Chaos and the dynamics of information in dissipative electronic systems

Abstract: The dynamics of chaotic systems are, by definition, exponentially sensitive to initial conditions and may appear rather random. In this work, we explore relations between the chaotic dynamics of an observable and the dynamics of information (entropy) contained in this observable, focussing on a disordered metal coupled to a dissipative, e.g. phononic, bath. The chaotic dynamics is characterised by Lyapunov exponents $\lambda$, the rates of growth of out-of-time order correlators (OTOCs), quantities of the form $\langle[\hat A(t),\hat B(0)]^{2}\rangle\propto\exp(2\lambda t)$, where $\hat A$ and $\hat B$ are the operators of, e.g., the total current of electrons in a metallic quantum dot. We demonstrate that the Lyapunov exponent $\lambda$ matches the rate of decay of information stored in the observable $\langle \hat A(t)\rangle$ after applying a small perturbation with a small classical uncertainty. This relation suggests a way to measure Lyapunov exponents in experiment. We compute both the Lyapunov exponent and the rate of decay of information microscopically in a disordered metal in the presence of a bosonic bath, which may, in particular, represent interactions in the system. For a sufficiently short range of the correlations in the bath, the exponent has the form $\lambda=\lambda_{0}-1/\tau$, where $\lambda_{0}$ is the (temperature-independent) Lyapunov exponent in the absence of the bath and $1/\tau$ is the inelastic scattering rate. Our results demonstrate also the existence of a transition between chaotic and non-chaotic behaviour at $\lambda_{0}=1/\tau$, which may be triggered, e.g., by changing the temperature of the bath.