Strong-coupling approaches to moiré superconductivity and competing orders

Superconductivity in the limit of a vanishing bandwidth in isolated bands is a classic example of a non-perturbative problem, where BCS theory does not apply. What sets the superconducting phase stiffness, and relatedly the transition temperature, in this limit is of both fundamental and practical interest. This question has become especially relevant with the discovery of superconductivity in moiré materials. I will present a non-perturbative framework to obtain the low-energy optical spectral weight for partially filled electronic flat bands with generic density-density interactions, and apply it to the problem of twisted bilayer graphene with screened Coulomb interactions at integer fillings to put upper bounds on the maximum superconducting transition temperature. I will also present numerically exact results obtained using sign-problem-free quantum Monte-Carlo methods for the interplay between superconductivity and various competing orders in models of interacting flat (non-)topological bands.