Light-cone spreading of perturbations and the butterfly effect in a classical spin chain

TL;DR

This paper investigates how localized perturbations in a classical chaotic spin chain spread ballistically, revealing connections to chaos indicators like Lyapunov exponents and the KPZ universality class.

Contribution

It demonstrates the ballistic spread of perturbations in a classical Heisenberg chain and links this to the growth of interfaces described by the KPZ equation, bridging classical chaos and statistical physics.

Findings

Perturbations spread ballistically with a finite speed.

The classical OTOC correlates with quantum chaos indicators.

Perturbation dynamics relate to KPZ interface growth.

Abstract

We find that localised perturbations in a chaotic classical many-body system-- the classical Heisenberg We find that the effects of a localised perturbation in a chaotic classical many-body system--the classical Heisenberg chain at infinite temperature--spread ballistically with a finite speed even when the local spin dynamics is diffusive. We study two complementary aspects of this butterfly effect: the rapid growth of the perturbation, and its simultaneous ballistic (light-cone) spread, as characterised by the Lyapunov exponents and the butterfly speed respectively. We connect this to recent studies of the out-of-time-ordered commutators (OTOC), which have been proposed as an indicator of chaos in a quantum system. We provide a straightforward identification of the OTOC with a natural correlator in our system and demonstrate that many of its interesting qualitative features are present in the classical system. Finally, by analysing the scaling forms, we relate the growth, spread and propagation of the perturbation with the growth of one-dimensional interfaces described by the Kardar-Parisi-Zhang (KPZ) equation.