Level statistics inside the core of a superconductive vortex

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/Δ<< τ<< 1/ω_0 = E_F/Δ^2). At sufficiently low energies E << ω_{Th}, the energy level statistics is described by the "zero-dimensional" limit of this supermatrix theory, with the effective "Thouless energy" ω_{Th} \sim (ω_0/τ)^{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 ω_0 by the relative amount of the order of [2\ln(1/ω_0τ)]^{-1}.