Statistics of correlations across the many-body localization transition

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

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

Exploring quantum correlations in a many-body localized system

Julian Léonard, Matthew Rispoli, Alexander Lukin +4 more

Bulletin of the American Physical SocietyMar 5, 20190 citations

Fock-space anatomy of eigenstates across the many-body localization transition

Sthitadhi Roy, David E. Logan

We explore the Fock-space structure of eigenstates across the many-body localization (MBL) transition in a disordered, interacting quantum spin- $\frac{1}{2}$ chain. Eigenstate expectation values of spatially local observables, which distinguish an MBL phase from an ergodic one, can be represented in terms of eigenstate amplitudes on the Fock space. Motivated by this, we introduce and study spatial correlations on the Fock space. From these, a correlation length emerges, which is found to vary discontinuously across the MBL transition, and is intimately connected to the discontinuous jump in the multifractal exponents characterizing the Fock-space wave functions. Exploiting the direct connection between the local observables and Fock-space correlations, we show that the discontinuity in the length scale also implies discontinuous behavior of the local observables across the transition. A scaling theory based on these Fock-space correlations is constructed, which is closely connected to that for the inverse participation ratio. It yields a volume scale in the ergodic phase and a length scale in the MBL phase, whose critical properties suggest a Kosterlitz-Thouless--type scenario for the MBL transition, as is predicted by recent phenomenological theories. Finally, we also show how correlation functions on the Fock space reveal the inhomogeneities in eigenstate amplitudes on the Fock space in the MBL phase.We explore the Fock-space structure of eigenstates across the many-body localization (MBL) transition in a disordered, interacting quantum spin-$\frac{1}{2}$ chain. Eigenstate expectation values of spatially local observables, which distinguish an MBL phase from an ergodic one, can be represented in terms of eigenstate amplitudes on the Fock space. Motivated by this, we introduce and study spatial correlations on the Fock space. From these, a correlation length emerges, which is found to vary discontinuously across the MBL transition, and is intimately connected to the discontinuous jump in the multifractal exponents characterizing the Fock-space wave functions. Exploiting the direct connection between the local observables and Fock-space correlations, we show that the discontinuity in the length scale also implies discontinuous behavior of the local observables across the transition. A scaling theory based on these Fock-space correlations is constructed, which is closely connected to that for the inverse participation ratio. It yields a volume scale in the ergodic phase and a length scale in the MBL phase, whose critical properties suggest a Kosterlitz-Thouless--type scenario for the MBL transition, as is predicted by recent phenomenological theories. Finally, we also show how correlation functions on the Fock space reveal the inhomogeneities in eigenstate amplitudes on the Fock space in the MBL phase.

arXiv:2106.09036Physical review. B./Physical review. BNov 1, 202146 citations

Dynamics at the Many-Body Localization Transition

Léa Ferreira dos Santos, Jonathan Torres-Herrera

Bulletin of the American Physical SocietyJun 11, 20150 citations

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 explore the Fock-space structure of eigenstates across the many-body localization (MBL) transition in a disordered, interacting quantum spin-

\frac{1}{2}

chain. Eigenstate expectation values of spatially local observables, which distinguish an MBL phase from an ergodic one, can be represented in terms of eigenstate amplitudes on the Fock space. Motivated by this, we introduce and study spatial correlations on the Fock space. From these, a correlation length emerges, which is found to vary discontinuously across the MBL transition, and is intimately connected to the discontinuous jump in the multifractal exponents characterizing the Fock-space wave functions. Exploiting the direct connection between the local observables and Fock-space correlations, we show that the discontinuity in the length scale also implies discontinuous behavior of the local observables across the transition. A scaling theory based on these Fock-space correlations is constructed, which is closely connected to that for the inverse participation ratio. It yields a volume scale in the ergodic phase and a length scale in the MBL phase, whose critical properties suggest a Kosterlitz-Thouless--type scenario for the MBL transition, as is predicted by recent phenomenological theories. Finally, we also show how correlation functions on the Fock space reveal the inhomogeneities in eigenstate amplitudes on the Fock space in the MBL phase.