Spatio-temporal spread of perturbations in power-law models at low temperatures: Exact results for OTOC

TL;DR

This paper provides exact and analytical results for the Out-of-Time-Order Commutator in power-law interacting particle systems, revealing distinct behaviors across different interaction regimes and connecting to integrability and field theory insights.

Contribution

It offers the first exact analytical results for OTOC in classical power-law models at low temperatures, including the effects of nonlinear dispersion and universality classes.

Findings

Excellent agreement between analytical and numerical OTOC results.

Identification of a transition at k=3 from non-Airy to Airy universality.

Rich features observed for the integrable case k=2.

Abstract

We present exact results for the classical version of the Out-of-Time-Order Commutator (OTOC) for a family of power-law models consisting of $N$ particles in one dimension and confined by an external harmonic potential. These particles are interacting via power-law interaction of the form $\propto \sum_{\substack{i, j=1 (i\neq j)}}^N|x_i-x_j|^{-k}$ $\forall$ $k>1$ where $x_i$ is the position of the $i^\text{th}$ particle. We present numerical results for the OTOC for finite $N$ at low temperatures and short enough times so that the system is well approximated by the linearized dynamics around the many body ground state. In the large-$N$ limit, we compute the ground-state dispersion relation in the absence of external harmonic potential exactly and use it to arrive at analytical results for OTOC. We find excellent agreement between our analytical results and the numerics. We further obtain analytical results in the limit where only linear and leading nonlinear (in momentum) terms in the dispersion relation are included. The resulting OTOC is in agreement with numerics in the vicinity of the edge of the "light cone". We find remarkably distinct features in OTOC below and above $k=3$ in terms of going from non-Airy behaviour ($1<k<3$) to an Airy universality class ($k>3$). We present certain additional rich features for the case $k=2$ that stem from the underlying integrability of the Calogero-Moser model. We present a field theory approach that also assists in understanding certain aspects of OTOC such as the sound speed. Our findings are a step forward towards a more general understanding of the spatio-temporal spread of perturbations in long-range interacting systems.