Diamagnetic response and phase stiffness for interacting isolated narrow bands

Superconductivity in electronic systems, where the non-interacting bandwidth for a set of isolated bands is small compared to the scale of the interactions, is a non-perturbative problem. Here we present a theoretical framework for computing the electromagnetic response in the limit of zero frequency and vanishing wavenumber for the interacting problem, which controls the superconducting phase stiffness, without resorting to any mean-field approximation. Importantly, 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 bands. We also obtain the electromagnetic response directly in the limit of an infinite gap to the remote bands, using the appropriate ``projected" gauge-transformations. These results can be used to obtain a conservative upper bound on the phase stiffness, and relatedly the superconducting transition temperature, with a few assumptions. We apply this formalism to a host of topologically (non-)trivial ``flat-band" systems, including twisted bilayer graphene.