Crossover from Poisson to Wigner-Dyson Level Statistics in Spin Chains with Integrability Breaking

TL;DR

This paper investigates how adding an integrability-breaking term to the XXZ spin chain causes a transition in energy-level statistics from Poisson to Wigner-Dyson, indicating a shift from integrable to chaotic behavior.

Contribution

It provides numerical evidence of the crossover in level statistics in finite chains and suggests that even infinitesimal integrability breaking could induce chaos in large systems.

Findings

Level statistics transition from Poisson to Wigner-Dyson with integrability breaking

Different measures of level statistics follow distinct crossover patterns

Evidence indicates infinitesimal integrability breaking may cause chaos in large systems

Abstract

We study numerically the evolution of energy-level statistics as an integrability-breaking term is added to the XXZ Hamiltonian. For finite-length chains, physical properties exhibit a cross-over from behavior resulting from the Poisson level statistics characteristic of integrable models to behavior corresponding to the Wigner-Dyson statistics characteristic of the random-matrix theory used to describe chaotic systems. Different measures of the level statistics are observed to follow different crossover patterns. The range of numerically accessible system sizes is too small to establish with certainty the scaling with system size, but the evidence suggests that in a thermodynamically large system an infinitesimal integrability breaking would lead to Wigner-Dyson behavior.