Dynamical phases in a "multifractal" Rosenzweig-Porter model

TL;DR

This paper develops a theory for the static and dynamic phases in a multifractal Rosenzweig-Porter model, identifying different decay regimes of survival probability and analyzing phase diagrams for related random matrix ensembles.

Contribution

It introduces a general framework for understanding survival probability decay in multifractal RP models and provides exact phase diagrams for specific ensembles like RRG and LN-RP.

Findings

Identified exponential, stretch-exponential, and frozen-dynamics phases.

Derived exact phase diagrams for RRG and LN-RP models.

Showed finite-size effects induce a multifractal phase near the localization transition.

Abstract

We consider the static and dynamic phases in a Rosenzweig-Porter (RP) random matrix ensemble with the tailed distribution of off-diagonal matrix elements of the form of the large-deviation ansatz. We present a general theory of survival probability in such a random-matrix model and show that the {\it averaged} survival probability may decay with time as the simple exponent, as the stretch-exponent and as a power-law or slower. Correspondingly, we identify the exponential, the stretch-exponential and the frozen-dynamics phases. As an example, we consider the mapping of the Anderson model on Random Regular Graph (RRG) onto the "multifractal" RP model and find exact values of the stretch-exponent $\kappa$ depending on box-distributed disorder in the thermodynamic limit. As another example we consider the logarithmically-normal RP (LN-RP) random matrix ensemble and find analytically its phase diagram and the exponent $\kappa$. In addition, our theory allows to compute the shift of apparent phase transition lines at a finite system size and show that in the case of RP associated with RRG and LN-RP with the same symmetry of distribution function of hopping, a finite-size multifractal "phase" emerges near the tricritical point which is also the point of localization transition.