Diamagnetic response and phase stiffness for interacting isolated narrow bands

A platform that serves as an ideal playground for realizing high'' temperature superconductors are materials where the electrons' kinetic energy is completely quenched, and interactions provide the only energy scale in the problem for $T_c$. However, when the non-interacting bandwidth for a set of isolated bands is small compared to the scale of the interactions, the problem is inherently non-perturbative and requires going beyond the traditional mean-field theory of superconductivity. In two spatial dimensions, $T_c$ is controlled by the superconducting phase stiffness. Here we present a general theoretical framework for computing the electromagnetic response for generic model Hamiltonians, which controls the maximum possible superconducting phase stiffness and thereby $T_c$, without resorting to any mean-field approximation. Importantly, our explicit computations demonstrate that the contribution to the phase stiffness arises from (i) integrating out’’ the remote bands that couple to the microscopic current operator, and (ii) the density-density interactions projected onto the isolated narrow bands. Our framework can be used to obtain an upper bound on the phase stiffness, and relatedly the superconducting transition temperature, for a range of physically inspired models involving both topological and non-topological narrow bands with arbitrary density-density interactions. We discuss a number of salient aspects of this formalism by applying it to a specific model of interacting flat bands and compare it against the known $T_c$ from independent numerically exact computations.