Largest Lyapunov exponent of long-range XY systems

TL;DR

This paper analytically computes the largest Lyapunov exponent for long-range XY Hamiltonian systems with power-law interactions, validating the theoretical predictions with numerical simulations.

Contribution

It introduces an analytical method to estimate the Lyapunov exponent in long-range interacting spin systems using a stochastic approach and microcanonical averages.

Findings

Excellent agreement between theoretical estimates and numerical simulations.

The stochastic approach effectively predicts Lyapunov exponents in disordered regimes.

Analytical expressions depend on statistical properties of the Hessian matrix.

Abstract

We calculate analytically the largest Lyapunov exponent of the so-called $\alpha XY$ Hamiltonian in the high energy regime. This system consists of a $d$-dimensional lattice of classical spins with interactions that decay with distance following a power-law, the range being adjustable. In disordered regimes the Lyapunov exponent can be easily estimated by means of the "stochastic approach", a theoretical scheme based on van Kampen's cumulant expansion. The stochastic approach expresses the Lyapunov exponent as a function of a few statistical properties of the Hessian matrix of the interaction that can be calculated as suitable microcanonical averages. We have verified that there is a very good agreement between theory and numerical simulations.