Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Non-Hermitian bulk–boundary correspondence in quantum dynamics

Abstract

Bulk–boundary correspondence, a guiding principle in topological matter, relates robust edge states to bulk topological invariants. Its validity, however, has so far been established only in closed systems. Recent theoretical studies indicate that this principle requires fundamental revisions for a wide range of open systems with effective non-Hermitian Hamiltonians. Therein, the intriguing localization of nominal bulk states at boundaries, known as the non-Hermitian skin effect, suggests a non-Bloch band theory in which non-Bloch topological invariants are defined in generalized Brillouin zones, leading to a general bulk–boundary correspondence beyond the conventional framework. Here, we experimentally observe this fundamental non-Hermitian bulk–boundary correspondence in discrete-time non-unitary quantum-walk dynamics of single photons. We demonstrate pronounced photon localizations near boundaries even in the absence of topological edge states, thus confirming the non-Hermitian skin effect. Facilitated by our experimental scheme of edge-state reconstruction, we directly measure topological edge states, which are in excellent agreement with the non-Bloch topological invariants. Our work unequivocally establishes the non-Hermitian bulk–boundary correspondence as a general principle underlying non-Hermitian topological systems and paves the way for a complete understanding of topological matter in open systems.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Quantum walks with the non-Hermitian skin effect.
Fig. 2: Experimental implementation.
Fig. 3: Experimental demonstration of the non-Hermitian skin effect.
Fig. 4: Non-Bloch bulk–boundary correspondence for edge states with ε = π.
Fig. 5: Evidence for non-Bloch bulk–boundary correspondence.

Data availability

The data represented in Figs. 35 are available as Source Data. All other data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

References

  1. 1.

    Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    ADS  Google Scholar 

  2. 2.

    Qi, X. L. & Zhang, S. C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    ADS  Google Scholar 

  3. 3.

    Lee, T. E. Anomalous edge state in a non-Hermitian lattice. Phys. Rev. Lett. 116, 133903 (2016).

    ADS  Google Scholar 

  4. 4.

    Yao, S. & Wang, Z. Edge states and topological invariants of non-Hermitian systems. Phys. Rev. Lett. 121, 086803 (2018).

    ADS  Google Scholar 

  5. 5.

    Yao, S., Song, F. & Wang, Z. Non-Hermitian Chern bands. Phys. Rev. Lett. 121, 136802 (2018).

    ADS  Google Scholar 

  6. 6.

    Kunst, F. K., Edvardsson, E., Budich, J. C. & Bergholtz, E. J. Biorthogonal bulk–boundary correspondence in non-Hermitian systems. Phys. Rev. Lett. 121, 026808 (2018).

    ADS  Google Scholar 

  7. 7.

    Yokomizo, K. & Murakami, S. Non-Bloch band theory of non-Hermitian systems. Phys. Rev. Lett. 123, 066404 (2019).

    ADS  MathSciNet  Google Scholar 

  8. 8.

    Alvarez, V. M., Vargas, J. B., Berdakin, M. & Torres, L. F. Topological states of non-Hermitian systems. Eur. Phys. J. Spec. Top. 227, 1295–1308 (2018).

    Google Scholar 

  9. 9.

    Lee, C. H. & Thomale, R. Anatomy of skin modes and topology in non-Hermitian systems. Phys. Rev. B 99, 201103(R) (2019).

    ADS  Google Scholar 

  10. 10.

    Ghatak, A. & Das, T. New topological invariants in non-Hermitian systems. J. Phys. Condens. Matter 31, 263001 (2019).

    ADS  Google Scholar 

  11. 11.

    Borgnia, D. S., Kruchkov, A. J. & Slager, R.-J. Non-Hermitian boundary modes. Phys. Rev. Lett. 124, 056802 (2020).

    ADS  Google Scholar 

  12. 12.

    Martinez Alvarez, V. M., Barrios Vargas, J. E. & Foa Torres, L. E. F. Non-Hermitian robust edge states in one dimension: anomalous localization and eigenspace condensation at exceptional points. Phys. Rev. B 97, 121401(R) (2018).

    ADS  Google Scholar 

  13. 13.

    Longhi, S. Probing non-Hermitian skin effect and non-Bloch phase transitions. Phys. Rev. Res. 1, 023013 (2019).

    Google Scholar 

  14. 14.

    McDonald, A., Pereg-Barnea, T. & Clerk, A. A. Phase-dependent chiral transport and effective non-Hermitian dynamics in a bosonic Kitaev–Majorana chain. Phys. Rev. X 8, 041031 (2018).

    Google Scholar 

  15. 15.

    Poli, C., Bellec, M., Kuhl, U., Mortessagne, F. & Schomerus, H. Selective enhancement of topologically induced interface states in a dielectric resonator chain. Nat. Commun. 6, 6710 (2015).

    ADS  Google Scholar 

  16. 16.

    Zeuner, J. M. et al. Observation of a topological transition in the bulk of a non-Hermitian system. Phys. Rev. Lett. 115, 040402 (2015).

    ADS  Google Scholar 

  17. 17.

    Xiao, L. et al. Observation of topological edge states in parity–time-symmetric quantum walks. Nat. Phys. 13, 1117–1123 (2017).

    Google Scholar 

  18. 18.

    Zhan, X. et al. Detecting topological invariants in nonunitary discrete-time quantum walks. Phys. Rev. Lett. 119, 130501 (2017).

    ADS  MathSciNet  Google Scholar 

  19. 19.

    Xiao, L. et al. Higher winding number in a nonunitary photonic quantum walk. Phys. Rev. A 98, 063847 (2018).

