Bounds on the superconducting transition temperature

I summarize recent progress on obtaining rigorous upper bounds on superconducting transition temperature Tc in two dimensions independent of pairing mechanism or interaction strength. These results are derived by finding a general upper bound for the superfluid stiffness for a multi-band system with arbitrary interactions, with the only assumption that the external vector potential couples to the kinetic energy and not to the interactions. This bound is then combined with the universal relation between the superfluid stiffness and the Berezinskii-Kosterlitz-Thouless Tc in 2D. For parabolic dispersion, one obtains the simple result that kBTc <= EF/8, which has been tested in recent experiments. More generally, the bounds are expressed in terms of the optical spectral weight and lead to stringent constraints for the Tc of low-density, strongly correlated superconductors. Results for Tc bounds for models of flat-band superconductors, where the kinetic energy vanishes and the vector potential must couple to interactions, are briefly summarized. Upper bounds on Tc in 3D remains an open problem, and I describe how questions of universality underlie the challenges in 3D.