[Paper Review] Superconductivity and normal state properties in flat bands

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

Physics of Superconductivity and MagnetismPhysics and Astronomy77 references135 citations

TL;DR

This paper resolves a critical flaw in the theoretical description of superfluid weight in flat band superconductors by deriving complete mean-field equations that properly account for order parameter dependence on the vector potential. It establishes that the minimal quantum metric—defined as the quantum metric with the smallest trace—is the correct geometric quantity governing superfluid weight in isolated flat bands, and shows that non-isolated flat bands with band touchings can actually enhance superconductivity.

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.

Motivation & Objective

• To correct an incomplete and flawed derivation of the mean-field superfluid weight in flat band systems that neglects the vector potential dependence of order parameters.
• To establish the minimal quantum metric as the fundamental geometric quantity determining superfluid weight in isolated flat bands.
• To investigate the impact of band gap closure and band touching points on superconductivity in multiband models.
• To demonstrate that non-isolated flat bands with linear band touchings can enhance superconductivity, challenging the assumption that isolation is necessary.

Proposed method

• Derives complete mean-field equations for superfluid weight by rigorously accounting for the gauge dependence of order parameters in multiband systems.
• Introduces a generalized S-matrix construction to model pairing and band touchings in mean-field theory.
• Performs exact calculations of the Cooper pair mass in attractive Hubbard models under uniform pairing conditions.
• Uses space group symmetries to constrain and identify the competing non-universal term in the effective mass beyond the quantum metric.
• Analyzes the role of orbital positions at high-symmetry lattice sites, showing they guarantee minimal quantum metric.
• Compares results across lattice models (e.g., Lieb lattice, square lattice) to validate the minimal quantum metric's role in superfluid weight.

Experimental results

Research questions

• RQ1 How does the standard derivation of superfluid weight in flat bands fail when the order parameter's gauge dependence is neglected?
• RQ2 What is the correct geometric quantity that determines superfluid weight in isolated flat bands, and why is the minimal quantum metric the appropriate choice?
• RQ3 Can non-isolated flat bands with band touchings actually enhance superconductivity rather than suppress it?
• RQ4 How do space group symmetries constrain the effective mass of Cooper pairs beyond the quantum metric?
• RQ5 What is the role of the S-matrix construction in modeling pairing and band touchings in mean-field theory?

Key findings

• The standard mean-field derivation of superfluid weight in flat bands is incomplete and can lead to incorrect, non-zero superfluid weight predictions when the vector potential dependence of order parameters is ignored.
• In isolated flat bands with time-reversal symmetry, the superfluid weight is exactly proportional to the minimal quantum metric—the quantum metric with the smallest possible trace for the given lattice model.
• When orbitals are located at high-symmetry positions, the quantum metric is guaranteed to be minimal, ensuring the superfluid weight is maximized under geometric constraints.
• The BKT transition temperature on a Lieb lattice with a half-filled flat band is highest when the band gap closes (δ = 0), indicating that linear band touching enhances superconductivity.
• Mean-field theory shows that non-isolated flat bands with band touchings can be beneficial for superconductivity, as the superfluid weight remains finite and even increases under certain conditions.
• Exact calculations of the Cooper pair mass confirm two contributions: the quantum metric and a competing non-universal term, with the latter constrained by space group symmetries to vanish at high-symmetry orbital positions.