Universal power law in crossover from integrability to quantum chaos

TL;DR

This paper investigates how one-dimensional interacting fermion models transition from integrability to quantum chaos, revealing a universal power-law scaling of the crossover perturbation with system size in gapless cases.

Contribution

It demonstrates that the crossover perturbation scales as a universal power law with system size in gapless systems, and suggests the exponent is characteristic of the underlying random matrix ensemble.

Findings

Crossover perturbation scales as ~L^{-3} in gapless systems.

Scaling appears faster than a power law in gapped systems.

Universal power-law behavior is robust across microscopic details.

Abstract

We study models of interacting fermions in one dimension to investigate the crossover from integrability to non-integrability, 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 $L \to \infty$ non-integrability sets in for an arbitrarily small integrability-breaking perturbation. The crossover value of the perturbation scales as a power law $\sim L^{-3}$ when the integrable system is gapless and the scaling 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 non-integrable system. For systems with a gap, the crossover scaling appears to be faster than a power law.