We study the spectral statistics of quantum systems with finite Hilbert spaces. We derive a theorem showing that eigenlevels in such systems cannot be globally uncorrelated, even in the case of fully integrable dynamics, as a consequence of the unfolding procedure. We provide an analytic expression for the power spectrum of the $\delta_n$ statistic for a model of intermediate statistics with level repulsion but independent spacings, and we show both numerically and analytically that the result is spoiled by the unfolding procedure. Then, we provide a simple model to account for this phenomenon, and test it by means of numerics on the disordered XXZ chain, the paradigmatic model of many-body localization, and the rational Gaudin-Richardson model, a prototypical model for quantum integrability.We study the spectral statistics of quantum systems with finite Hilbert spaces. We derive a theorem showing that eigenlevels in such systems cannot be globally uncorrelated, even in the case of fully integrable dynamics, as a consequence of the unfolding procedure. We provide an analytic expression for the power spectrum of the $\delta_n$ statistic for a model of intermediate statistics with level repulsion but independent spacings, and we show both numerically and analytically that the result is spoiled by the unfolding procedure. Then, we provide a simple model to account for this phenomenon, and test it by means of numerics on the disordered XXZ chain, the paradigmatic model of many-body localization, and the rational Gaudin-Richardson model, a prototypical model for quantum integrability.

Oct 13, 20200 citations

Stringent Test on Power Spectrum of Quantum Integrable and Chaotic Systems

Ángel L. Corps, Armando Relaño

Quantum chaotic and integrable systems are known to exhibit a characteristic $1/f$ and $1/f^{2}$ noise, respectively, in the power spectrum associated to their spectral fluctuations. A recent work [R. Riser, V. A. Osipov, and E. Kanzieper, \textit{Power Spectrum of Long Eigenlevel Sequences in Quantum Chaotic Systems}, Phys. Rev. Lett. \textbf{118}, 204101 (2017)] calls into question the approximations used to derive these results from random matrix theory. In this paper we show that such approximations do remain valid under almost any circumstances. For the integrable limit, we devise a protocol to exactly recover the original results. As a corollary, we show that the theoretical predictions for other statistics are bound for failure regarding long-range correlations, due to unavoidable spurious effects emerging from the analysis. By means of a rigorous statistical test, we also show that the corrections for the chaotic case introduced in the aforementioned paper require huge statistics to become relevant ---averages over more than $1000$ realizations are mandatory. As an application, we study a paradigmatic model for the crossover from the thermal to the many-body localized phase. We show that our protocol succeeds in describing the crossover. Furthermore, it also succeeds in proving that the Gaussian $β$ -ensemble fails to account for long-range correlations between the energy levels of this paradigmatic model.Quantum chaotic and integrable systems are known to exhibit a characteristic $1/f$ and $1/f^{2}$ noise, respectively, in the power spectrum associated to their spectral fluctuations. A recent work [R. Riser, V. A. Osipov, and E. Kanzieper, \textit{Power Spectrum of Long Eigenlevel Sequences in Quantum Chaotic Systems}, Phys. Rev. Lett. \textbf{118}, 204101 (2017)] calls into question the approximations used to derive these results from random matrix theory. In this paper we show that such approximations do remain valid under almost any circumstances. For the integrable limit, we devise a protocol to exactly recover the original results. As a corollary, we show that the theoretical predictions for other statistics are bound for failure regarding long-range correlations, due to unavoidable spurious effects emerging from the analysis. By means of a rigorous statistical test, we also show that the corrections for the chaotic case introduced in the aforementioned paper require huge statistics to become relevant ---averages over more than $1000$ realizations are mandatory. As an application, we study a paradigmatic model for the crossover from the thermal to the many-body localized phase. We show that our protocol succeeds in describing the crossover. Furthermore, it also succeeds in proving that the Gaussian $β$-ensemble fails to account for long-range correlations between the energy levels of this paradigmatic model.

arXiv:1908.09285Aug 25, 20191 citation

Level statistics across the many-body localization transition

Piotr Sierant, Jakub Zakrzewski

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.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.

arXiv:1808.02795Physical review. B./Physical review. BMar 22, 2019103 citations

Model of level statistics for disordered interacting quantum many-body systems

Piotr Sierant, Jakub Zakrzewski

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 $h$ 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 $h$ 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.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 $h$ 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 $h$ 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.

arXiv:1907.10336Physical review. B./Physical review. BMar 11, 202065 citations

Weighted models for level statistics across the many--body localization transition

Piotr Sierant, Jakub Zakrzewski

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 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.

Aug 8, 20183 citations

Spectral statistics across the many-body localization transition

Maksym Serbyn, Joel E. Moore

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.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.

arXiv:1508.07293Physical review. B./Physical review. BJan 29, 2016217 citations

\delta_n

statistic for a model of intermediate statistics with level repulsion but independent spacings, and we show both numerically and analytically that the result is spoiled by the unfolding procedure. Then, we provide a simple model to account for this phenomenon, and test it by means of numerics on the disordered XXZ chain, the paradigmatic model of many-body localization, and the rational Gaudin-Richardson model, a prototypical model for quantum integrability.

Quantum chaotic and integrable systems are known to exhibit a characteristic

1/f

and

1/f^{2}

noise, respectively, in the power spectrum associated to their spectral fluctuations. A recent work [R. Riser, V. A. Osipov, and E. Kanzieper, \textit{Power Spectrum of Long Eigenlevel Sequences in Quantum Chaotic Systems}, Phys. Rev. Lett. \textbf{118}, 204101 (2017)] calls into question the approximations used to derive these results from random matrix theory. In this paper we show that such approximations do remain valid under almost any circumstances. For the integrable limit, we devise a protocol to exactly recover the original results. As a corollary, we show that the theoretical predictions for other statistics are bound for failure regarding long-range correlations, due to unavoidable spurious effects emerging from the analysis. By means of a rigorous statistical test, we also show that the corrections for the chaotic case introduced in the aforementioned paper require huge statistics to become relevant ---averages over more than

1000

realizations are mandatory. As an application, we study a paradigmatic model for the crossover from the thermal to the many-body localized phase. We show that our protocol succeeds in describing the crossover. Furthermore, it also succeeds in proving that the Gaussian

β

-ensemble fails to account for long-range correlations between the energy levels of this paradigmatic model.

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

h

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

h

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.