Quantum to classical crossover in many-body chaos and scrambling from relaxation in a glass

Abstract

Chaotic quantum systems with Lyapunov exponent $\hat{I}{»}_{\mathrm{L}}$ obey an upper bound $\hat{I}{»}_{\mathrm{L}}\le 2\ddot{I}€k_{\mathrm{B}}T/\hslash$ at temperature $T$, implying a divergence of the bound in the classical limit $\hslash \to 0$. Following this trend, does a quantum system necessarily become `more chaotic' when quantum fluctuations are reduced? Moreover, how do symmetry breaking and associated non-trivial dynamics influence the interplay of quantum mechanics and chaos? We explore these questions by computing $\hat{I}{»}_{\mathrm{L}}(\hslash ,T)$ in the quantum spherical $p$-spin glass model, where $\hslash$ can be continuously varied. We find that quantum fluctuations, in general, make paramagnetic phase less and the replica symmetry-broken spin glass phase more chaotic. We show that the approach to the classical limit could be non-trivial, with non-monotonic dependence of $\hat{I}{»}_{\mathrm{L}}$ on $\hslash$ close to the dynamical glass transition temperature $T_d$. Our results in the classical limit ($\hslash \to 0$) naturally describe chaos in super-cooled liquid in structural glasses. We find a maximum in $\hat{I}{»}_{\mathrm{L}}(T)$ substantially above $T_d$, concomitant with the crossover from simple to slow glassy relaxation. We further show that $\hat{I}{»}_{\mathrm{L}}\sim T^{\hat{I}}\pm$, with the exponent $\hat{I}\pm$ varying between 2 and 1 from quantum to classical limit, at low temperatures in the spin glass phase.

6 pages, 4 figures