Upper bounds on superconducting and excitonic phase-stiffness for interacting isolated narrow bands

TL;DR

This paper develops a non-perturbative method to determine upper bounds on the superconducting and excitonic phase-stiffness in interacting flat-band systems, relevant for understanding maximum critical temperatures in moiré materials.

Contribution

It introduces a novel formalism that calculates phase-stiffness without Wannierization or pairing symmetry assumptions, applicable directly in momentum space.

Findings

Derived upper bounds on phase-stiffness for sign-problem-free models.

Extended the formalism to excitonic superfluids.

Analyzed bounds for flat bands with Landau-level-like wavefunctions.

Abstract

Inspired by the discovery of superconductivity in moir\'e materials with isolated narrow bandwidth electronic bands, here we analyze critically the question of what is the maximum attainable $T_c$ in interacting flat-band systems. We focus specifically on the low-energy effective theory, where the density-density interactions are projected to the set of partially-filled flat bands. The resulting problem is inherently non-perturbative, where the standard mean-field approximation is not applicable. Here we develop further our recent Schrieffer-Wolff transformation based approach (PNAS, 120 (11), e2217816120 (2023)) to compute the effective electromagnetic response and the superconducting phase-stiffness in terms of "projected" gauge-transformations, and extend the formalism to compute the stiffness for excitonic superfluids. Importantly, our method requires neither any "wannierization" for the narrow bands of interest, regardless of their (non-)topological character, nor any knowledge of an underlying pairing-symmetry, and can be setup directly in momentum-space. We use this formalism to derive upper bounds on the phase-stiffness for sign-problem-free models, where their values are known independently from numerically exact quantum Monte-Carlo computations. We also illustrate the analytical structure of these bounds for the superconducting and excitonic phase-stiffness for perfectly flat-bands that have Landau-level-like wavefunctions.