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arXiv:2112.09700 · Quantum

Non-ergodic delocalized phase with Poisson level statistics

Weichen Tang, Ivan M. Khaymovich

Published Jun 9, 2022• 39 citations
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Abstract

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.

DOI
10.22331/q-2022-06-09-733

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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.
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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 .
The study of many-body localized (MBL) phases intrinsically links spectral properties with eigenstate characteristics: localized systems exhibit Poisson level statistics and area-law entanglement entropy, while ergodic systems display volume-law entanglement and follow random matrix theory predictions, including level repulsion. Here, we introduce the interacting Anderson Quantum Sun model, which significantly deviates from these conventional expectations. In addition to standard localized and ergodic phases, we identify a regime that exhibits volume-law entanglement coexisting with intermediate spectral statistics. We also identify another nonstandard regime marked by Poisson level statistics, sub-volume entanglement growth, and rare-event-dominated correlations, indicative of emerging ergodic instabilities. These results highlight unconventional routes of ergodicity breaking and offer fresh perspectives on how Anderson localization may be destabilized.
In contrast with Anderson localization where a genuine localization is observed in real space, the many-body localization (MBL) problem is much less understood in Hilbert space, the support of the eigenstates. In this Letter, using exact diagonalization techniques we address the ergodicity properties in the underlying N-dimensional complex networks spanned by various computational bases for up to L=24 spin-1/2 particles (i.e., Hilbert space of size N≃2.7×10^{6}). We report fully ergodic eigenstates in the delocalized phase (irrespective of the computational basis), while the MBL regime features a generically (basis-dependent) multifractal behavior, delocalized but nonergodic. The MBL transition is signaled by a nonuniversal jump of the multifractal dimensions.
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.
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.