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arXiv:2006.04827 · Physical Review Research

Fragile extended phases in the log-normal Rosenzweig-Porter model

Ivan M. Khaymovich, V. E. Kravtsov, B. L. Altshuler, L. B. Ioffe

Published Dec 9, 2020• 107 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevresearch.2.043346

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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 (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.
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 ${N}^{{D}_{1}\ensuremath{-}1}$. 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.
Models with correlated disorder are rather common in physics. In some of them, like the Aubry-André (AA) model, the localization phase diagram can be found from the (self)duality with respect to the Fourier transform. In others, like the all-to-all translation-invariant Rosenzweig-Porter (TI RP) ensemble or the many-body localization (MBL), one needs to develop more sophisticated and usually phenomenological methods to obtain the localization transition. In addition, such models contain not only localization, but also the ergodicity-breaking transition in the Hilbert-space eigenstate structure, giving way to the nonergodic (i.e., MBL) phase of states with nontrivial fractal dimensions D q > 0 . In this work, we suggest a method to calculate both the above transitions and a lower bound to the fractal dimensions D 2 and D ∞ , relevant for the physical observables. To verify this method, we apply it to the class of long-range (self-)dual models, interpolating between a tight-binding ballistic model and TI RP via both power-law dependencies of the hopping terms and the on-site disorder correlations, and, thus, being out of the validity range of the previously developed methods. We show that the interplay of the correlated disorder and the power-law decaying hopping terms leads to the emergence of the two types of nonergodic phases in the entire range of parameters, even without having any quasiperiodicity of the AA potential. The analytical results of the above method are in full agreement with the extensive numerical calculations.
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.
Motivated by the problem of Many-Body Localization and the recent numerical results for the level and eigenfunction statistics on the random regular graphs, a generalization of the Rosenzweig-Porter random matrix model is suggested that possesses two localization transitions as the parameter γγγ of the model varies from 0 to ∞\infty∞. One of them is the Anderson transition from the localized to the extended states that happens at γ=2γ=2γ=2. The other one at γ=1γ=1γ=1 is the transition from the extended non-ergodic (multifractal) states to the extended ergodic states similar to the eigenstates of the Gaussian Orthogonal Ensemble. We computed the two-level spectral correlation function, the spectrum of multifractality f(α)f(α)f(α) and the wave function overlap which all show the transitions at γ=1γ=1γ=1 and γ=2γ=2γ=2.