Many-body localization due to random interactions

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Many-body localization due to random interactions

Piotr Sierant, Dominique Delande, and Jakub Zakrzewski

Phys. Rev. A 95, 021601(R) – Published 9 February 2017

Abstract

The possibility of observing many-body localization of ultracold atoms in a one-dimensional optical lattice is discussed for random interactions. In the noninteracting limit, such a system reduces to single-particle physics in the absence of disorder, i.e., to extended states. In effect, the observed localization is inherently due to interactions and is thus a genuine many-body effect. In the system studied, many-body localization manifests itself in a lack of thermalization visible in temporal propagation of a specially prepared initial state, in transport properties, in the logarithmic growth of entanglement entropy, and in statistical properties of energy levels.

Received 6 July 2016
Revised 28 November 2016

DOI:https://doi.org/10.1103/PhysRevA.95.021601

Physics Subject Headings (PhySH)

Cold gases in optical lattices Many-body localization

Atomic, Molecular & Optical

Authors & Affiliations

Piotr Sierant¹, Dominique Delande², and Jakub Zakrzewski^1,3,*

¹Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, ulica Łojasiewicza 11, 30-348 Kraków, Poland
²Laboratoire Kastler Brossel, UPMC, Sorbonne Universités, CNRS, ENS, PSL Research University, Collège de France, 4 Place Jussieu, 75005 Paris, France
³Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński, ulica Łojasiewicza 11, 30-348 Kraków, Poland

^*jakub.zakrzewski@uj.edu.pl

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Issue

Vol. 95, Iss. 2 — February 2017

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Images

Figure 1
Many-body localization for a 1D Bose-Hubbard model with random interaction strength. An initial state with a density wave profile is temporally propagated using a tDMRG algorithm. (a) The magnetization $M$ (initially unity) rapidly decays to a nonzero quasistationary value, a clear-cut proof of the absence of thermalization, i.e., of MBL. (b) The corresponding entropy of entanglement $S$ grows logarithmically with time $t$ after the initial transient. The magnetization increases with the disorder strength $U$ correlating with a slower increase of the entanglement entropy. Results are averaged over ten disorder realizations, for system size $L = 60$ and $N = 90$ bosons. The additional curve, labeled “ $U = 40$ Spin model,” is for the spin model discussed in the text. This simplified model exhibits Anderson localization and gives rise to a magnetization comparable to the one of the full model at large $U$ (see Fig. 2), but the entanglement entropy saturates at long time, emphasizing the difference between single-particle and many-body localization.
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Figure 2
Quasistationary magnetization $M$ versus $U$ for different system sizes $L$ indicated in the figure. Data are averaged over time for long times $t \in [10, 40]$ to remove the oscillatory behavior visible in Fig. 1. Data for small system sizes, obtained from exact diagonalization, are averaged over several hundred realizations of disorder. Results for $L = 60$ ( $L = 1000$ ) are obtained using the tDMRG algorithm and are averaged over 20 (4) realizations. For readability, the error bars (one standard deviation) are shown for a single $U$ value, but are very similar for all $U$ values. The quasicoalescence of the $L = 60$ and $L = 1000$ results indicates the absence of finite-size effects. The simplified spin model (dashed curve), discussed in the text, slightly overestimates the magnetization, but catches correctly the asymptotic behavior at large $U$ . The two-site model, with its analytic prediction $M \approx 1 - 8 π / U$ (see the text), reproduces quantitatively the results at large $U$ . The black bar in the range $U \in [25, 35]$ indicates the region where the transition to MBL occurs; see the discussion in the text.
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Figure 3
Average ratio of adjacent energy gaps $\bar{r}$ as a function of the interaction strength amplitude $U$ for different system sizes $L$ and number of bosons $N$ with a fixed density 3/2. Data are averaged over many disorder realizations. Only the energy range where eigenstates significantly overlap with the initial state is taken into account to facilitate a relevant comparison of time and spectral information. The dashed horizontal lines are the GOE and Poisson predictions.
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Figure 4
Level-spacing distributions for $N = 9$ particles on $L = 6$ sites with open boundary conditions after averaging over several realizations of disorder with strength $U$ . The $U = 15$ data are accurately described by the semi-Poisson distribution $P (s) \propto s^{β} exp [- (1 + β) s]$ with $β \approx 0.508$ , as expected in the critical region. At lower $U$ (7 and 10) there is a transition towards the GOE distribution, where the data are well reproduced by a $P (s) \propto s^{β} exp (- C_{2} s^{2 - γ})$ distribution proposed in [24]. The inset shows two limiting cases: The small (large) $U$ distribution is well reproduced by the GOE (Poisson) prediction.
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