Signatures of the many-body localized regime in two dimensions


Lessons from Anderson localization highlight the importance of the dimensionality of real space for localization due to disorder. More recently, studies of many-body localization have focused on the phenomenon in one dimension using techniques of exact diagonalization and tensor networks. On the other hand, experiments in two dimensions have provided concrete results going beyond the previously numerically accessible limits while posing several challenging questions. We present the large-scale numerical examination of a disordered Bose–Hubbard model in two dimensions realized in cold atoms, which shows entanglement-based signatures of many-body localization. By generalizing a low-depth quantum circuit to two dimensions, we approximate eigenstates in the experimental parameter regimes for large systems, which is beyond the scope of exact diagonalization. A careful analysis of the eigenstate entanglement structure provides an indication of the putative phase transition marked by a peak in the fluctuations of entanglement entropy in a parameter range consistent with experiments.

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Fig. 1: Quantum circuits in the diagrammatic representation of tensor networks.
Fig. 2: Benchmark calculations for a 4 × 4 system.
Fig. 3: Entropy and on-site occupation distributions.
Fig. 4: Variance of the entropy as a function of Δ for 30 disorder realizations.
Fig. 5: Mobility edges obtained as explained in the main text.
Fig. 6: Dependence of correlation length ξ on disorder strength Δ for nmax = 1.

Data availability

The data that support the plots within this article and other findings of this study are available from the corresponding author upon reasonable request.


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The authors thank A. Chandran and C. Laumann for stimulating discussions and a careful reading of the manuscript, and A. Abadal, I. Bloch, J.-Y. Choi and C. Gross for detailed discussions related to the experiments. The authors also thank D. Huse, J. Imbrie, V. Oganesyan, W. De Roeck, A. Scardicchio, S. Sondhi and T. Spencer for fruitful discussions. S.H.S. and T.B.W. are both supported by TOPNES (EPSRC grant no. EP/I031014/1). S.H.S. is also supported by EPSRC grant no. EP/N01930X/1. T.B.W. acknowledges use of the University of Oxford Advanced Research Computing (ARC) facility in carrying out this work ( A.P. is supported by the Glasstone Fellowship and thanks the Aspen Center for Physics, which is supported by National Science Foundation grant no. PHY-1607611, and the Simons Center for Geometry and Physics, Stonybrook University, for their hospitality, where part of this work was performed. T.B.W. is grateful for support by the European Commission under the Marie Curie Programme. Statement of compliance with EPSRC policy framework on research data: this publication is theoretical work that does not require supporting research data.

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T.B.W. performed all the numerical simulations. The theoretical analysis and writing of the manuscript were jointly performed by A.P., T.B.W. and S.H.S.

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Correspondence to Arijeet Pal.

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Wahl, T.B., Pal, A. & Simon, S.H. Signatures of the many-body localized regime in two dimensions. Nature Phys 15, 164–169 (2019).

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