Crossover from Poisson to Wigner-Dyson level statistics in spin chains with integrability breaking

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 crossover from behavior corresponding to 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.