Critical Properties of the Many-Body Localization Transition

Khemani, Vedika; Lim, S. P.; Sheng, D. N.; Huse, David A.

doi:10.1103/physrevx.7.021013

Cited by 260 publications

(311 citation statements)

References 37 publications

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“…. Within either phase, σ E is small, while near the transition, where we find both MBL-like and ETH-like states in the energy interval that is probed, σ E is enhanced [39][40][41]43]. (iv) The level spacings in the entanglement spectrum follow distinct statistical distributions in the ETH and MBL regimes.…”

Section: Many-body Localization In the Heisenberg Chain And Entamentioning

confidence: 98%

“…The entanglement entropy is expected to be larger in the volume-law entangled ETH regime than in the area law entangled MBL regime. Thus, in the transition region, where some entanglement spectra are MBL-like and some are ETH-like, the entanglement entropy will vary most strongly from one eigenstate to the next [39][40][41]43]. The maximum in the variance can thus be associated with the transition between the two regimes.…”

Section: B Single Disorder Realizationmentioning

confidence: 99%

“…The network correctly picks up the power-law structure of the entanglement spectrum in the MBL phase, as can be deduced from its nearly linear slope in this logarithmic plot. evolution of the local structure of the quantum states across the transition [40,[46][47][48]. As the transition is approached from the MBL side, ETH-like regions emerge in rare places where the random field variations are small.…”

Section: Spatial Structure Of Individual Eigenstatesmentioning

confidence: 99%

“…[8] has characterized the transition using ground state properties of the disordered Heisenberg chain via a neural network-based approach of classifying entanglement spectra). For a specific disorder configuration, this allows, for instance, to trace the evolution of individual ETH states deep in the MBL regime [40,[46][47][48]. We achieve this by considering the spectra from multiple real-space entanglement cuts as input for the (7) trained with cost function (8) on entanglement spectra obtained from an exact diagonalization on N = 16 sites.…”

Section: Introductionmentioning

confidence: 99%

“…They have been used to study the ETH-MBL transition in finite size numerical simulations, in particular for an extensive analysis of the Heisenberg model in a random field. These characterizing quantities include energy level statistics [30][31][32][33][34][35], level statistics [25,36] as well as density of states [37] analyses of the entanglement spectrum and studies of the distribution of the entanglement entropy over a region of energy eigenstates [18,[38][39][40][41][42][43]. Necessarily, these methods rely on a physical understanding of the nature of either regime or of the transition.…”

Section: Introductionmentioning

confidence: 99%

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Probing many-body localization with neural networks

2017

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We show that a simple artificial neural network trained on entanglement spectra of individual states of a many-body quantum system can be used to determine the transition between a many-body localized and a thermalizing regime. Specifically, we study the Heisenberg spin-1/2 chain in a random external field. We employ a multilayer perceptron with a single hidden layer, which is trained on labeled entanglement spectra pertaining to the fully localized and fully thermal regimes. We then apply this network to classify spectra belonging to states in the transition region. For training, we use a cost function that contains, in addition to the usual error and regularization parts, a term that favors a confident classification of the transition region states. The resulting phase diagram is in good agreement with the one obtained by more conventional methods and can be computed for small systems. In particular, the neural network outperforms conventional methods in classifying individual eigenstates pertaining to a single disorder realization. It allows us to map out the structure of these eigenstates across the transition with spatial resolution. Furthermore, we analyze the network operation using the dreaming technique to show that the neural network correctly learns by itself the power-law structure of the entanglement spectra in the many-body localized regime.

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Section: Many-body Localization In the Heisenberg Chain And Entamentioning

confidence: 98%

Section: B Single Disorder Realizationmentioning

confidence: 99%

Section: Spatial Structure Of Individual Eigenstatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Probing many-body localization with neural networks

2017

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The ergodic side of the many‐body localization transition

2017

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Recent studies point towards nontriviality of the ergodic phase in systems exhibiting many-body localization (MBL), which shows subexponential relaxation of local observables, subdiffusive transport and sublinear spreading of the entanglement entropy. Here we review the dynamical properties of this phase and the available numerically exact and approximate methods for its study. We discuss in which sense this phase could be considered ergodic and present possible phenomenological explanations of its dynamical properties. We close by analyzing to which extent the proposed explanations were verified by numerical studies and present the open questions in this field.

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One‐particle density matrix characterization of many‐body localization

et al. 2017

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We study interacting fermions in one dimension subject to random, uncorrelated onsite disorder, a paradigmatic model of many-body localization (MBL). This model realizes an interaction-driven quantum phase transition between an ergodic and a many-body localized phase, with the transition occurring in the many-body eigenstates. We propose a single-particle framework to characterize these phases by the eigenstates (the natural orbitals) and the eigenvalues (the occupation spectrum) of the one-particle density matrix (OPDM) in individual many-body eigenstates. As a main result, we find that the natural orbitals are localized in the MBL phase, but delocalized in the ergodic phase. This qualitative change in these single-particle states is a many-body effect, since without interactions the single-particle energy eigenstates are all localized. The occupation spectrum in the ergodic phase is thermal in agreement with the eigenstate thermalization hypothesis, while in the MBL phase the occupations preserve a discontinuity at an emergent Fermi edge. This suggests that the MBL eigenstates are weakly dressed Slater determinants, with the eigenstates of the underlying Anderson problem as reference states. We discuss the statistical properties of the natural orbitals and of the occupation spectrum in the two phases and as the transition is approached. Our results are consistent with the existing picture of emergent integrability and localized integrals of motion, or quasiparticles, in the MBL phase. We emphasize the close analogy of the MBL phase to a zero-temperature Fermi liquid: in the studied model, the MBL phase is adiabatically connected to the Anderson insulator and the occupation-spectrum discontinuity directly indicates the presence of quasiparticles localized in real space. Finally, we show that the same picture emerges for interacting fermions in the presence of an experimentally-relevant bichromatic lattice and thereby demonstrate that our findings are not limited to a specific model

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Critical Properties of the Many-Body Localization Transition

Cited by 260 publications

References 37 publications

Probing many-body localization with neural networks

Probing many-body localization with neural networks

The ergodic side of the many‐body localization transition

One‐particle density matrix characterization of many‐body localization

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