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arXiv:2012.12841 · Physical review. B./Physical review. B

Lévy-Rosenzweig-Porter random matrix ensemble

Giulio Biroli, Marco Tarzia

Published Mar 25, 2021• 66 citations
arXivHTMLPDF

Abstract

In this paper, we consider an extension of the Rosenzweig-Porter model, the L\'evy-RP (L-RP) model, in which the off-diagonal matrix elements are broadly distributed, providing a more realistic benchmark to develop an effective description of nonergodic extended (NEE) states in interacting many-body disordered systems. We put forward a simple, general, and intuitive argument that allows one to unveil the multifractal structure of the minibands in the local spectrum when hybridization is due to anomalously large transition amplitudes in the tails of the distribution. The idea is that the energy spreading of the minibands can be determined self-consistently by requiring that the maximal hybridization rate ${\mathcal{H}}_{ij}$ between a site $i$ and the other ${N}^{{D}_{1}}$ sites of the support set is of the same order of the Thouless energy itself ${N}^{{D}_{1}\ensuremath{-}1}$. This argument yields the fractal dimensions that characterize the statistics of the multifractal wave functions in the NEE phase, as well as the whole phase diagram of the L-RP ensemble. Its predictions are confirmed both analytically, by a thorough investigation of the self-consistent equation for the local density of states obtained using the cavity approach, and numerically, via extensive exact diagonalizations.

