Superconductivity and normal state properties in flat bands

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.