Signatures of the many-body localized regime in two dimensions

Abstract

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.

References

  1. 1.

    Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958).

  2. 2.

    Basko, D. M., Aleiner, I. L. & Altshuler, B. L. Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states. Ann. Phys. 321, 1126–1205 (2006).

  3. 3.

    Gornyi, I., Mirlin, A. & Polyakov, D. Interacting electrons in disordered wires: Anderson localization and low-T transport. Phys. Rev. Lett. 95, 206603 (2005).

  4. 4.

    Pal, A. & Huse, D. A. Many-body localization phase transition. Phys. Rev. B 82, 174411 (2010).

  5. 5.

    Oganesyan, V. & Huse, D. A. Localization of interacting fermions at high temperature. Phys. Rev. B 75, 155111 (2007).

  6. 6.

    Nandkishore, R. & Huse, D. A. Many-body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015).

  7. 7.

    Bardarson, J. H., Pollmann, F. & Moore, J. E. Unbounded growth of entanglement in models of many-body localization. Phys. Rev. Lett. 109, 017202 (2012).

  8. 8.

    Huse, D. A., Nandkishore, R., Oganesyan, V., Pal, A. & Sondhi, S. L. Localization-protected quantum order. Phys. Rev. B 88, 014206 (2013).

  9. 9.

    Pekker, D., Refael, G., Altman, E., Demler, E. & Oganesyan, V. Hilbert-glass transition: new universality of temperature-tuned many-body dynamical quantum criticality. Phys. Rev. X 4, 011052 (2014).

  10. 10.

    Bahri, Y., Vosk, R., Altman, E. & Vishwanath, A. Localization and topology protected quantum coherence at the edge of hot matter. Nat. Commun. 6, 7341 (2015).

  11. 11.

    Chandran, A., Khemani, V., Laumann, C. R. & Sondhi, S. L. Many-body localization and symmetry-protected topological order. Phys. Rev. B 89, 144201 (2014).

  12. 12.

    Kjäll, J. A., Bardarson, J. H. & Pollmann, F. Many-body localization in a disordered quantum Ising chain. Phys. Rev. Lett. 113, 107204 (2014).

  13. 13.

    Vosk, R., Huse, D. A. & Altman, E. Theory of the many-body localization transition in one-dimensional systems. Phys. Rev. X 5, 031032 (2015).

  14. 14.

    Potter, A. C., Vasseur, R. & Parameswaran, S. A. Universal properties of many-body delocalization transitions. Phys. Rev. X 5, 031033 (2015).

  15. 15.

    Chandran, A., Laumann, C. R. & Oganesyan, V. Finite size scaling bounds on many-body localized phase transitions. Preprint at https://arxiv.org/abs/1509.04285 (2015).

  16. 16.

    Lim, S. P. & Sheng, D. N. Many-body localization and transition by density matrix renormalization group and exact diagonalization studies. Phys. Rev. B 94, 045111 (2016).

  17. 17.

    Khemani, V., Lim, S. P., Sheng, D. N. & Huse, D. A. Critical properties of the many-body localization transition. Phys. Rev. X 7, 021013 (2017).

  18. 18.

    Dumitrescu, P. T., Vasseur, R. & Potter, A. C. Scaling theory of entanglement at the many-body localization transition. Phys. Rev. Lett. 119, 110604 (2017).

  19. 19.

    Lee, P. A. & Ramakrishnan, T. V. Disordered electronic systems. Rev. Mod. Phys. 57, 287–337 (1985).

  20. 20.

    Imbrie, J. Z. On many-body localization for quantum spin chains. J. Stat. Phys. 163, 998–1048 (2016).

  21. 21.

    Schreiber, M. et al. Observation of many-body localization of interacting fermions in a quasirandom optical lattice. Science 349, 842–845 (2015).

  22. 22.

    Chandran, A., Pal, A., Laumann, C. R. & Scardicchio, A. Many-body localization beyond eigenstates in all dimensions. Phys. Rev. B 94, 144203 (2016).

  23. 23.

    De Roeck, W. & Huveneers, F. Stability and instability towards delocalization in many-body localization systems. Phys. Rev. B 95, 155129 (2017).

  24. 24.

    Roeck, W. D. & Imbrie, J. Z. Many-body localization: stability and instability. Phil. Trans. R. Soc. A 375, 20160422 (2017).

  25. 25.

    Agarwal, K. et al. Rare-region effects and dynamics near the many-body localization transition. Ann. Phys. 529, 1600326 (2017).

  26. 26.

    Luitz, D. J., Huveneers, F. & De Roeck, W. How a small quantum bath can thermalize long localized chains. Phys. Rev. Lett. 119, 150602 (2017).

  27. 27.

    Ponte, P., Laumann, C. R., Huse, D. A. & Chandran, A. Thermal inclusions: how one spin can destroy a many-body localized phase. Phil. Trans. R. Soc. A 375, 20160428 (2017).

