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.

Spectral statistics of a minimal quantum glass model

“…This enhancement shows up in realistic systems such as the Bunimovich stadium [58,59], which have different sectors of their Hamiltonian which behave differently under reflection symmetry. It has recently been shown [46] that at times sub-exponential in system size, all-to-all spin glasses exhibit an enhancement of the ramp equal to the number of effective "sectors", i.e., regions of configuration space rendered dynamically disconnected by large energy barriers. This work serves as an extension of that result for a toy model of spin glasses, going out to late times.…”

“…Motivated by our recent results for the early time ramp of the SFF for a quantum spin glass [46], we construct a minimal quantum model which can also exhibit glassy behavior. We create this model by generalizing the RP model examined in the previous section.…”

“…In this phase the system will be confined to the subspaces corresponding to the blocks of A instead of individual states. This generalization of the RP model is motivated by our recent work examining the SFF of a canonical quantum spin glass model [46]. In that work we found that the SFF has a linear ramp at times sub-exponential in the system size, as would a chaotic system.…”

The Loschmidt spectral form factor

“…In chaotic systems, the SFF exhibits a dip-ramp-plateau structure (figure 1), while in integrable systems, the ramp is not present. It can be shown that in systems with conserved modes, or even slow modes or nearly conserved quantities, the ramp is significantly enhanced [4,[16][17][18].…”

Probing chaos in the spherical p-spin glass model

“…The inclusion of quantum fluctuations usually opens up a new route for thermalization in the equilibrium dynamics due to tunnelling between metastable states [21][22][23][24], even though counter-intuitive effects of quantum fluctuations inducing glassiness has been found in quantum quench protocols [25] or more recently in presence of more complex energy landscapes [26]. Within such a rich variety of phenomena, the recent theoretical [27][28][29][30][31] and experimental [32] observation of many-body chaos in quantum spin-glasses has drawn particular attention. Specifically, in Refs.…”