Decay rate of the generalized survival probability as a signature of localization transitions

The survival probability measures the probability that a system taken out of equilibrium remains in its initial state. It is given by the power spectrum of the energy distribution of the initial state, which is also known as the local density of states (LDOS). The width of the LDOS gives the short-time decay rate of the survival probability. Inspired by the generalized entropies, we introduce a generalized version of the survival probability. We show that the width of the generalized LDOS, σ_q , can be used to detect the transition from a phase with extended states to a localized phase. In contrast with existing quantities to detect this transition, σ_q is self-averaging. Our studies are done for the following four systems: the power-law banded random matrix ensemble, the one-dimensional Aubry-Andr'e model with and without interactions, and the one-dimensional Heisenberg model with on-site disorder.