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Observation of energy-resolved many-body localization

Nature Physics volume 17, pages 234–239 (2021)Cite this article

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Abstract

Many-body localization (MBL) describes a quantum phase where an isolated interacting system subject to sufficient disorder displays non-ergodic behaviour, evading thermal equilibrium that occurs under its own dynamics. Previously, the thermalization–MBL transition has been largely characterized with the growth of disorder. Here, we explore a new axis, reporting on an energy-resolved MBL transition using a 19-qubit programmable superconducting processor, which enables precise control and flexibility of both disorder strength and initial state preparation. We observe that the onset of localization occurs at different disorder strengths, with distinguishable energy scales, by measuring time-evolved observables and quantities related to many-body wave functions. Our results open avenues for the experimental exploration of many-body mobility edges in MBL systems, whose existence is widely debated due to the finiteness of the system size, and where exact simulations in classical computers become unfeasible.

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Fig. 1: Energy-resolved MBL and its measurement scheme.
Fig. 2: Quantum processor and experimental pulse sequence.
Fig. 3: MBL phase diagram—experiment versus simulation.
Fig. 4: Real-time dynamics of imbalance—experiment versus simulation.
Fig. 5: Wave-function-related measurement and minimal entanglement quantification.

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

Code availability

The codes used for the numerical simulation are available from the corresponding authors on reasonable request.

References

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

    ADS  Google Scholar 

  2. Altman, E. Many-body localization and quantum thermalization. Nat. Phys. 14, 979–983 (2018).

    Google Scholar 

  3. Abanin, D. A., Altman, E., Bloch, I. & Serbyn, M. Colloquium: many-body localization, thermalization, and entanglement. Rev. Mod. Phys. 91, 021001 (2019).

    ADS  MathSciNet  Google Scholar 

  4. Alet, F. & Laflorencie, N. Many-body localization: an introduction and selected topics. C. R. Phys. 19, 498–525 (2018).

    ADS  Google Scholar 

  5. Deutsch, J. M. Quantum statistical mechanics in a closed system. Phys. Rev. A 43, 2046–2049 (1991).

    ADS  Google Scholar 

  6. Srednicki, M. Chaos and quantum thermalization. Phys. Rev. E 50, 888–901 (1994).

    ADS  Google Scholar 

  7. Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854–858 (2008).

    ADS  Google Scholar 

  8. D’Alessio, L., Kafri, Y., Polkovnikov, A. & Rigol, M. From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics. Adv. Phys. 65, 239–362 (2016).

    ADS  Google Scholar 

  9. Chiaro, B. et al. Direct measurement of non-local interactions in the many-body localized phase. Preprint at https://arxiv.org/abs/1910.06024 (2019).

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

    ADS  Google Scholar 

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

    ADS  MATH  Google Scholar 

  12. Imbrie, J. Z. Diagonalization and many-body localization for a disordered quantum spin chain. Phys. Rev. Lett. 117, 027201 (2016).

    ADS  Google Scholar 

  13. Žnidarič, M., Prosen, T. & Prelovšek, P. Many-body localization in the Heisenberg XXZ magnet in a random field. Phys. Rev. B 77, 064426 (2008).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  15. Mondaini, R. & Rigol, M. Many-body localization and thermalization in disordered Hubbard chains. Phys. Rev. A 92, 041601(R) (2015).

    ADS  Google Scholar 

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

    Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  18. Bordia, P. et al. Coupling identical one-dimensional many-body localized systems. Phys. Rev. Lett. 116, 140401 (2016).

    ADS  Google Scholar 

  19. Lüschen, H. P. et al. Observation of slow dynamics near the many-body localization transition in one-dimensional quasiperiodic systems. Phys. Rev. Lett. 119, 260401 (2017).

    ADS  Google Scholar 

  20. Kohlert, T. et al. Observation of many-body localization in a one-dimensional system with a single-particle mobility edge. Phys. Rev. Lett. 122, 170403 (2019).

    ADS  Google Scholar 

  21. Rispoli, M. et al. Quantum critical behaviour at the many-body localization transition. Nature 573, 385–389 (2019).

    ADS  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  23. Smith, J. et al. Many-body localization in a quantum simulator with programmable random disorder. Nat. Phys. 12, 907–911 (2016).

    Google Scholar 

  24. Xu, K. et al. Emulating many-body localization with a superconducting quantum processor. Phys. Rev. Lett. 120, 050507 (2018).

    ADS  Google Scholar 

  25. Roushan, P. et al. Spectroscopic signatures of localization with interacting photons in superconducting qubits. Science 358, 1175–1179 (2017).

    ADS  MathSciNet  Google Scholar 

  26. Neill, C. et al. A blueprint for demonstrating quantum supremacy with superconducting qubits. Science 360, 195–199 (2018).

