Survival probability in Generalized Rosenzweig-Porter random matrix ensemble

TL;DR

This paper analyzes the survival probability dynamics in the generalized Rosenzweig-Porter random matrix model, identifying distinct decay behaviors across ergodic, multifractal, and localized phases to serve as phase indicators.

Contribution

It introduces the survival probability as a dynamical indicator to distinguish phases in the generalized Rosenzweig-Porter model, combining analytical and numerical methods.

Findings

Ergodic phase shows power-law decay with oscillations.

Multifractal phase exhibits exponential decay with a vanishing rate.

Localized phase has a finite saturation value of survival probability.

Abstract

We study analytically and numerically the dynamics of the generalized Rosenzweig-Porter model, which is known to possess three distinct phases: ergodic, multifractal and localized phases. Our focus is on the survival probability $R(t)$, the probability of finding the initial state after time $t$. In particular, if the system is initially prepared in a highly-excited non-stationary state (wave packet) confined in space and containing a fixed fraction of all eigenstates, we show that $R(t)$ can be used as a dynamical indicator to distinguish these three phases. Three main aspects are identified in different phases. The ergodic phase is characterized by the standard power-law decay of $R(t)$ with periodic oscillations in time, surviving in the thermodynamic limit, with frequency equals to the energy bandwidth of the wave packet. In multifractal extended phase the survival probability shows an exponential decay but the decay rate vanishes in the thermodynamic limit in a non-trivial manner determined by the fractal dimension of wave functions. Localized phase is characterized by the saturation value of $R(t\to\infty)=k$, finite in the thermodynamic limit $N\rightarrow\infty$, which approaches $k=R(t\to 0)$ in this limit.