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

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.