Diamagnetic response and phase stiffness for interacting isolated narrow bands

Superconductivity is a macroscopic manifestation of a quantum phenomenon where pairs of electrons delocalize and develop phase coherence over a long distance. A long-standing quest has been to address the underlying microscopic mechanisms that fundamentally limit the superconducting transition temperature, T. A platform which serves as an ideal playground for realizing "high"-temperature superconductors are materials where the electrons' kinetic energy is quenched and interactions provide the only energy scale in the problem. However, when the noninteracting bandwidth for a set of isolated bands is small compared to the interactions, the problem is inherently nonperturbative. In two spatial dimensions, T is controlled by superconducting phase stiffness. Here, we present a theoretical framework for computing the electromagnetic response for generic model Hamiltonians, which controls the maximum possible superconducting phase stiffness and thereby T, without resorting to any mean-field approximation. 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 on to the isolated narrow bands. Our framework can be used to obtain an upper bound on the phase stiffness and relatedly T for a range of physically inspired models involving both topological and nontopological narrow bands with 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 the upper bound against the known T from independent numerically exact computations.