    ADS  MathSciNet  Google Scholar 

  27. Wei, X., Cheng, C., Xianlong, G. & Mondaini, R. Investigating many-body mobility edges in isolated quantum systems. Phys. Rev. B 99, 165137 (2019).

    ADS  Google Scholar 

  28. Song, C. et al. 10-qubit entanglement and parallel logic operations with a superconducting circuit. Phys. Rev. Lett. 119, 180511 (2017).

    ADS  Google Scholar 

  29. Song, C. et al. Generation of multicomponent atomic Schrödinger cat states of up to 20 qubits. Science 365, 574–577 (2019).

    ADS  MathSciNet  Google Scholar 

  30. Atas, Y. Y., Bogomolny, E., Giraud, O. & Roux, G. Distribution of the ratio of consecutive level spacings in random matrix ensembles. Phys. Rev. Lett. 110, 084101 (2013).

    ADS  Google Scholar 

  31. Griffiths, R. B. Nonanalytic behavior above the critical point in a random Ising ferromagnet. Phys. Rev. Lett. 23, 17–19 (1969).

    ADS  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  34. Luitz, D. J., Laflorencie, N. & Alet, F. Extended slow dynamical regime close to the many-body localization transition. Phys. Rev. B 93, 060201(R) (2016).

    ADS  Google Scholar 

  35. 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).

    ADS  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  37. Kim, H. & Huse, D. A. Ballistic spreading of entanglement in a diffusive nonintegrable system. Phys. Rev. Lett. 111, 127205 (2013).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  39. Lukin, A. et al. Probing entanglement in a many-body–localized system. Science 364, 256–260 (2019).

    ADS  Google Scholar 

  40. De Roeck, W., Huveneers, F., Müller, M. & Schiulaz, M. Absence of many-body mobility edges. Phys. Rev. B 93, 014203 (2016).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  42. Geraedts, S. D., Bhatt, R. N. & Nandkishore, R. Emergent local integrals of motion without a complete set of localized eigenstates. Phys. Rev. B 95, 064204 (2017).

    ADS  Google Scholar 

  43. Roy, S., Chalker, J. T. & Logan, D. E. Percolation in Fock space as a proxy for many-body localization. Phys. Rev. B 99, 104206 (2019).

    ADS  Google Scholar 

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Acknowledgements

We thank W. Liu, C. Song and K. Xu for technical support. R.M. thanks R. Fazio and M. Rigol for discussions. Devices were made at the Nanofabrication Facilities at Institute of Physics in Beijing and National Center for Nanoscience and Technology in Beijing. This research was supported by the National Natural Science Foundation of China (grant nos. 11674021, 11974039, 11851110757, 11725419 and 11904145), NSAF-U1930402, the National Key Research and Development Program of China (grant nos. 2016YFA0300600, 2017YFA0304300 and 2019YFA0308100), and the Zhejiang Province Key Research and Development Program (grant no. 2020C01019).

Author information

Author notes

  1. These authors contributed equally: Qiujiang Guo, Chen Cheng, Zheng-Hang Sun.

Authors and Affiliations

  1. Interdisciplinary Center for Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics, Zhejiang University, Hangzhou, China

    Qiujiang Guo, Zixuan Song, Hekang Li, Zhen Wang, Wenhui Ren, Hang Dong & H. Wang

  2. School of Physical Science and Technology, Lanzhou University, Lanzhou, China

    Chen Cheng

  3. Beijing Computational Science Research Center, Beijing, China

    Chen Cheng & Rubem Mondaini

  4. Institute of Physics, Chinese Academy of Sciences, Beijing, China

    Zheng-Hang Sun, Dongning Zheng & Heng Fan

  5. CAS Center for Excellence in Topological Quantum Computation, School of Physical Sciences, UCAS, Beijing, China

    Zheng-Hang Sun, Dongning Zheng & Heng Fan

  6. Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama, Japan

    Yu-Ran Zhang

Authors
  1. Qiujiang Guo
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Contributions

C.C. and R.M. proposed the idea. C.C. and Z.-H.S. performed the numerical simulation. Q.G. conducted the experiment. H.L. and D.Z. fabricated the device. R.M., C.C., Q.G., Z.-H.S., H.F. and H.W. co-wrote the manuscript, and all authors contributed to the experimental set-up, discussions of the results and development of the manuscript.

Corresponding authors

Correspondence to Rubem Mondaini, Heng Fan or H. Wang.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–10, discussion and Table 1, nine sections and references.

Source data

Source Data Fig. 3

Including both experimental and numerical values.

Source Data Fig. 4

Including both experimental and numerical values.

Source Data Fig. 5

Including both experimental and numerical values.

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Guo, Q., Cheng, C., Sun, ZH. et al. Observation of energy-resolved many-body localization. Nat. Phys. 17, 234–239 (2021). https://doi.org/10.1038/s41567-020-1035-1

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