Lyapunov growth in quantum spin chains

TL;DR

This paper investigates Lyapunov growth in quantum spin chains, showing that higher spins can exhibit exponential growth of commutators, and connects quantum and classical Lyapunov exponents in the infinite-spin limit.

Contribution

The study extends the Ising spin chain model to higher spins, demonstrating the emergence of Lyapunov growth and establishing a classical-quantum correspondence for the Lyapunov exponent.

Findings

Exponential growth window appears for large spins.

A quantum Lyapunov exponent is numerically extracted.

Classical and quantum exponents agree in the infinite-spin limit.

Abstract

The Ising spin chain with longitudinal and transverse magnetic fields is often used in studies of quantum chaos, displaying both chaotic and integrable regions in its parameter space. However, even at a strongly chaotic point this model does not exhibit Lyapunov growth of the commutator squared of spin operators, as this observable saturates before exponential growth can manifest itself (even in situations where a spatial suppression factor makes the initial commutator small). We extend this model from the spin 1/2 Ising model to higher spins, demonstrate numerically that a window of exponential growth opens up for sufficiently large spin, and extract a quantity which corresponds to a notion of a Lyapunov exponent. In the classical infinite-spin limit, we identify and compute the appropriate classical analogue of the commutator squared, and show that the corresponding exponent agrees with the infinite-spin limit extracted from the quantum spin chain.