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arXiv:2501.06563 · Physical review. B./Physical review. B

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

Federico Balducci, Giacomo Bracci Testasecca, Jacopo Niedda, Antonello Scardicchio, Carlo Vanoni

Published May 12, 2025• 3 citations
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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.

DOI
10.1103/64m5-m9ty

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We perform a thorough and complete analysis of the Anderson localization transition on several models of random graphs with regular and random connectivity. The unprecedented precision and abundance of our exact diagonalization data (both spectra and eigenstates), together with new finite size scaling and statistical analysis of the graph ensembles, unveils a universal behavior which is described by two simple, integer, scaling exponents. A by-product of such analysis is a reconciliation of the tension between the results of perturbation theory coming from strong disorder and earlier numerical works, which seemed to suggest that there should be a non-ergodic region above a given value of disorder WEW_{E}WE​ which is strictly less than the Anderson localization critical disorder WCW_CWC​, and that of other works which suggest that there is no such region. We find that, although no separate WEW_{E}WE​ exists from WCW_CWC​, the length scale at which fully developed ergodicity is found diverges like ∣W−WC∣−1|W-W_C|^{-1}∣W−WC​∣−1, while the critical length over which delocalization develops is ∼∣W−WC∣−1/2\sim |W-W_C|^{-1/2}∼∣W−WC​∣−1/2. The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. In addition, by looking at eigenstates and studying leading and sub-leading terms in system size-dependence of participation entropies, we show that the former contain information about the non-ergodicity volume which becomes non-trivial already deep in the delocalized regime. We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition.
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