Scaling analysis and renormalization group on the mobility edge in the quantum random energy model

TL;DR

This paper investigates the localization-delocalization transition in the quantum Random Energy Model using renormalization group analysis, revealing universal scaling behaviors and fixed points relevant to many-body localization phenomena.

Contribution

It introduces a scaling theory of localization in the QREM, identifying fixed points and universal properties, and connects these to Anderson transitions on expander graphs.

Findings

RG trajectories flow toward ergodic phase at zero energy density

Localization transition at the spectrum center with properties similar to Anderson transition

Scaling behavior at finite energy density analogous to expander graph transitions

Abstract

Building on recent progress in the study of Anderson and many-body localization via the renormalization group (RG), we examine the scaling theory of localization in the quantum Random Energy Model (QREM). The QREM is known to undergo a localization-delocalization transition at finite energy density, while remaining fully ergodic at the center of the spectrum. At zero energy density, we show that RG trajectories consistently flow toward the ergodic phase, and are characterized by an unconventional scaling of the fractal dimension near the ergodic fixed point. When the disorder amplitude is rescaled, as suggested by the forward scattering approximation approach, a localization transition emerges also at the center of the spectrum, with properties analogous to the Anderson transition on expander graphs. At finite energy density, a localization transition takes place without disorder rescaling, and yet it exhibits a scaling behavior analogous to the one observed on expander graphs. The universality class of the model remains unchanged under the rescaling of the disorder, reflecting the independence of the RG from microscopic details. Our findings demonstrate the robustness of the scaling behavior of random graphs and offer new insights into the many-body localization transition.