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

Chaotic quantum systems with Lyapunov exponent $\lambda_\mathrm{L}$ obey an upper bound $\lambda_\mathrm{L}\leq 2\pi k_\mathrm{B}T/\hbar$ at temperature $T$, implying a divergence of the bound in the classical limit $\hbar\to 0$. 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 $\lambda_\mathrm{L}(\hbar,T)$ in the quantum spherical $p$-spin glass model, where $\hbar$ 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 $\lambda_\mathrm{L}$ on $\hbar$ close to the dynamical glass transition temperature $T_d$. Our results in the classical limit ($\hbar\to 0$) naturally describe chaos in super-cooled liquid in structural glasses. We find a maximum in $\lambda_\mathrm{L}(T)$ substantially above $T_d$, concomitant with the crossover from simple to slow glassy relaxation. We further show that $\lambda_\mathrm{L}\sim T^\alpha$, with the exponent $\alpha$ varying between 2 and 1 from quantum to classical limit, at low temperatures in the spin glass phase.