Inhomogeneous Pairing in Highly Disordered s-wave Superconductors

We study a simple model of a two-dimensional s-wave superconductor in the presence of a random potential in the regime of large disorder. We first use the Bogoliubov-de Gennes (BdG) approach to show that, with increasing disorder the pairing amplitude becomes spatially inhomogeneous, and the system cannot be described within conventional approaches for studying disordered superconductors which assume a uniform order parameter. In the high disorder regime, we find that the system breaks up into superconducting islands (with large pairing amplitude) separated by an insulating sea. We show that this inhomogeneity has important implications for the physical properties of this system, such as superfluid density and the density of states. We find that a finite spectral gap persists in the density of states for all values of disorder and we provide a detailed understanding of this remarkable result. We next generalize Anderson's idea of the pairing of exact eigenstates to include an inhomogeneous pairing amplitude, and show that it is able to qualitatively capture many of the nontrivial features of the full BdG analysis. Finally, we study the transition to a gapped insulating state driven by the quantum phase fluctuations about the inhomogeneous superconducting state.Comment: 15 pages, 16 figure

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

Understanding the material parameters that control the superconducting transition temperature $T_c$ is a problem of fundamental importance. In many novel superconductors, phase fluctuations determine $T_c$, 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 $T_c$ 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 $E_F$, we find that $k_BT_c \leq E_F/8$, 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 $T_c$ to be close to the experimentally observed value. Finally, we discuss the question of deriving rigorous upper bounds on $T_c$ in 3D.Comment: Revised figures, includes estimates from another model of MA-TBG, published version of manuscrip