Many-body localization and thermalization: Insights from the entanglement spectrum

Abstract

We study the entanglement spectrum in the many body localizing and thermalizing phases of one and two dimensional Hamiltonian systems, and periodically driven 'Floquet' systems. We focus on the level statistics of the entanglement spectrum as obtained through numerical diagonalization, finding structure beyond that revealed by more limited measures such as entanglement entropy. In the thermalizing phase the entanglement spectrum obeys level statistics governed by an appropriate random matrix ensemble. For Hamiltonian systems this can be viewed as evidence in favor of a strong version of the eigenstate thermalization hypothesis (ETH). Similar results are also obtained for Floquet systems, where they constitute a result 'beyond ETH', and show that the corrections to ETH governing the Floquet entanglement spectrum have statistical properties governed by a random matrix ensemble. The particular random matrix ensemble governing the Floquet entanglement spectrum depends on the symmetries of the Floquet drive, and therefore can depend on the choice of origin of time. In the many body localized phase the entanglement spectrum is also found to show level repulsion, following a semi-Poisson distribution (in contrast to the energy spectrum, which follows a Poisson distribution). This semi-Poisson distribution is found to come mainly from states at high entanglement energies. The observed level repulsion only occurs for interacting localized phases. We also demonstrate that equivalent results can be obtained by calculating with a single typical eigenstate, or by averaging over a microcanonical energy window -a surprising result in the localized phase. This discovery of new structure in the pattern of entanglement of localized and thermalizing phases may open up new lines of attack on many body localization, thermalization, and the localization transition.

Probing many-body localization with neural networks

“…In this appendix, we report our observations when training instead with (a) the spectrum of the reduced density matrix ρ A and (b) the differences of the H e eigenvalues. The motivation for input data of type (a) is that diagonalizing ρ A , instead of its logarithm, requires less preprocessing, while the motivation for (b) is that the differences of the H e eigenvalues have been shown [25,36] to be statistically distributed in a unique fashion depending on whether the regime is MBL or ETH.…”

“…(iv) The level spacings in the entanglement spectrum follow distinct statistical distributions in the ETH and MBL regimes. A statistical analysis of the level distributions thus allows to identify the nature of individual eigenstates [25,36].…”

“…According to the ETH, local observables in a typical many-body eigenstate should take the values that pertain to the observables in a thermal ensemble, with the whole system acting as a heat bath for its subsystems in the thermodynamic limit. A well-studied class of systems that violate the ETH are those exhibiting many-body localization (MBL) [18][19][20][21][22][23][24][25], meaning that partial memory of initial conditions is preserved for infinite times. Due to this property, which is intimately related to the emergence of an extensive number of integrals of motion [23,[26][27][28], MBL systems have been envisioned as particularly robust quantum memories [29].…”

“…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.…”

Rare‐region effects and dynamics near the many‐body localization transition

“…The MBL phase resembles noninteracting Anderson insulators in some ways (e.g., spatial correlations decay exponentially, and eigenstates have area-law entanglement [35]). However, there are also important distinctions in entanglement dynamics [36,37], dephasing [38][39][40], linear [41] and nonlinear [42][43][44][45][46][47][48] response, and the entanglement spectrum [49,50]. These developments (reviewed in Refs.…”