Dynamical phases in a ``multifractal'' Rosenzweig-Porter model

Ivan Khaymovich,Vladimir Kravtsov

doi:10.21468/scipostphys.11.2.045

Abstract

We consider the static and the dynamical phases in a Rosenzweig-Porter (RP) random matrix ensemble with a 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 averaged survival probability may decay with time as a simple exponent, as a 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 localization model on Random Regular Graph onto the RP model and find exact values of the stretch-exponent \kappaκ in the thermodynamic limit. As another example we consider the logarithmically-normal RP random matrix ensemble and find analytically its phase diagram and the exponent \kappaκ. Our theory allows to describe analytically the finite-size multifractality and to compute the critical length with the exponent \nu_{MF}=1νMF=1 associated with it.

Highlights

We show that the survival probability Rav(t) in the RP ensemble corresponding to regular graph (RRG) is stretch-exponential in the entire extended phase [47], like it was earlier conjectured in Ref. [23]
In order to confirm the correspondence between RRG and MF-RP developed in the previous section, here we present the results of exact diagonalization for Rav(t) both in the logarithmicallynormal RP ensemble, Eq (47) with the ACTA symmetry, p = 1, N = 65536, and for the Anderson model on RRG with W = 12, N = 131072, Fig. 8
The consideration of such models is motivated by the fact that similar models emerge in the mapping of the sparse matrix Hamiltonians of disordered interacting many-body systems [17] and of the Anderson localization model on hierarchical graphs [35] onto the dense random matrix models

Summary

Introduction

Random matrix theory (RMT) has tremendous range of applications in physics spanning from the spectra of heavy nuclei to physics of glassy matter. Since the diffusion on a hierarchical graph (like in a well-known example of a Cayley tree) results in the simple exponential decay of survival probability, the stretch-exponential behavior should be associated with the sub-diffusion This observation is important because even on the extended side of the MBL transition in some many-body systems the diffusive character of transport is not completely confirmed by several numerical calculations [36,37,38,39,40,41,42,43,44]. We derive analytically the lines of phase transitions between the exponential (E) and stretch-exponential (SE) dynamics, as well as the expression for the stretch-exponent κ for a generic “multifractal” RP model

The model

Dynamical phases

Finite-size multifractality

Overlap correlation function and Levy stable distribution

Correspondence between Anderson model on RRG and MF-RP

Conclusions and Discussion