Spectral Form Factor of a Quantum Spin Glass

Abstract

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.

38 pages, comments welcome

References in corpus (5)

• Thermalization and its mechanism for generic isolated quantum systems
• Periodic-Orbit Theory of Universality in Quantum Chaos
• Many-body mobility edge in a mean-field quantum spin glass
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Cited by in corpus (3)

• Dynamical quantum ergodicity from energy level statistics
• The Loschmidt Spectral Form Factor
• Nonergodic delocalized paramagnetic states in quantum neural networks