It is widely expected that systems which fully thermalize are chaotic in the sense of exhibiting random-matrix statistics of their energy level spacings, whereas integrable systems exhibit Poissonian statistics. In this paper, we investigate a third class: spin glasses. These systems are partially chaotic but do not achieve full thermalization due to large free energy barriers. We examine the level spacing statistics of a canonical infinite-range quantum spin glass, the quantum p-spherical model, using an analytic path integral approach. We find statistics consistent with a direct sum of independent random matrices, and show that the number of such matrices is equal to the number of distinct metastable configurations -- the exponential of the spin glass "complexity" as obtained from the quantum Thouless-Anderson-Palmer equations. We also consider the statistical properties of the complexity itself and identify a set of contributions to the path integral which suggest a Poissonian distribution for the number of metastable configurations. Our results show that level spacing statistics can probe the ergodicity-breaking in quantum spin glasses and provide a way to generalize the notion of spin glass complexity beyond models with a semi-classical limit.

10aDisordered Systems and Neural Networks (cond-mat.dis-nn)10aFOS: Physical sciences10aHigh Energy Physics - Theory (hep-th)10aStatistical Mechanics (cond-mat.stat-mech)10aStrongly Correlated Electrons (cond-mat.str-el)1 aWiner, Michael1 aBarney, Richard1 aBaldwin, Christopher, L.1 aGalitski, Victor1 aSwingle, Brian uhttps://arxiv.org/abs/2203.12753