Quantum-to-Classical Crossover in Many-Body Chaos and Scrambling from Relaxation in a Glass
Surajit Bera, K. Y. Venkata Lokesh, Sumilan Banerjee
Abstract
Chaotic quantum systems with Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ obey an upper bound ${\ensuremath{\lambda}}_{L}\ensuremath{\le}2\ensuremath{\pi}{k}_{B}T/\ensuremath{\hbar}$ at temperature $T$, implying a divergence of the bound in the classical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$. Following this trend, does a quantum system necessarily become ``more chaotic'' when quantum fluctuations are reduced? Moreover, how do symmetry breaking and associated nontrivial dynamics influence the interplay of quantum mechanics and chaos? We explore these questions by computing ${\ensuremath{\lambda}}_{L}(\ensuremath{\hbar},T)$ in the quantum spherical $p$-spin glass model, where $\ensuremath{\hbar}$ 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 nontrivial, with nonmonotonic dependence of ${\ensuremath{\lambda}}_{L}$ on $\ensuremath{\hbar}$ close to the dynamical glass transition temperature ${T}_{d}$. Our results in the classical limit ($\ensuremath{\hbar}\ensuremath{\rightarrow}0$) naturally describe chaos in supercooled liquid in structural glasses. We find a maximum in ${\ensuremath{\lambda}}_{L}(T)$ substantially above ${T}_{d}$, concomitant with the crossover from simple to slow glassy relaxation. We further show that ${\ensuremath{\lambda}}_{L}\ensuremath{\sim}{T}^{\ensuremath{\alpha}}$, with the exponent $\ensuremath{\alpha}$ varying between 2 and 1 from quantum to classical limit, at low temperatures in the spin glass phase.