# Integrability to chaos in one dimensional quantum systems: Perspectives from transport and energy level statistics

## Subroto Mukerjee

The notion of integrability (operatively defined as the existence of as many conservation laws as degrees of freedom of a system) is an important one for both classical and quantum systems. In particular, integrability strongly constrains the dynamics of a system thereby possibly preventing thermalization. Here, we study the onset of such thermal (or non-integrable) behaviour by perturbing integrable quantum systems. The particular systems studied are one dimensional interacting and/or disordered systems of fermions, bosons and spins. We will show that transport in non-integrable systems differs siginficantly from that in integrable systems boith at zero and finite frequencies. In particular, we will use transport and energy level statistics to study the crossover from integrability to non-integrability and will show  that integrability can be destroyed by infinitesimally small perturbations in the thermodynamic limit. The crossover scale from integrability to non-integrability appears to go to zero as a power law in system size, the exponent of which is independent of microscopic details. We will provide evidence that the exponent appears to depend primarily on the random matrix ensembles  involved in the crossover from integrability to non-integrability.