Effect of the coupling to a superconductor on the level statistics of a metal grain in a magnetic field

TL;DR

This paper develops a theoretical framework for understanding how coupling a disordered metal grain to a superconductor affects its energy level statistics, especially under a magnetic field that breaks time-reversal symmetry.

Contribution

It introduces a microscopic justification for a new random-matrix ensemble describing the spectral statistics of such systems with superconductor coupling.

Findings

An excitation gap of order δ can open under certain conditions.

The theory supports the validity of the Altland-Zirnbauer ensemble.

Magnetic field fully breaks time-reversal symmetry in the model.

Abstract

A theory is presented for the statistics of the excitation spectrum of a disordered metal grain in contact with a superconductor. A magnetic field is applied to fully break time-reversal symmetry in the grain. Still, an excitation gap of the order of $\delta$ opens up provided $N\Gamma^2\gtrsim 1$. Here $\delta$ is the mean level spacing in the grain, $\Gamma$ the tunnel probability through the contact with the superconductor, and $N$ the number of transverse modes in the contact region. This provides a microscopic justification for the new random-matrix ensemble of Altland and Zirnbauer.