Bounds on the Superconducting Transition Temperature: Applications to Twisted Bilayer Graphene and Cold Atoms

Abstract

Understanding the material parameters that control the superconducting transition temperature T_{c} is a problem of fundamental importance. In many novel superconductors phase fluctuations determine T_{c}, rather than the collapse of the pairing amplitude. We derive rigorous upper bounds on the superfluid phase stiffness for multiband systems, valid in any dimension. This in turn leads to an upper bound on T_{c} in two dimensions, which holds irrespective of pairing mechanism, interaction strength, or order-parameter symmetry. Our bound is particularly useful for the strongly correlated regime of low-density and narrow-band systems, where mean-field theory fails. For a simple parabolic band in 2D with Fermi energy E_{F}, we find that k_{B}T_{c}≤E_{F}/8, an exact result that has direct implications for the 2D BCS-BEC crossover in ultracold Fermi gases. Applying our multiband bound to magic-angle twisted bilayer graphene, we find that band structure results constrain the maximum T_{c} to be close to the experimentally observed value. Finally, we discuss the question of deriving rigorous upper bounds on T_{c} in 3D.