  28. 28.

    Potirniche, I.-D., Banerjee, S. & Altman, E. On the stability of many-body localization in d > 1. Preprint at https://arxiv.org/abs/1805.01475 (2018).

  29. 29.

    Choi, J.-y et al. Exploring the many-body localization transition in two dimensions. Science 352, 1547–1552 (2016).

  30. 30.

    Bordia, P. et al. Probing slow relaxation and many-body localization in two-dimensional quasi-periodic systems. Phys. Rev. X 7, 041047 (2017).

  31. 31.

    Wahl, T. B., Pal, A. & Simon, S. H. Efficient representation of fully many-body localized systems using tensor networks. Phys. Rev. X 7, 021018 (2017).

  32. 32.

    Geraedts, S. D., Nandkishore, R. & Regnault, N. Manybody localization and thermalization: insights from the entanglement spectrum. Phys. Rev. B 93, 174202 (2016).

  33. 33.

    Inglis, S. & Pollet, L. Accessing many-body localized states through the generalized Gibbs ensemble. Phys. Rev. Lett. 117, 120402 (2016).

  34. 34.

    Thomson, S. J. & Schiró, M. Time evolution of many-body localized systems with the flow equation approach. Phys. Rev. B 97, 060201(R) (2018).

  35. 35.

    Luitz, D. J., Laorencie, N. & Alet, F. Many-body localization edge in the random-field Heisenberg chain. Phys. Rev. B 91, 081103 (2015).

  36. 36.

    Rubio-Abadal, A. et al. Probing many-body localization in the presence of a quantum bath. Preprint at https://arxiv.org/abs/1805.00056 (2018).

  37. 37.

    Serbyn, M., Papić, Z. & Abanin, D. A. Local conservation laws and the structure of the many-body localized states. Phys. Rev. Lett. 111, 127201 (2013).

  38. 38.

    Huse, D. A., Nandkishore, R. & Oganesyan, V. Phenomenology of fully many-body-localized systems. Phys. Rev. B 90, 174202 (2014).

  39. 39.

    Ros, V., Mueller, M. & Scardicchio, A. Integrals of motion in the many-body localized phase. Nucl. Phys. B 891, 420–465 (2015).

  40. 40.

    Imbrie, J. Z., Ros, V. & Scardicchio, A. Local integrals of motion in many-body localized systems. Ann. Phys. 529, 1600278 (2017).

  41. 41.

    Bauer, B. & Nayak, C. Area laws in a many-body localized state and its implications for topological order. J. Stat. Mech. 2013, P09005 (2013).

  42. 42.

    Pekker, D. & Clark, B. K. Encoding the structure of manybody localization with matrix product operators. Phys. Rev. B 95, 035116 (2017).

  43. 43.

    Chandran, A., Carrasquilla, J., Kim, I. H., Abanin, D. A. & Vidal, G. Spectral tensor networks for many-body localization. Phys. Rev. B 92, 024201 (2015).

  44. 44.

    Pollmann, F., Khemani, V., Cirac, J. I. & Sondhi, S. L. Efficient variational diagonalization of fully many-body localized Hamiltonians. Phys. Rev. B 94, 041116 (2016).

  45. 45.

    Friesdorf, M., Werner, A. H., Brown, W., Scholz, V. B. & Eisert, J. Many-body localization implies that eigenvectors are matrix-product states. Phys. Rev. Lett. 114, 170505 (2015).

  46. 46.

    Yu, X., Pekker, D. & Clark, B. K. Finding matrix product state representations of highly excited eigenstates of many-body localized hamiltonians. Phys. Rev. Lett. 118, 017201 (2017).

  47. 47.

    Khemani, V., Pollmann, F. & Sondhi, S. L. Obtaining highly excited eigenstates of many-body localized hamiltonians by the density matrix renormalization group approach. Phys. Rev. Lett. 116, 247204 (2016).

  48. 48.

    Wahl, T. B. Tensor networks demonstrate the robustness of localization and symmetry protected topological phases. Phys. Rev. B 98, 054204 (2018).

  49. 49.

    Kulshreshtha, A., Pal, A., Wahl, T. B. & Simon, S. H. Behavior of l-bits near the many-body localization transition. Preprint at https://arxiv.org/abs/1707.05362 (2018).

  50. 50.

    Yu, X., Luitz, D. J. & Clark, B. K. Bimodal entanglement entropy distribution in the many-body localization transition. Phys. Rev. B 94, 184202 (2016).

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Acknowledgements

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 (https://doi.org/10.5281/zenodo.22558). 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.

Correspondence to Arijeet Pal.

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Supplementary Text, Supplementary Figures 1–7

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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). https://doi.org/10.1038/s41567-018-0339-x

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