Power-law random banded matrices and ultrametric matrices: Eigenvector distribution in the intermediate regime

Bogomolny, E.; Sieber, Martin

doi:10.1103/physreve.98.042116

Cited by 35 publications

(33 citation statements)

References 24 publications

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“…A slight discrepancy compared to the GOE prediction, is observed for W < 1, as an excess of small values of the eigenstate coefficients compared to GOE. Although it is consistent with a so-called weak ergodic phase, where the wavefunction occupies a finite, but tiny fraction of the Hilbert space, observed in several single-particle and many-body systems [75][76][77], we argue that this discrepancy is a result of proximity to an integrable point at W = 0, and should -if this is indeed the case -disappear either in the thermodynamic limit, or if integrability is broken. We leave the verification of this prediction to a future study.…”

Section: Summary and Discussionsupporting

confidence: 76%

Multifractality and its role in anomalous transport in the disordered XXZ spin-chain

2020

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The disordered XXZ model is a prototype model of the many-body localization (MBL) transition. Despite numerous studies of this model, the available numerical evidence of multifractality of its eigenstates is not very conclusive due to severe finite size effects. Moreover it is not clear if similarly to the case of single-particle physics, multifractal properties of the many-body eigenstates are related to anomalous transport, which is observed in this model. In this work, using a state-of-the-art, massively parallel, numerically exact method, we study systems of up to 24 spins and show that a large fraction of the delocalized phase flows towards ergodicity in the thermodynamic limit, while a region immediately preceding the MBL transition appears to be multifractal in this limit. We discuss the implication of our finding on the mechanism of subdiffusive transport.

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Section: Summary and Discussionsupporting

confidence: 76%

Multifractality and its role in anomalous transport in the disordered XXZ spin-chain

2020

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“…This implies that the standard random matrix classes (GOE, GUE) may not be the optimal random-matrix models for describing the mid-spectrum eigenstates. Other random matrix classes, such as the power-law-banded random matrices [17,38,97,[121][122][123], might be fruitful to examine as models of eigenstates of non-integrable many-body Hamiltonians.…”

Section: Discussionmentioning

confidence: 99%

Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems

2019

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Multifractal dimensions allow for characterizing the localization properties of states in complex quantum systems. For ergodic states the finite-size versions of fractal dimensions converge to unity in the limit of large system size. However, the approach to the limiting behavior is remarkably slow. Thus, an understanding of the scaling and finite-size properties of fractal dimensions is essential. We present such a study for random matrix ensembles, and compare with two chaotic quantum systems -the kicked rotor and a spin chain. For random matrix ensembles we analytically obtain the finite-size dependence of the mean behavior of the multifractal dimensions, which provides a lower bound to the typical (logarithmic) averages. We show that finite statistics has remarkably strong effects, so that even random matrix computations deviate from analytic results (and show strong sample-to-sample variation), such that restoring agreement requires exponentially large sample sizes. For the quantized standard map (kicked rotor) the multifractal dimensions are found to follow the random matrix predictions closely, with the same finite statistics effects. For a XXZ spin-chain we find significant deviations from the random matrix prediction -the large-size scaling follows a system-specific path towards unity. This suggests that local many-body Hamiltonians are "weakly ergodic", in the sense that their eigenfunction statistics deviate from random matrix theory. arXiv:1905.03099v2 [cond-mat.stat-mech]

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“…In the case of noninteracting particles in a disordered potential, the Mott argument for hybridization states that the metal-insulator transition occurs when the number of sites in resonance with a given site i stays finite in the thermodynamic limit [46][47][48]. Based on the analogy with single-particle Anderson localization, one can thus characterize the localized phase by the condition (Mott criterion)…”

Section: Estimate Of the Out-of-equilibrium Phase Diagram Within The Forward-scattering Approximationmentioning

confidence: 99%

“…and from this estimate the point at which many-body localization of the QREM takes place. Analogously, the ergodicity-breaking transition can be estimated from the Fermi golden rule [28,[46][47][48]. In fact, the spreading amplitude N ε (x)|M(ε, x)| 2 quantifies the escape rate of a particle created at a given site i.…”

Section: Estimate Of the Out-of-equilibrium Phase Diagram Within The Forward-scattering Approximationmentioning

confidence: 99%

Out-of-equilibrium phase diagram of the quantum random energy model

et al. 2021

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In this paper we study the out-of-equilibrium phase diagram of the quantum version of Derrida's random energy model, which is the simplest model of mean-field spin glasses. We interpret its corresponding quantum dynamics in Fock space as a one-particle problem in very high dimension to which we apply different theoretical methods tailored for high-dimensional lattices: the forward-scattering approximation, a mapping to the Rosenzweig-Porter model, and the cavity method. Our results indicate the existence of two transition lines and three distinct dynamical phases: a completely many-body localized phase at low energy, a fully ergodic phase at high energy, and a multifractal "bad metal" phase at intermediate energy. In the latter, eigenfunctions occupy a diverging volume yet an exponentially vanishing fraction of the total Hilbert space. We discuss the limitations of our approximations and the relationship with previous studies.

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Power-law random banded matrices and ultrametric matrices: Eigenvector distribution in the intermediate regime

Cited by 35 publications

References 24 publications

Multifractality and its role in anomalous transport in the disordered XXZ spin-chain

Multifractality and its role in anomalous transport in the disordered XXZ spin-chain

Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems

Out-of-equilibrium phase diagram of the quantum random energy model

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