Upper bounds on the superconducting critical temperature: Applications to Twisted-Bilayer Graphene

Understanding the material parameters that control the superconducting (SC) transition temperature T_c is a problem of fundamental importance. In many novel superconductors of interest, fluctuations of the phase of the SC order parameter determines T_c, rather than the BCS collapse of the amplitude due to pair breaking. We derive rigorous upper bounds on the superfluid density or phase stiffness D_s valid in any dimension, essentially controlled by the non-interacting band structure. This in turn leads to rigorous upper bounds on T_c in 2D, which holds irrespective of the form or strength of the pairing interactions, mechanism or order-parameter symmetry. Although these bounds are of completely general validity, they are most illuminating for narrow-band, strongly interacting systems. Our results lead to strong restrictions for magic-angle twisted bilayer graphene (MA-TBG), where we show that the maximum possible T_c is quite close to the experimental observations. This shows that MA-TBG must be a strongly interacting system where phase stiffness rather than pair breaking is responsible for T_c. We also discuss the question of deriving rigorous upper bounds on T_c in 3D.