Are there bounds on the superconducting transition temperature?*

Understanding limits on the superconducting transition temperature Tc is a question of fundamental and practical importance. I will begin by describing developments in quantum materials and in the BCS-BEC crossover in ultracold atoms that challenge conventional wisdom on what controls Tc. I will then describe exact upper bounds [1] on the BKT Tc for two dimensional (2D) systems expressed in terms of the optical spectral weight that are valid for any pairing mechanism or strength. The 2D bound takes a particularly simple form for parabolic dispersion -- Tc cannot exceed one-eighth the Fermi temperature – which has been realized in recent experiments on Li:ZrNCl.  I will next describe results for multi-band systems with complex band structures and discuss applications to 2D superconducting materials including monolayer FeSe/STO and magic angle twisted bilayer graphene. Finally, I will describe the generalization [2] of the optical spectral weight bounds for trivial and topological flat bands in 2D which involve the quantum geometry of electronic wave functions. I will conclude by describing the challenges in deriving bounds on Tc in 3D, which remains an open problem.