Light-Cone Spreading of Perturbations and the Butterfly Effect in a Classical Spin Chain

Abstract

We find that the effects of a localized 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 characterized 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 analyzing 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 equation.

Out of time ordered quantum dissipation

“…Such comparisons may be useful in classifying systems according to the behaviour of their OTOCs. In this work, we saw that the OTO couplings are not just a feature of the quantum mechanical theory of the particle, but they show up in out of time ordered Poisson brackets in the classical limit as well (Similar classical limits of OTOCs have been discussed in [73][74][75][76]). A stochastic interpretation of this classical OTO behaviour would be useful in understanding the significance of such OTOCs in the quantum mechanical framework (where OTO dynamics has been mostly studied up till now) as well as devising experiments to measure them.…”

Kardar-Parisi-Zhang universality from soft gauge modes

“…It follows that in the infinite-size limit l → ∞ (or equivalently, at asymptotically long times) the hydrodynamic Higgs mechanism is only robust for integrable systems. This explains the observation of superdiffusive scaling of spin autocorrelation functions in isotropic, non-integrable magnets at short times 20 , and also indicates why it should cross over to the nonlinear fluctuating hydrodynamics prediction of diffusive scaling at long times 21,52,53 .…”

Regularization dependence of the OTOC. Which Lyapunov spectrum is the physical one?

“…al. [9] use the canonical thermal OTOC C(t; β) (0,0) in a numerical study, whereas a cold atom experiment measures a Loschmidt echo [10], which can be related to C(t; β) ( 1 2 ,0) . As the theoretical prediction for these two correlation functions is different due to the regulator dependence, these two experimental results cannot be compared to each other.…”