Understanding the material parameters that control the superconducting transition temperature Tc is a problem of fundamental importance. In many novel superconductors phase fluctuations determine Tc, rather than the collapse of the pairing amplitude. We derive rigorous upper bounds on the superfluid phase stiffness for multi-band systems, valid in any dimension. This in turn leads to an upper bound on Tc in two dimensions (2D), 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 EF , we find that kBTc ≤ EF /8, an exact result that has direct implications for the 2D BCS-BEC crossover in ultra-cold Fermi gases. Applying our multi-band bound to magic-angle twisted bilayer graphene (MA-TBG), we find that band structure results constrain the maximum Tc to be close to the experimentally observed value. Finally, we discuss the question of deriving rigorous upper bounds on Tc in 3D. arXiv:1811.12428v3 [cond-mat.supr-con]

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We predict two topological superconducting phases in microscopic models arising from the Berry phase associated with the valley degree of freedom in gapped Dirac honeycomb systems. The first one is a topological helical spin-triplet superconductor with a nonzero center-of-mass momentum that does not break time-reversal symmetry. We also find a topological chiral-triplet superconductor with Chern number ±1 with equal-spin-pairing in one valley and opposite-spin-triplet pairing in the other valley. Our results are obtained for the Kane-Mele model in which we have explored the effect of three different interactions, onsite attraction U, nearest-neighbor density-density attraction V , and nearest-neighbor antiferromagnetic exchange J, within self-consistent Bogoliubov-de Gennes theory. Transition metal dichalcogenides and cold atom experiments are promising platforms to explore these phases. arXiv:1806.08795v3 [cond-mat.supr-con]

We present exact results that give insight into how interactions lead to transport and superconductivity in a flat band where the electrons have no kinetic energy. We obtain bounds for the optical spectral weight for flat-band superconductors that lead to upper bounds for the superfluid stiffness and the two-dimensional (2D) Tc. We focus on on-site attraction |U| on the Lieb lattice with trivial flat bands and on the π-flux model with topological flat bands. For trivial flat bands, the low-energy optical spectral weight D̃low≤ñ|U|Ω/2 with ñ=minn,2−n , where n is the flat-band density and Ω is the Marzari–Vanderbilt spread of the Wannier functions (WFs). We also obtain a lower bound involving the quantum metric. For topological flat bands, with an obstruction to localized WFs respecting all symmetries, we again obtain an upper bound for D̃low linear in |U|. We discuss the insights obtained from our bounds by comparing them with mean-field and quantum Monte Carlo results.