Upper bounds on the superfluid stiffness and superconducting $T_c$: Applications to twisted-bilayer graphene and ultra-cold Fermi gases

Understanding the material parameters that control the superconducting (SC) transition temperature $T_c$ is a problem of fundamental importance. In many novel superconductors, phase fluctuations of the SC order parameter determine $T_c$, rather than the mean field collapse of the amplitude due to pair breaking. We derive rigorous upper bounds on the superfluid phase stiffness $D_s$ valid in any dimension. This in turn leads to an upper bound on $T_c$ in 2D, which holds irrespective of mechanism, strength of pairing interaction, or order-parameter symmetry. These bounds lead to stringent constraints for the strongly correlated regime of low-density and narrow-band systems. We show that $k_BT_c leq E_F/8$ across the 2D BCS-BEC crossover in ultra-cold Fermi gas. For magic-angle twisted bilayer graphene (MA-TBG), the band structure constrains the maximum possible $T_c$ to be close to the experimentally observed value, demonstrating that MA-TBG is in a phase-fluctuation dominated regime. Finally, we discuss the question of deriving rigorous upper bounds on $T_c$ in 3D