container> quasiparticle weight decreases continuously as the transition is approached from either side, vanishing only at the KB QCP. Therefore, the quasiparticle weight of the <mjx-container ctxtmenu_counter=\"284\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"f\" data-semantic-type=\"identifier\"><mjx-c>𝑓</mjx-c></mjx-mi></mjx-math></mjx-container> band is nonzero not only in the Kondo phase, but also in the RKKY phase; hence, the FL quasiparticles comprise <mjx-container ctxtmenu_counter=\"285\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"c\" data-semantic-type=\"identifier\"><mjx-c>𝑐</mjx-c></mjx-mi></mjx-math></mjx-container> <i>and</i> <mjx-container ctxtmenu_counter=\"286\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"f\" data-semantic-type=\"identifier\"><mjx-c>𝑓</mjx-c></mjx-mi></mjx-math></mjx-container> electrons in both phases. The Fermi surface (FS) volumes in the two phases differ, implying a FS reconstruction at the KB QCP. Whereas the large-FS Kondo phase has a two-band structure as expected, the small-FS RKKY phase unexpectedly has a three-band structure. We provide a detailed analysis of quasiparticle properties of both the Kondo and, for the first time, also the RKKY phase and uncover their differences. The FS reconstruction is accompanied by the appearance of a Luttinger surface on which the <mjx-container ctxtmenu_counter=\"287\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"f\" data-semantic-type=\"identifier\"><mjx-c>𝑓</mjx-c></mjx-mi></mjx-math></mjx-container> self-energy diverges. The volumes of the Luttinger and Fermi surfaces are related to the charge density by a generalized Luttinger sum rule. We interpret the small FS volume and the emergent Luttinger surface as evidence for <mjx-container ctxtmenu_counter=\"288\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" overflow=\"linebreak\" role=\"tree\" sre-explorer- style=\"font-size: 100.7%;\" tabindex=\"0\"><mjx-math data-semantic-structure=\"0\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"f\" data-semantic-type=\"identifier\"><mjx-c>𝑓</mjx-c></mjx-mi></mjx-math></mjx-container>-electron fractionalization in the RKKY phase. Finally, we compute the temperature dependence of the Hall coefficient and the specific heat, finding good qualitative agreement with experiments.","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":null,"pages":null},"PeriodicalIF":11.6000,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physical Review X","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1103/physrevx.14.041036","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
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Abstract

We study paramagnetic quantum criticality in the periodic Anderson model (PAM) using cellular dynamical mean-field theory (CDMFT), with the numerical renormalization group (NRG) as a cluster impurity solver. The PAM describes itinerant 𝑐 electrons hybridizing with a lattice of localized 𝑓 electrons. At zero temperature, it exhibits a much-studied quantum phase transition from a Kondo phase to a Ruderman-Kittel-Kasuya-Yosida (RKKY) phase when the hybridization is decreased through a so-called Kondo breakdown quantum critical point (KB QCP). There, Kondo screening of 𝑓 spins by 𝑐 electrons breaks down, so that 𝑓 excitations change their character from somewhat itinerant to mainly localized, while 𝑐 excitations remain itinerant. Building on Phys. Rev. Lett. 101, 256404 (2008), which interpreted the KB transition as an orbital-selective Mott transition, we here elucidate its nature in great detail by performing a high-resolution, real-frequency study of various dynamical quantities (susceptibilities, self-energies, and spectral functions). NRG allows us to study the quantum critical regime governed by the QCP and located between two temperature scales, 𝑇FL<𝑇NFL. In this regime, we find fingerprints of non-Fermi-liquid (NFL) behavior in several dynamical susceptibilities. Surprisingly, CDMFT self-consistency is essential to stabilize the QCP and the NFL regime. The Fermi-liquid (FL) scale 𝑇FL decreases toward and vanishes at the KB QCP; at temperatures below 𝑇FL, FL behavior emerges. At 𝑇=0, we find the following properties. The KB transition is continuous. The 𝑓 quasiparticle weight decreases continuously as the transition is approached from either side, vanishing only at the KB QCP. Therefore, the quasiparticle weight of the 𝑓 band is nonzero not only in the Kondo phase, but also in the RKKY phase; hence, the FL quasiparticles comprise 𝑐 and 𝑓 electrons in both phases. The Fermi surface (FS) volumes in the two phases differ, implying a FS reconstruction at the KB QCP. Whereas the large-FS Kondo phase has a two-band structure as expected, the small-FS RKKY phase unexpectedly has a three-band structure. We provide a detailed analysis of quasiparticle properties of both the Kondo and, for the first time, also the RKKY phase and uncover their differences. The FS reconstruction is accompanied by the appearance of a Luttinger surface on which the 𝑓 self-energy diverges. The volumes of the Luttinger and Fermi surfaces are related to the charge density by a generalized Luttinger sum rule. We interpret the small FS volume and the emergent Luttinger surface as evidence for 𝑓-electron fractionalization in the RKKY phase. Finally, we compute the temperature dependence of the Hall coefficient and the specific heat, finding good qualitative agreement with experiments.

