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arXiv:1807.06983

Intermediate spectral statistics in the many--body localization transition

Piotr Sierant, Jakub Zakrzewski

Published Jul 18, 2018• 4 citations
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Abstract

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.

DOI
10.48550/arxiv.1807.06983

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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.
Level 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 in the case of 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 considered weighted short-range plasma model yields a very good agreement both for level spacing distribution including its exponential tail and the number variance up to tens of level spacings outperforming previously proposed models. In particular, our model grasps the critical level statistics which arise at disorder strength for which the inter-sample fluctuations are the strongest. Going beyond the paradigmatic examples of many-body localization in spin systems, we show that the considered model also grasps the level statistics of disordered Bose- and Fermi-Hubbard models. The remaining deviations for long-range spectral correlations are discussed and attributed mainly to the intricacies of level unfolding.
Level statistics of systems that undergo many-body localization transition are studied. An analysis of the gap ratio statistics from the perspective of inter- and intrasample randomness allows us to pin point differences between transitions in random and quasirandom disorder, showing the effects due to Griffiths rare events for the former case. It is argued that the transition in the case of 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 considered weighted short-range plasma model yields a very good agreement both for level spacing distribution including its exponential tail and the number variance up to tens of level spacings outperforming previously proposed models. In particular, our model grasps the critical level statistics which arise at disorder strength for which the intersample fluctuations are the strongest. Going beyond the paradigmatic examples of many-body localization in spin systems, we show that the considered model also grasps the level statistics of disordered Bose- and Fermi-Hubbard models. The remaining deviations for long-range spectral correlations are discussed and attributed mainly to the intricacies of level unfolding.
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.
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.
We study a one-dimensional XXZ spin chain in a random field on the metallic side of the many-body localization transition by level statistics. For a fixed interaction, and intermediate disorder below the many-body localization transition, we find that, asymptotically, the number variance grows faster than linear with a disorder-dependent exponent. This is consistent with the existence of an anomalous Thouless energy in the spectrum. In noninteracting disordered metals, this is an energy scale related to the typical time for a particle to diffuse across the sample. In the interacting case, it seems related to a more intricate anomalous diffusion process. This interpretation is not fully consistent with recent claims that for intermediate disorder, level statistics are described by a plasma model with power-law decaying interactions whose number variance grows slower than linear. As disorder is further increased, still on the metallic side, the Thouless energy is gradually washed out. In the range of sizes we can explore, level statistics are scale invariant and approach Poisson statistics at the many-body localization transition. Slightly below the many-body localization transition, spectral correlations, well described by critical statistics, are quantitatively similar to those of a high-dimensional, noninteracting, disordered conductor at the Anderson transition.
We study the transitions between ergodic and many-body localized phases in spin systems, subject to quenched disorder, including the Heisenberg chain and the central spin model. In both cases systems with common spin lengths 1/21/21/2 and 1 are investigated via exact numerical diagonalization and random matrix techniques. Particular attention is paid to the sample-to-sample variance ${({\mathrm{\ensuremath{\Delta}}}_{s}r)}^{2}$ of the averaged consecutive-gap ratio $\ensuremath{\langle}r\ensuremath{\rangle}$ for different disorder realizations. For both types of systems and spin lengths we find a maximum in ${\mathrm{\ensuremath{\Delta}}}_{s}r$ as a function of disorder strength, accompanied by an inflection point of $\ensuremath{\langle}r\ensuremath{\rangle}$, signaling the transition from ergodicity to many-body localization. The critical disorder strength is found to be somewhat smaller than the values reported in the recent literature. Further information about the transitions can be gained from the probability distribution of expectation values within a given disorder realization.