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arXiv:2108.11654 · Journal of Physics A Mathematical and Theoretical

On intermediate statistics across many-body localization transition

Bitan De, Piotr Sierant, Jakub Zakrzewski

Published Nov 16, 2021• 11 citations
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Abstract

Abstract The level statistics in the transition between delocalized and localized phases of many body interacting systems is considered. We recall the joint probability distribution for eigenvalues resulting from the statistical mechanics for energy level dynamics as introduced by Pechukas and Yukawa. The resulting single parameter analytic distribution is probed numerically via Monte Carlo method. The resulting higher order spacing ratios are compared with data coming from different quantum many body systems. It is found that this Pechukas–Yukawa distribution compares favorably with β–Gaussian ensemble—a single parameter model of level statistics proposed recently in the context of disordered many-body systems. Moreover, the Pechukas–Yukawa distribution is also only slightly inferior to the two-parameter β–h ansatz shown earlier to reproduce level statistics of physical systems remarkably well.

DOI
10.1088/1751-8121/ac39cd

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Abstract The level statistics in the transition between delocalized and localized phases of many body interacting systems is considered. We recall the joint probability distribution for eigenvalues resulting from the statistical mechanics for energy level dynamics as introduced by Pechukas and Yukawa. The resulting single parameter analytic distribution is probed numerically via Monte Carlo method. The resulting higher order spacing ratios are compared with data coming from different quantum many body systems. It is found that this Pechukas–Yukawa distribution compares favorably with β–Gaussian ensemble—a single parameter model of level statistics proposed recently in the context of disordered many-body systems. Moreover, the Pechukas–Yukawa distribution is also only slightly inferior to the two-parameter β–h ansatz shown earlier to reproduce level statistics of physical systems remarkably well.
We numerically study level statistics of disordered interacting quantum many-body systems. A two-parameter plasma model which controls the level repulsion exponent $\ensuremath{\beta}$ and range hhh of interactions between eigenvalues is shown to reproduce accurately features of level statistics across the transition from the ergodic to many-body localized phase. Analysis of higher-order spacing ratios indicates that the considered $\ensuremath{\beta}\ensuremath{-}h$ model accounts even for long-range spectral correlations and allows us to obtain a clear picture of the flow of level statistics across the transition. Comparing the spectral form factors of the $\ensuremath{\beta}\ensuremath{-}h$ model and of a system across the ergodic-many-body-localized transition, we show that the range of effective interactions between eigenvalues hhh is related to the Thouless time which marks the onset of quantum chaotic behavior of the system. Analysis of level statistics of the random quantum circuit which hosts chaotic and localized phases supports the claim that the $\ensuremath{\beta}\ensuremath{-}h$ model grasps universal features of level statistics in transition between ergodic and many-body-localized phases also for systems breaking time-reversal invariance.
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.
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 .
From the random matrix theory all the energy levels should be strongly correlated due to the presence of all off-diagonal entries.In this work we introduce two new statistics to more accurately characterize these long-distance interactions in the disordered many-body systems with only short-range interaction. In the (p,q)(p, q)(p,q) statistics, we directly measure the long distance energy level spacings, while in the second approach, we randomly eliminate some of the energy levels, and then measure the reserved η%η\%η% energy levels using nearest-neighbor level spacings. We benchmark these results using the results in standard Gaussian ensembles. Some analytical distribution functions with extremely high accuracy are derived, which automatically satisfy the inverse relation and duality relation. These two measurements satisfy the same universal scaling law during the transition from the Gaussian ensembles to the Poisson ensemble, with critical disorder strength and corresponding exponent are independent of these measurements. These results shade new insight into the stability of many-body localized phase and their universal properties in the disordered many-body systems.
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.
Spectral statistics of systems that undergo many--body localization transition are studied. An analysis of the gap ratio statistics from the perspective of inter- and intra-sample randomness allows us to pin point differences between transitions in random and quasi-random disorder, showing the effects due to Griffiths rare events for the former case. It is argued that the transition for a random disorder exhibits universal features that are identified by constructing an appropriate model of intermediate spectral statistics which is a generalization of the family of short-range plasma models. The model incorporates the inter- and intra-sample fluctuations and faithfully reproduces level spacing distributions as well as number variance during the transition from ergodic to many--body localized phase. In particular, it grasps the critical level statistics which arise at disorder strength for which the fluctuations are the strongest.