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周期性安德森模型的新兴特性:重费米子量子临界的高分辨率实频研究
我们使用蜂窝动力学均场理论(CDMFT)研究了周期性安德森模型(PAM)中的顺磁性量子临界,并将数值重正化群(NRG)作为簇杂质求解器。PAM 描述了巡回𝑐电子与局部𝑓电子晶格的杂化。在零温条件下,当杂化程度降低到所谓的 Kondo 击穿量子临界点(KB QCP)时,它表现出从 Kondo 相到 Ruderman-Kittel-Kasuya-Yosida (RKKY) 相的量子相变,这种相变已被广泛研究。在那里,𝑓 自旋对𝑐 电子的 Kondo 屏蔽被打破,因此𝑓 激发改变了它们的特性,从一定程度上的巡回变为主要的局部化,而𝑐 激发则保持巡回。基于《物理评论快报》(Phys.101, 256404 (2008)将 KB 转变解释为轨道选择性莫特转变的基础上,我们在这里通过对各种动力学量(感度、自能和光谱函数)进行高分辨率、实频研究,详细阐明了它的性质。NRG 使我们能够研究 QCP 所支配的、位于两个温度标度 𝑇FL<𝑇NFL 之间的量子临界体系。在这一机制中,我们在几种动力学易感性中发现了非费米液体(NFL)行为的指纹。令人惊讶的是,CDMFT 的自洽性对于稳定 QCP 和 NFL 体系至关重要。费米液体(FL)尺度 𝑇FL 向 KB QCP 减小并消失;在温度低于 𝑇FL 时,FL 行为出现。在 𝑇=0 时,我们发现以下特性。KB 过渡是连续的。𝑓准粒子重量在从两侧接近转变时持续减少,仅在 KB QCP 处消失。因此,𝑓 带的准粒子权重不仅在 Kondo 相中不为零,在 RKKY 相中也不为零;因此,FL 准粒子在这两个相中都包括𝑐 和 𝑓 电子。两相中的费米面(FS)体积不同,这意味着在 KB QCP 上有 FS 重构。大费米面 Kondo 相具有预期的双带结构,而小费米面 RKKY 相却意外地具有三带结构。我们详细分析了 Kondo 相以及 RKKY 相的准粒子特性,并首次发现了它们之间的差异。在 FS 重构的同时,还出现了一个卢廷格面,𝑓 的自能在该面上发散。鲁丁格表面和费米表面的体积通过广义鲁丁格和则与电荷密度相关。我们将较小的费米面体积和出现的卢廷格表面解释为 RKKY 相中 𝑓 电子分化的证据。最后,我们计算了霍尔系数和比热的温度依赖性,发现与实验的定性一致。
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来源期刊
Physical Review X
Physical Review X PHYSICS, MULTIDISCIPLINARY-
CiteScore
24.60
自引率
1.60%
发文量
197
审稿时长
3 months
期刊介绍: Physical Review X (PRX) stands as an exclusively online, fully open-access journal, emphasizing innovation, quality, and enduring impact in the scientific content it disseminates. Devoted to showcasing a curated selection of papers from pure, applied, and interdisciplinary physics, PRX aims to feature work with the potential to shape current and future research while leaving a lasting and profound impact in their respective fields. Encompassing the entire spectrum of physics subject areas, PRX places a special focus on groundbreaking interdisciplinary research with broad-reaching influence.
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