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arXiv:1510.08322 · Journal of Statistical Mechanics Theory and Experiment

Level repulsion exponent β for many-body localization transitions and for Anderson localization transitions via Dyson Brownian motion

Cécile Monthus

Published Mar 1, 2016• 11 citations
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Abstract

Abstract The generalization of the Dyson Brownian motion approach of random matrices to Anderson localization (AL) models (Chalker et al 1996 Phys. Rev. Lett . 77 554) and to many-body localization (MBL) Hamiltonians (Serbyn and Moore 2015 arXiv:1508.07293) is revisited to extract the level repulsion exponent β , where in the delocalized phase governed by the Wigner–Dyson statistics, , in the localized phase governed by the Poisson statistics, and at the critical point. The idea is that the Gaussian disorder variables h i are promoted to Gaussian stationary processes h i ( t ) in order to sample the disorder stationary distribution with some time correlation τ . The statistics of energy levels can then be studied via Langevin and Fokker–Planck equations. For the MBL quantum spin Hamiltonian with random fields h i , we obtain in terms of the Edwards–Anderson matrix for the same eigenstate m = n and for consecutive eigenstates m = n + 1. For the Anderson localization tight-binding Hamiltonian with random on-site energies h i , we find in terms of the density correlation matrix for consecutive eigenstates m = n + 1, while the diagonal element m = n corresponds to the inverse participation ratio of the eigenstate .

DOI
10.1088/1742-5468/2016/03/033113

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Abstract The generalization of the Dyson Brownian motion approach of random matrices to Anderson localization (AL) models (Chalker et al 1996 Phys. Rev. Lett . 77 554) and to many-body localization (MBL) Hamiltonians (Serbyn and Moore 2015 arXiv:1508.07293) is revisited to extract the level repulsion exponent β , where in the delocalized phase governed by the Wigner–Dyson statistics, , in the localized phase governed by the Poisson statistics, and at the critical point. The idea is that the Gaussian disorder variables h i are promoted to Gaussian stationary processes h i ( t ) in order to sample the disorder stationary distribution with some time correlation τ . The statistics of energy levels can then be studied via Langevin and Fokker–Planck equations. For the MBL quantum spin Hamiltonian with random fields h i , we obtain in terms of the Edwards–Anderson matrix for the same eigenstate m = n and for consecutive eigenstates m = n + 1. For the Anderson localization tight-binding Hamiltonian with random on-site energies h i , we find in terms of the density correlation matrix for consecutive eigenstates m = n + 1, while the diagonal element m = n corresponds to the inverse participation ratio of the eigenstate .
Motivated by the many-body localization (MBL) phase in generic interacting disordered quantum systems, we develop a model simulating the same eigenstate structure like in MBL, but in the random-matrix setting. Demonstrating the absence of energy level repulsion (Poisson statistics), this model carries non-ergodic eigenstates, delocalized over the extensive number of configurations in the Hilbert space. On the above example, we formulate general conditions to a single-particle and random-matrix models in order to carry such states, based on the transparent generalization of the Anderson localization of single-particle states and multiple resonances.
The space of one-dimensional disordered interacting quantum models displaying a many-body localization (MBL) transition seems sufficiently rich to produce critical points with level statistics interpolating continuously between the Poisson statistics of the localized phase and the Wigner–Dyson statistics of the delocalized phase. In this paper, we consider the strong disorder limit of the MBL transition, where the level statistics at the MBL critical point is close to the Poisson statistics. We analyze a one-dimensional quantum spin model, in order to determine the statistical properties of the rare extensive resonances that are needed to destabilize the MBL phase. At criticality, we find that the entanglement entropy can grow with an exponent 0 < α < 1 anywhere between the area law α = 0 and the volume law α = 1 , as a function of the resonances properties, while the entanglement spectrum follows the strong multifractality statistics. In the MBL phase near criticality, we obtain the simple value ν = 1 for the correlation length exponent. Independently of the strong disorder limit, we explain why, for the many-body localization transition concerning individual eigenstates, the correlation length exponent ν is not constrained by the usual Harris inequality ν ≥ 2 / d , so that there is no theoretical inconsistency with the best numerical measure ν = 0 . 8 ( 3 ) obtained by Luitz et al. (2015).
The many-body localization transition (MBLT) between ergodic and many-body localized phases in disordered interacting systems is a subject of much recent interest. The statistics of eigenenergies is known to be a powerful probe of crossovers between ergodic and integrable systems in simpler examples of quantum chaos. We consider the evolution of the spectral statistics across the MBLT, starting with mapping to a Brownian motion process that analytically relates the spectral properties to the statistics of matrix elements. We demonstrate that the flow from Wigner-Dyson to Poisson statistics is a two-stage process. First, a fractal enhancement of matrix elements upon approaching the MBLT from the delocalized side produces an effective power-law interaction between energy levels, and leads to a plasma model for level statistics. At the second stage, the gas of eigenvalues has local interactions and the level statistics belongs to a semi-Poisson universality class. We verify our findings numerically on the XXZ spin chain. We provide a microscopic understanding of the level statistics across the MBLT and discuss implications for the transition that are strong constraints on possible theories.
For short-ranged disordered quantum models in one dimension, the Many-Body-Localization is analyzed via the adaptation to the Many-Body context [M. Serbyn, Z. Papic and D.A. Abanin, PRX 5, 041047 (2015)] of the Thouless point of view on the Anderson transition : the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (Delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (Many-Body-Localized Phase with an area-law entanglement). The central object is thus the level of Hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the Strong-Multifractality Spectrum f(0≤α≤2)=α2f(0 \leq \alpha \leq 2)=\frac{\alpha}{2}f(0≤α≤2)=2α​, well known in the context of Anderson Localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. As a consequence, the level statistics at criticality is the closest possible to the Poisson statistics of the Localized phase, and the Delocalized phase near the critical point is not ergodic.
Abstract For short-ranged disordered quantum models in one dimension, the many-body-localization is analyzed via the adaptation to the many-body context (Serbyn et al 2015 Phys. Rev . X 5 041047) of the Thouless point of view on the Anderson transition: the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (many-body-localized phase with an area-law entanglement). The central object is thus the level of hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the strong-multifractality spectrum f ( 0 ⩽ α ⩽ 2 ) = α 2 , well known in the context of Anderson localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. Finally, the possibility of a delocalized non-ergodic phase near criticality is discussed.
Abstract We study the critical level statistics at the many-body localization (MBL) transition region in random spin systems. By employing the inter-sample randomness as indicator, we manage to locate the MBL transition point in both orthogonal and unitary models. We further count the n th order gap ratio distributions at the transition region up to n = 4, and find they fit well with the short-range plasma model with inverse temperature β = 1 for orthogonal model and β = 2 for unitary. These critical level statistics are argued to be universal by comparing results from systems both with and without total S z conservation. We also point out that these critical distributions can emerge from the spectrum of a Poisson ensemble, which indicates the thermal-MBL transition point is more affected by the MBL phase rather than thermal phase.