Understanding limits on the superconducting transition temperature Tc is a question of fundamental and practical importance. I will begin by describing quantum materials and ultracold atoms experiments that challenge conventional wisdom on what controls Tc. I will present exact upper bounds [1] on the superfluid stiffness, which in turn lead to bounds on the BKT Tc in 2D independent of pairing strength or mechanism. The bound takes a particularly simple form for parabolic dispersion in 2D – Tc cannot exceed one-eighth the Fermi temperature – which has been realized in recent experiments [2]. 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 twisted bilayer graphene. I will then describe progress on topological flat bands, where the Tc bound [3] involves 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.
[1] T. Hazra, N. Verma, M. Randeria, Phys. Rev. X 9, 031049 (2019)
[2] Y. Nakagawa et al., Science 372, 190 (2021)
[3] N. Verma, T. Hazra, M. Randeria, Proc. Nat. Acad. Sci. 118, e2106744118 (2021)
