Spectral Physics of Spin Glasses*

It is widely established that fast-thermalizing chaotic systems exhibit random-matrix-like statistics in their energy level spacings, while integrable systems exhibit Poissonian statistics. In this paper we investigate a third class of systems: spin glasses. These systems are chaotic don't achieve full thermalization due to energy or entropy barriers between regions of phase space. We examine their spectral statistics analytically using a mean-field theory approach, and find statistics consistent with independent random matrices for each connected component of the phase space. Our techniques show that spectral statistics are sensitive to this ergodicity breaking, and can be generalized to diagnose glassiness in strongly quantum systems without a classical analog, as well as proving novel results about the complexity distribution in our toy model.