Chaos in a quantum rotor model

TL;DR

This paper investigates quantum chaos in a model of coupled quantum rotors, revealing exponential growth of chaos indicators with a temperature-dependent Lyapunov exponent, and characterizes phase transition effects.

Contribution

It provides the first calculation of quantum chaos growth rates in a quantum rotor model with random interactions, highlighting temperature dependence and phase transition effects.

Findings

Lyapunov exponent proportional to 1/M

Exponential growth of squared commutator

Temperature dependence of chaos growth rate

Abstract

We study scrambling in a model consisting of a number $N$ of $M$-component quantum rotors coupled by random infinite-range interactions. This model is known to have both a paramagnetic phase and a spin glass phase separated by second order phase transition. We calculate in perturbation theory the squared commutator of rotor fields at different sites in the paramagnetic phase, to leading non-trivial order at large $N$ and large $M$. This quantity diagnoses the onset of quantum chaos in this system, and we show that the squared commutator grows exponentially with time, with a Lyapunov exponent proportional to $\frac{1}{M}$. At high temperature, the Lyapunov exponent limits to a value set by the microscopic couplings, while at low temperature, the exponent exhibits a $T^4$ dependence on temperature $T$.