Level statistics inside the core of a superconductive vortex

TL;DR

This paper develops a microscopic supermatrix sigma-model to analyze low-energy electron states in the vortex core of a superconductor with disorder, revealing a connection to random matrix theory and quantifying level spacing modifications.

Contribution

It introduces a supermatrix sigma-model approach for vortex core states in disordered superconductors, linking microscopic theory to random matrix models and analyzing energy level statistics.

Findings

Level statistics follow the zero-dimensional sigma-model limit at low energies.

Density of states matches the Altland-Zirnbauer class C random matrix results.

Nonzero modes slightly increase the mean interlevel distance.

Abstract

Microscopic theory of the type of Efetov's supermatrix sigma-model is constructed for the low-lying electron states in a mixed superconductive-normal system with disorder. The developed technique is used for the study of the localized states in the core of a vortex in a moderately clean superconductor (1/\Delta << \tau << 1/\omega_0 = E_F/\Delta^2). At sufficiently low energies E << \omega_{Th}, the energy level statistics is described by the "zero-dimensional" limit of this supermatrix theory, with the effective "Thouless energy" \omega_{Th} \sim (\omega_0/\tau)^{1/2}. Within this energy range the result for the density of states is equivalent to that obtained within Altland-Zirnbauer random matrix model of class C. Nonzero modes of the sigma-model increase the mean interlevel distance \omega_0 by the relative amount of the order of [2\ln(1/\omega_0\tau)]^{-1}.