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arXiv:1407.4251 · Physical Review Letters

Purification and Many-Body Localization in Cold Atomic Gases

F. Andraschko, Tilman Enss, Jesko Sirker

Published Nov 17, 2014• 98 citations
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Abstract

We propose to observe many-body localization in cold atomic gases by realizing a Bose-Hubbard chain with binary disorder and studying its nonequilibrium dynamics. In particular, we show that measuring the difference in occupation between even and odd sites, starting from a prepared density-wave state, provides clear signatures of localization. Furthermore, we confirm as hallmarks of the many-body localized phase a logarithmic increase of the entanglement entropy in time and Poissonian level statistics. Our numerical density-matrix renormalization group calculations for infinite system size are based on a purification approach; this allows us to perform the disorder average exactly, thus producing data without any statistical noise and with maximal simulation times of up to a factor 10 longer than in the clean case.

DOI
10.1103/physrevlett.113.217201

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We propose to observe many-body localization in cold atomic gases by realizing a Bose-Hubbard chain with binary disorder and studying its nonequilibrium dynamics. In particular, we show that measuring the difference in occupation between even and odd sites, starting from a prepared density-wave state, provides clear signatures of localization. Furthermore, we confirm as hallmarks of the many-body localized phase a logarithmic increase of the entanglement entropy in time and Poissonian level statistics. Our numerical density-matrix renormalization group calculations for infinite system size are based on a purification approach; this allows us to perform the disorder average exactly, thus producing data without any statistical noise and with maximal simulation times of up to a factor 10 longer than in the clean case.
Many-body localization for a system of bosons trapped in a one-dimensional lattice is discussed. Two models that may be realized for cold atoms in optical lattices are considered. The model with a random on-site potential is compared with previously introduced random interactions model. While the origin and character of the disorder in both systems is different they show interesting similar properties. In particular, many-body localization appears for a sufficiently large disorder as verified by a time evolution of initial density wave states as well as using statistical properties of energy levels for small system sizes. Starting with different initial states, we observe that the localization properties are energy-dependent which reveals an inverted many-body localization edge in both systems (that finding is also verified by statistical analysis of energy spectrum). Moreover, we consider computationally challenging regime of transition between many body localized and extended phases where we observe a characteristic algebraic decay of density correlations which may be attributed to subdiffusion (and Griffiths-like regions) in the studied systems. Ergodicity breaking in the disordered Bose–Hubbard models is compared with the slowing-down of the time evolution of the clean system at large interactions.
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.
We numerically investigate one-dimensional Bose-Hubbard chains with on-site disorder by means of exact diagonalization. A primary focus of our work is on characterizing Fock-space localization in this model from the single-particle perspective. For this purpose, we compute the one-particle density matrix (OPDM) in many-body eigenstates. We show that the natural orbitals (the eigenstates of the OPDM) are extended in the ergodic phase and real-space localized when one enters the many-body localization (MBL) phase. Furthermore, the distributions of occupations of the natural orbitals can be used as measures of Fock-space localization in the respective basis. Consistent with previous studies, we observe signatures of a transition from the ergodic to the MBL regime when increasing the disorder strength. We further demonstrate that Fock-space localization, albeit weaker, is also evidently present in the distribution of the physical densities in the MBL regime, for both soft- and hard-core bosons. Moreover, the full distribution of the densities of the physical particles provides a one-particle measure for the detection of the ergodic-MBL transition which could be directly accessed in experiments with ultracold gases.
We are able to detect clear signatures of dephasing---a distinct trait of many-body localization (MBL)---via the dynamics of two-site entanglement, quantified through the concurrence. Using the protocol implemented by M. Schreiber et al. [Science 349, 842 (2015)], we show that in the MBL phase the average two-site entanglement decays in time as a power law, while in the Anderson localized phase it tends to a plateau. The power-law exponent is not universal and displays a clear dependence on the interaction strength. This behavior is also qualitatively different from the ergodic phase, where the two-site entanglement decays exponentially. All the results are obtained by means of time-dependent density matrix renormalization-group simulations and further corroborated by analytical calculations on an effective model. Two-site entanglement has been measured in cold atoms: our analysis paves the way for the first direct experimental test of many-body dephasing in the MBL phase.
An interacting quantum system that is subject to disorder may cease to thermalize owing to localization of its constituents, thereby marking the breakdown of thermodynamics. The key to understanding this phenomenon lies in the system's entanglement, which is experimentally challenging to measure. We realize such a many-body-localized system in a disordered Bose-Hubbard chain and characterize its entanglement properties through particle fluctuations and correlations. We observe that the particles become localized, suppressing transport and preventing the thermalization of subsystems. Notably, we measure the development of nonlocal correlations, whose evolution is consistent with a logarithmic growth of entanglement entropy, the hallmark of many-body localization. Our work experimentally establishes many-body localization as a qualitatively distinct phenomenon from localization in noninteracting, disordered systems.
Many-body localized systems in which interactions and disorder come together defy the expectations of quantum statistical mechanics: In contrast to ergodic systems, they do not thermalize when undergoing nonequilibrium dynamics. What is less clear, however, is how topological features interplay with many-body localized phases as well as the nature of the transition between a topological and a trivial state within the latter. In this paper, we numerically address these questions using a combination of extensive tensor network calculations, specifically density matrix renormalization-group-X, as well as exact diagonalization, leading to a comprehensive characterization of Hamiltonian spectra and eigenstate entanglement properties.