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

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.

Emergent fractal phase in energy stratified random models

“…This direction has immediate applications in the description of short-range graph models and their mapping to the random-matrix ensembles (see, e.g. [49]).…”

“…In addition, the disordered many-body systems considered in the Hilbert space [6] and their counterparts on the hierarchical graph structures [22,23,49,50] have been recently mapped (within some approximations) to the above Rosenzweig-Porter-like models. In these mappings the all-to-all hops which amplitudes depend on the Hilbert space dimension N emerge naturally from the short-range models in the hierarchical (Hilbert) space.…”

“…Indeed, the all-to-all coupling is obtained in these models from the consideration of long series of quantum transitions to (at least) exponentially growing number of available configurations at the certain hopping distance, while the size-dependence of the hopping amplitudes is related to the exponentially decaying propagators [6]. Proliferating this process up to the Hilbert space diameter, one immediately restores the complete graph with the N -dependent amplitudes [49].…”

“…However, with all the above mentioned success in applications [6,8,22,23,[29][30][31][32][33]49], the RP model has some caveats. Indeed, it is formed by a complete graph with the independent identically distributed (i.i.d.)…”

Fluctuation-driven transitions in localized insulators: Intermittent metallicity and path chaos precede delocalization

“…This transition entails a nonanalytic behavior of the spatial decay constant γ L governing the decay rates of excitations in the thermodynamic limit (akin to localization lengths or Lyapunov exponents in noninteracting systems). In contrast, the putative transition between an ergodic and possibly nonergodic delocalized phase, which was suggested to map the unfreezing of an associated effective polymer problem [86], takes place within the metallic phase where excitations are delocalized, even though transport appears to be anomalously slow [89,90].…”