Abstract

Understanding the material parameters that control the superconducting (SC) transition temperature Tc is a problem of fundamental importance. In many novel superconductors of interest, fluctuations of the phase of the SC order parameter determines Tc, rather than the BCS collapse of the amplitude due to pair breaking. We derive rigorous upper bounds on the superfluid stiffness Ds in terms of the optical conductivity sum-rule, valid in any dimension, essentially controlled by the non-interacting band structure. This in turn leads to rigorous upper bounds on Tc in 2D, which holds irrespective of the form or strength of the pairing interactions, mechanism or order-parameter symmetry. We estimate the upper bounds on Tc for FeSe on SrTiO3, cold atom systems and lattice Hubbard models and find that they are quite close to the maximum observed Tc in these strongly-correlated systems. For magic-angle twisted bilayer graphene, we not only calculated these bounds for lattice models of the flat bands, but we also present rigorous upper bounds on the low energy optical spectral weight that determines the maximum Tc in isolated flat bands in a multi-band system. The latter bounds are directly controlled by the Marzari-Vanderbilt spread functional of the flat band Wannier functions - pointing to a deep connection between the quantum geometry of the flat band eigenstates and the low energy optical response.

[1] Hazra, T., Verma, N. & Randeria, M. Bounds on the Superconducting Transition Temperature: Applications to Twisted Bilayer Graphene and Cold Atoms. Phys. Rev. X 9, 031049 (2019).
[2] Verma, N., Hazra, T. & Randeria, M. Optical spectral weight, phase stiffness, and Tc bounds for trivial and topological flat band superconductors. PNAS 118, (2021).