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 the standard theory of superconducting Tc and why recent developments in quantum materials and ultra-cold atoms challenge conventional wisdom. I will outline recent work on exact upper bounds [1] on Tc for two dimensional (2D) systems. This bound is particularly simple for quadratic dispersion - Tc cannot exceed one-eighth the Fermi temperature - which has been tested in recent experiments.  I will next describe results for systems with complex band structures like twisted bilayer graphene. I will briefly describe Tc bounds [2] for topological flat bands which involve the “quantum geometry” of wave functions. I will conclude with the challenges in deriving bounds on Tc in 3D, which remains an open problem. 

References:
[1] T. Hazra, N. Verma, M. Randeria, Phys. Rev. X 9, 031049 (2019);  
[2] N. Verma, T. Hazra, M. Randeria, Proc. Nat. Acad. Sci. 118, e2106744118 (2021)