Superfluid stiffness, optical spectral weight, and T_c bounds in multi-band superconductors

Recently we have used the optical sum rule for multi-band systems to bound the superfluid stiffness and derived upper bounds on the superconducting T_c in two dimensions. These bounds were shown [PRX 9, 031049 (2019)] to be particularly important for strongly interacting, narrow band systems, and found to give useful estimates for a variety of systems ranging from cold atoms to twisted bilayer graphene. We explore here various situations where one is forced to include bands far from the chemical potential in order to have localized Wannier functions, simple interactions, or to describe flat bands. Our goal is to compute the low-energy optical spectral weight and obtain tighter bounds on the superfluid stiffness, compared to the full spectral weight that includes all interband transitions. We also gain insight into how band topology impacts optical spectral weight. We will present results for model systems with flat bands, including Lieb and Kagome lattices, and estimate T_c bounds for monolayer FeSe on SrTiO₃ and for magic-angle twisted bilayer graphene.