Universal power law in crossover from integrability to quantum chaos

Abstract

We study models of interacting fermions in one dimension to investigate the crossover from integrability to nonintegrability, i.e., quantum chaos, as a function of system size. Using exact diagonalization of finite-sized systems, we study this crossover by obtaining the energy level statistics and Drude weight associated with transport. Our results reinforce the idea that for system size $L\ensuremath{\rightarrow}\ensuremath{\infty}$ nonintegrability sets in for an arbitrarily small integrability-breaking perturbation. The crossover value of the perturbation scales as a power law $\ensuremath{\sim}{L}^{\ensuremath{-}\ensuremath{\eta}}$ when the integrable system is gapless. The exponent $\ensuremath{\eta}\ensuremath{\approx}3$ appears to be robust to microscopic details and the precise form of the perturbation. We conjecture that the exponent in the power law is characteristic of the random matrix ensemble describing the nonintegrable system. For systems with a gap, the crossover scaling appears to be faster than a power law.