DOI
10.1103/physrevb.103.104205

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In this paper, we consider an extension of the Rosenzweig-Porter model, the L'evy-RP (L-RP) model, in which the off-diagonal matrix elements are broadly distributed, providing a more realistic benchmark to develop an effective description of nonergodic extended (NEE) states in interacting many-body disordered systems. We put forward a simple, general, and intuitive argument that allows one to unveil the multifractal structure of the minibands in the local spectrum when hybridization is due to anomalously large transition amplitudes in the tails of the distribution. The idea is that the energy spreading of the minibands can be determined self-consistently by requiring that the maximal hybridization rate Hij{\mathcal{H}}_{ij}Hij​ between a site iii and the other ND1{N}^{{D}_{1}}ND1​ sites of the support set is of the same order of the Thouless energy itself ND1\ensuremath−1{N}^{{D}_{1}\ensuremath{-}1}ND1​\ensuremath−1. This argument yields the fractal dimensions that characterize the statistics of the multifractal wave functions in the NEE phase, as well as the whole phase diagram of the L-RP ensemble. Its predictions are confirmed both analytically, by a thorough investigation of the self-consistent equation for the local density of states obtained using the cavity approach, and numerically, via extensive exact diagonalizations.
The Rosenzweig--Porter (RP) random matrix ensemble has emerged as a minimal model for the integrability-to-chaos crossover in quantum many-body systems. Its phase diagram features a region with fractal eigenstates, exhibiting intermediate spectral and localization properties between the fully localized and fully delocalized regimes. In this work, we explore several generalizations of the RP model and determine their level statistics at the scale of the Thouless energy ETE_TET​, which characterizes the crossover. Using tools from free probability theory and the replica method, we compute the full counting statistics in the limit of large system size, and show that it takes a simple, universal scaling form around ETE_TET​, shared across all variations of the model. We validate our analytical predictions using exact numerical diagonalization of large samples, and large-deviation algorithms that resolve the full counting statistics down to probabilities as low as 10−4010^{-40}10−40. We also contrast our predictions with measurements on the quantum random energy model, which is the simplest model displaying many-body localization.
Motivated by a series of recent works, an interest in multifractal phases has risen as they are believed to be present in the Many-Body Localized (MBL) phase and are of high demand in quantum annealing and machine learning. Inspired by the success of the Rosenzweig-Porter (RP) model with Gaussian-distributed hopping elements, several RP-like ensembles with the fat-tailed distributed hopping terms have been proposed, with claims that they host the desired multifractal phase. In the present work, we develop a general (graphical) approach allowing a self-consistent analytical calculation of fractal dimensions for a generic RP model and investigate what features of the RP Hamiltonians can be responsible for the multifractal phase emergence. We conclude that the only feature contributing to a genuine multifractality is the on-site energies’ distribution, meaning that no random matrix model with a statistically homogeneous distribution of diagonal disorder and uncorrelated off-diagonal terms can host a multifractal phase.
The Rosenzweig-Porter (RP) model has garnered much attention in the last decade, as it is a simple analytically tractable model showing both ergodic-nonergodic extended and Anderson localization transitions. Thus, it is a good toy model to understand the Hilbert-space structure of many-body localization phenomenon. In our Letter, we present analytical evidence, supported by exact numerics, that demonstrates the controllable tuning of the phase diagram in the RP model by employing on-site potentials with a nontrivial fractal dimension instead of the conventional random disorder. We demonstrate that such disorder extends the fractal phase and creates an unusual dependence of fractal dimensions of the eigenfunctions. Furthermore, we study the fate of level statistics in such a system to understand how these changes are reflected in the eigenvalue statistics.
Abstract In recent years, the Rosenzweig–Porter (RP) ensemble, obtained by adding a diagonal matrix with independent and identically distributed elements to a Gaussian random matrix, has been widely used as a minimal model for the emergence of fractal eigenstates in complex many-body systems. A key open question concerns the robustness of its phase diagram when the assumption of independent and uncorrelated entries is relaxed—an assumption that simplifies its analysis but is generally violated in realistic quantum systems. In this work, we take a first step in this direction by considering a deformed Wishart (rather than Gaussian) random matrix, which we dub the ‘Wishart–RP’ ensemble. Using perturbation theory, as well as the cavity and replica methods and the Dyson Brownian motion approach, we characterize its phase diagram and localization properties. Remarkably, we show that the level compressibility, which quantifies spectral correlations in the fractal phase, coincides with that of the Gaussian RP model, thereby extending the universality conjectured in Venturelli et al (2023 SciPost Phys. 14 110) beyond the fully uncorrelated setting. We confirm our results with numerical tests.
In this paper, we suggest an extension of the Rosenzweig-Porter (RP) model, the LN-RP model, in which the off-diagonal matrix elements have a wide, log-normal distribution.We argue that this model is more suitable to describe a generic many-body localization problem.In contrast to RP model, in LN-RP model, a fragile weakly ergodic phase appears that is characterized by broken basis-rotation symmetry which the fully ergodic phase, also present in this model, strictly respects in the thermodynamic limit.Therefore, in addition to the localization and ergodic transitions in LN-RP model, there exists also the transition between the two ergodic phases (FWE transition).We suggest new criteria of stability of the nonergodic phases that give the points of localization and ergodic transitions and prove that the Anderson localization transition in LN-RP model involves a jump in the fractal dimension of the egenfunction support set.We also formulate the criterion of FWE transition and obtain the full phase diagram of the model.We show that truncation of the log-normal tail shrinks the region of weakly ergodic phase and restores the multifractal and the fully ergodic phases.
The Rosenzweig-Porter model has seen a resurgence in interest as it exhibits a nonergodic extended phase between the ergodic extended metallic phase and the localized phase. Such a phase is relevant to many physical models from the Sachdev-Ye-Kitaev model in high-energy physics and quantum gravity to the interacting many-body localization in condensed-matter physics and quantum computing. This phase is characterized by fractal behavior of the wave functions and a postulated correlated miniband structure of the energy spectrum. Here we will seek evidence for the latter in the spectrum. Since this behavior is expected on intermediate energy scales, spectral rigidity or number variance is a natural way to tease it out. Nevertheless, due to the Thouless energy and ambiguities in the unfolding procedure, the results are inconclusive. On the other hand, by using the singular-value decomposition method, clear evidence for a super-Poissonian behavior in this regime emerges, consistent with a picture of correlated minibands.