Self-consistent-field ensembles of disordered Hamiltonians: Efficient solver and application to superconducting films

Abstract

Our general interest is in self-consistent-field (scf) theories of disordered fermions. They generate physically relevant sub-ensembles ("scf-ensembles") within a given Altland-Zirnbauer class. We are motivated to investigate such ensembles (i) by the possibility to discover new fixed points due to (long-range) interactions; (ii) by analytical scf-theories that rely on partial self-consistency approximations awaiting a numerical validation; (iii) by the overall importance of scf-theories for the understanding of complex interaction-mediated phenomena in terms of effective single-particle pictures. In this paper we present an efficient, parallelized implementation solving scf-problems with spatially local fields by applying a kernel-polynomial approach. Our first application is the Boguliubov-deGennes (BdG) theory of the attractive-$U$ Hubbard model in the presence of on-site disorder; the scf-fields are the particle density $n(\mathbf{r})$ and the gap function $\Delta (\mathbf{r})$. For this case, we reach system sizes unprecedented in earlier work. They allow us to study phenomena emerging at scales substantially larger than the lattice constant, such as the interplay of multifractality and interactions, or the formation of superconducting islands. For example, we observe that the coherence length exhibits a non-monotonic behavior with increasing disorder strength already at moderate $U$. With respect to methodology our results are important because we establish that partial self-consistency ("energy-only") schemes as typically employed in analytical approaches tend to miss qualitative physics such as island formation.

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• Quantum breakdown of superconductivity in low-dimensional materials
• Multifractally-enhanced superconductivity in thin films
• Interplay of superconductivity and localization near a 2D ferromagnetic quantum critical point
• Multifractally-enhanced superconductivity in two-dimensional systems with spin-orbit coupling
• Generalized multifractality in the spin quantum Hall symmetry class with interaction
• Generalized surface multifractality in 2D disordered systems
• Local density of state fluctuations in 2D superconductor as a probe of quantum diffusion
• Friedel Oscillations and superconducting-gap enhancement by impurity scattering
• Quantum Fluctuations and Multifractally-Enhanced Superconductivity in Disordered Thin Films
• Distribution of the order parameter in strongly disordered superconductors: An analytic theory
• Boundary multifractality in the spin quantum Hall symmetry class with interaction
• Multifractal correlations of the local density of states in dirty superconducting films