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 first derive rigorous upper bounds on the superfluid phase stiffness Ds valid in any dimension. This in turn leads to a rigorous upper bound on Tc in 2D, which holds irrespective of the form or strength of the pairing interactions, mechanism or order-parameter symmetry. These bounds are of completely general validity, and we show that they can lead to stringent constraints for narrow-band and strongly correlated systems. We apply our bounds to diverse systems. We show that kBTc ≤ EF/8 across the 2D BCS-BEC crossover in ultra-cold Fermi gas. For magic-angle twisted bilayer graphene (MA-TBG), we show that existing band structure results constrain the maximum possible Tc 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 Tc in 3D.
