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.