[论文解读] Superconductivity and normal state properties in flat bands

Kukka-Emilia Huhtinen|arXiv (Cornell University)|Mar 7, 2023

Physics of Superconductivity and Magnetism参考文献 77被引用 135

一句话总结

本文通过推导完整的有效场方程，纠正了在平坦能带超导体中对超流权重理论描述的关键缺陷，这些方程正确考虑了序参量对规范势的依赖性。研究确立了最小量子度量（即在给定晶格模型下迹最小的量子度量）是孤立平坦能带中超流权重的正确几何量，并表明存在能带简并的非孤立平坦能带实际上可以增强超导性。

ABSTRACT

Bloch bands with a constant dispersion, called flat bands, have attracted great recent interest. Due to the lack of kinetic energy, even small interactions are dominant, and the formation of exotic correlated phases can be favored. Flat bands are particularly promising for superconductivity, where Bardeen-Cooper Schrieffer theory predicts that the superconducting critical temperature can be greatly enhanced compared to a dispersive band. In this dissertation, we explore the properties of flat band models in the attractive Hubbard model, with particular focus on the superconducting phase and the normal state at temperatures above the superconducting transition. The first part presents the theoretical background on superconductivity in multiband Hubbard models, including a linear response computation of the mean-field superfluid weight. In the second part, we explore the role of quantum geometry in flat band superconductivity. In the third part, we discuss the normal state in flat band models, which differs from a usual Fermi liquid. This thesis consists of four publications. In publication II, we showed that the superfluid weight in isolated flat bands is related to the minimal quantum metric, which is the integrated quantum metric with the smallest possible trace. Furthermore, by studying models with dispersive bands touching the flat bands, we showed that such band touchings can be beneficial for superconductivity. In publication III, the many-body problem was solved beyond the mean-field level in isolated flat bands models fulfilling the uniform pairing condition. This study showed that the quadratic excitation of the Cooper pair bound states is determined by the minimal quantum metric. In publication IV, we considered the relationship between quantum geometry and the real part of the optical conductivity in non-interacting flat bands. While the conductivity at non-zero inelastic scattering rate is related to the components of the quantum metric, the DC conductivity vanishes at low enough temperatures due to the localization of the particles. We also showed that a connection to the quantum metric could appear when applying the Kubo-Streda formula at exactly zero temperature in systems with (partially) flat bands, but this effect was absent when evaluating the conductivity using the Kubo-Greenwood formula. In publication I, we studied the normal state properties in the Lieb lattice flat band using dynamical mean field theory. The normal state was found to exhibit a crossover between two different types of non-Fermi liquids. At intermediate interaction strengths, a pseudogap phase with preformed pairs was found. As the interaction was lowered, the state developed insulating characteristics, which could be attributed to the localization of particles.

研究动机与目标

• 纠正在平坦能带系统中对有效场超流权重的不完整且有缺陷的推导，该推导忽略了序参量对规范势的依赖性。
• 确立最小量子度量为决定孤立平坦能带中超流权重的基本几何量。
• 研究能带隙关闭和多带模型中能带简并点对超导性的影响。
• 证明具有线性能带简并的非孤立平坦能带实际上可以增强超导性，从而挑战了‘能带孤立性为必要条件’的假设。

提出的方法

• 通过严格考虑多带系统中序参量的规范依赖性，推导出超流权重的完整有效场方程。
• 提出一种广义的S矩阵构造方法，用于在有效场理论中建模配对和能带简并。
• 在均匀配对条件下，对吸引性 Hubbard 模型中的库珀对质量进行精确计算。
• 利用空间群对称性来约束和识别有效质量中超出量子度量的非普遍项。
• 分析高对称晶格位置处轨道位置的作用，表明其可保证量子度量最小化。
• 通过比较不同晶格模型（如 Lieb 晶格、正方晶格）的结果，验证最小量子度量在超流权重中的作用。

实验结果

研究问题

• RQ1 当忽略序参量的规范依赖性时，标准平坦能带中超流权重的推导为何会失效？
• RQ2 在孤立平坦能带中，决定超流权重的正确几何量是什么？为何最小量子度量是合适的选择？
• RQ3 具有能带简并的非孤立平坦能带是否真的能增强超导性而非抑制它？
• RQ4 空间群对称性如何约束超出量子度量的库珀对有效质量？
• RQ5 S矩阵构造在有效场理论中建模配对和能带简并时起到什么作用？

主要发现

• 在平坦能带中，标准有效场推导忽略了序参量对规范势的依赖性，导致超流权重预测不完整，甚至在不应出现非零值时得出非零结果。
• 在具有时间反演对称性的孤立平坦能带中，超流权重与最小量子度量精确成正比——即在给定晶格模型下迹最小的量子度量。
• 当轨道位于高对称位置时，量子度量被保证为最小，从而在几何约束下确保超流权重最大化。
• 在半满平坦能带的 Lieb 晶格上，当能带隙关闭（δ = 0）时，BKT 转变温度最高，表明线性能带简并可增强超导性。
• 有效场理论表明，具有能带简并的非孤立平坦能带对超导性有益，因为在此类系统中，超流权重可保持有限，甚至在特定条件下增加。
• 对库珀对质量的精确计算确认了两个贡献：量子度量和一个竞争性的非普遍项，后者在高对称轨道位置处受空间群对称性约束而消失。