Optical Spectral Weight and Tc Bounds for 2D Flat Band Superconductors

Motivated by Moiré materials, we investigate flat band superconductivity in 2D and obtain exact analytical results without making mean-field approximations. We review exact bounds for the 2D Tc in terms of the optical sum rule for multi-band superconductors [1] with arbitrary interactions, with the single assumption that the external vector potential couples to the kinetic energy. We then generalize these results to models with local attractive interactions where the low-energy physics is described by effective Hamiltonians without any kinetic energy. We first analyze trivial flat band systems for which we obtain upper and lower bounds [2] on the optical spectral weight that involve the "quantum geometry" of the flat band wavefunctions. We next extend our analysis to topological flat bands where one has an obstruction to constructing localized Wannier functions respecting all symmetries. Finally, we will discuss generalizations of our results to systems with strong-spin orbit coupling.