Quantum Lyapunov exponents

Classical chaotic systems exhibit exponential divergence of initially close trajectories, which is characterized by the Lyapunov exponent.  This sensitivity to initial conditions is popularly known as the butterfly effect – a metaphor coined by the meteorologist, Edward Lorenz, who observed that hurricanes in his weather model were sensitive to tiny changes in initial data, comparable to an effect of flapping of butterfly wings.  Of great recent interest has been to understand how/if the butterfly effect and Lyapunov exponent generalize to quantum mechanics, where the notion of a trajectory does not exist.  In this talk, I will introduce the measure of quantum chaoticity – so called out-of-time-ordered four-point correlator (whose semi-classical limit reproduces classical Lyapunov growth,) and use it to describe quantum chaotic dynamics and its eventual disappearance in various models of quantum chaos.  I will also discuss our recent results on interacting disordered metals, which exhibit an interaction-induced transition from quantum chaotic to non-chaotic dynamics, which may manifest itself as a sharp change in the distribution of energy levels from Wigner-Dyson to Poisson statistics.