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arXiv:1606.01260 · Physical review. B./Physical review. B

Bimodal entanglement entropy distribution in the many-body localization transition

Xiongjie Yu, David J. Luitz, Bryan K. Clark

Published Nov 15, 2016• 86 citations
arXivHTMLPDF

Abstract

We introduce the cut-averaged entanglement entropy in disordered periodic spin chains and prove it to be a concave function of subsystem size for individual eigenstates. This allows us to identify the entanglement scaling as a function of subsystem size for individual states in inhomogeneous systems. Using this quantity, we probe the critical region between the many-body localized (MBL) and ergodic phases in finite systems. In the middle of the spectrum, we show evidence for bimodality of the entanglement distribution in the MBL critical region, finding both volume law and area law eigenstates over disorder realizations as well as within single disorder realizations. The disorder-averaged entanglement entropy in this region then scales as a volume law with a coefficient below its thermal value. We discover in the critical region, as we approach the thermodynamic limit, that the cut-averaged entanglement entropy density falls on a one-parameter family of curves. Finally, we also show that without averaging over cuts the slope of the entanglement entropy vs subsystem size can be negative at intermediate and strong disorder, caused by rare localized regions in the system.

DOI
10.1103/physrevb.94.184202

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Abstract We study the details of the distribution of the entanglement spectrum (eigenvalues of the reduced density matrix) of a disordered spin chain exhibiting a many-body localization (MBL) transition. In the thermalizing region we identify the evolution under increasing system size of the eigenvalue distribution function, whose thermodynamic limit is close to (but possibly different from) the Marchenko–Pastur distribution. From the analysis we extract a correlation length L s ( h ) determining the minimum system size to enter the asymptotic region. We find that L s ( h ) diverges at the MBL transition. We discuss the nature of the subleading corrections to the entanglement spectrum distribution and to the entanglement entropy.
Many-body localization of a disorder interacting boson system in one dimension is studied numerically at the filling factor being one-half, in terms of level statistics, local compressibility, correlation function and entanglement entropies. The von Neumann entanglement entropy is decoupled into a particle number entropy and a configuration entropy. An anomalous volume-law behavior is found for the configuration entanglement entropy to confirm a recent experimental observation [A. Lukin, M. Rispoli, R. Schittko, et al., Science 364, 256 (2019)] for sufficient strong disorder, while the particle number entropy fulfills an area-law corresponding to the total entropy for disordered spin chain. The localization length are extracted from a two-body correlation function for many-body localization states and corresponding time-evolutions states as well. A phase diagrams is established with consisting of an ergodic thermalized region and a many-body-localization region in a parameter space of the disorder strength and the energy density. Two regions are separated by a many-body mobility edge deducted from the standard deviation of the particle-number entanglement entropy, which appears consistent with that based on the localization length. Slow dynamics characterized by a logarithmic time-dependence is explicitly shown for both the particle number entropy and the configuration entropy in an intermediate regime of their time-evolutions, which does not show up in the Anderson localization case, i.e. non-interacting disorder systems.