    ADS  Google Scholar 

  20. 20.

    Wang, K. et al. Simulating dynamic quantum phase transitions in photonic quantum walks. Phys. Rev. Lett. 122, 020501 (2019).

    ADS  Google Scholar 

  21. 21.

    Wang, K. et al. Observation of emergent momentum–time skyrmions in parity–time-symmetric non-unitary quench dynamics. Nat. Commun. 10, 2293 (2019).

    ADS  Google Scholar 

  22. 22.

    Xiao, L. et al. Observation of critical phenomena in parity–time-symmetric quantum dynamics. Phys. Rev. Lett. 123, 230401 (2019).

    ADS  Google Scholar 

  23. 23.

    Weimann, S. et al. Topologically protected bound states in photonic parity–time-symmetric crystals. Nat. Mater. 16, 433–438 (2017).

    ADS  Google Scholar 

  24. 24.

    Parto, M. et al. Edge-mode lasing in 1D topological active arrays. Phys. Rev. Lett. 120, 113901 (2018).

    ADS  Google Scholar 

  25. 25.

    Zhou, H. et al. Observation of bulk Fermi arc and polarization half charge from paired exceptional points. Science 359, 1009–1012 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  26. 26.

    Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).

    ADS  MathSciNet  Google Scholar 

  27. 27.

    Bandres, M. A. et al. Topological insulator laser: experiments. Science 359, eaar4005 (2018).

    Google Scholar 

  28. 28.

    Zhu, W. et al. Simultaneous observation of a topological edge state and exceptional point in an open and non-Hermitian acoustic system. Phys. Rev. Lett. 121, 124501 (2018).

    ADS  Google Scholar 

  29. 29.

    Wu, Y. et al. Observation of parity–time symmetry breaking in a single-spin system. Science 364, 878–880 (2019).

    ADS  MathSciNet  MATH  Google Scholar 

  30. 30.

    Li, J. et al. Observation of parity–time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms. Nat. Commun. 10, 855 (2019).

    ADS  Google Scholar 

  31. 31.

    Shen, H., Zhen, B. & Fu, L. Topological band theory for non-Hermitian Hamiltonians. Phys. Rev. Lett. 120, 146402 (2018).

    ADS  MathSciNet  Google Scholar 

  32. 32.

    Leykam, D., Bliokh, K. Y., Huang, C., Chong, Y. D. & Nori, F. Edge modes, degeneracies, and topological numbers in non-Hermitian systems. Phys. Rev. Lett. 118, 040401 (2017).

    ADS  MathSciNet  Google Scholar 

  33. 33.

    Gong, Z. et al. Topological phases of non-Hermitian systems. Phys. Rev. X 8, 031079 (2018).

    Google Scholar 

  34. 34.

    El-Ganainy, R. et al. Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11–19 (2018).

    Google Scholar 

  35. 35.

    Kawabata, K., Shiozaki, K., Ueda, M. & Sato, M. Symmetry and topology in non-Hermitian physics. Phys. Rev. X 9, 041015 (2019).

    Google Scholar 

  36. 36.

    Zhou, H. & Lee, J. Y. Periodic table for topological bands with non-Hermitian symmetries. Phys. Rev. B 99, 235112 (2019).

    ADS  Google Scholar 

  37. 37.

    Rudner, M. S. & Levitov, L. S. Topological transition in a non-Hermitian quantum walk. Phys. Rev. Lett. 102, 065703 (2009).

    ADS  Google Scholar 

  38. 38.

    Esaki, K., Sato, M., Hasebe, K. & Kohmoto, M. Edge states and topological phases in non-Hermitian systems. Phys. Rev. B 84, 205128 (2011).

    ADS  Google Scholar 

  39. 39.

    Zhu, B., Lü, R. & Chen, S. PT symmetry in the non-Hermitian Su–Schrieffer–Heeger model with complex boundary potentials. Phys. Rev. A 89, 062102 (2014).

    ADS  Google Scholar 

  40. 40.

    Rudner, M. S., Lindner, N. H., Berg, E. & Levin, M. Anomalous edge states and the bulk–edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X 3, 031005 (2013).

    Google Scholar 

  41. 41.

    Asbóth, J. K. & Obuse, H. Bulk–boundary correspondence for chiral symmetric quantum walks. Phys. Rev. B 88, 121406(R) (2013).

    ADS  Google Scholar 

  42. 42.

    Deng, T. & Yi, W. Non-Bloch topological invariants in a non-Hermitian domain-wall system. Phys. Rev. B 100, 035102 (2019).

    ADS  Google Scholar 

  43. 43.

    Yao, S., Yan, Z. & Wang, Z. Topological invariants of Floquet systems: general formulation, special properties, and Floquet topological defects. Phys. Rev. B 96, 195303 (2017).

    ADS  Google Scholar 

  44. 44.

    Fruchart, M. Complex classes of periodically driven topological lattice systems. Phys. Rev. B 93, 115429 (2016).

    ADS  Google Scholar 

  45. 45.

    Longhi, S. Non-Bloch PT symmetry breaking in non-Hermitian photonics quantum walks. Opt. Lett. 44, 5804–5807 (2019).

    ADS  Google Scholar 

  46. 46.

    Helbig, T. et al. Observation of bulk boundary correspondence breakdown in topolectrical circuits. Preprint at https://arxiv.org/abs/1907.11562 (2019).

  47. 47.

    Ghatak, A., Brandenbourger, M., van Wezel, J. & Coulais, C. Observation of non-Hermitian topology and its bulk–edge correspondence. Preprint at https://arxiv.org/abs/1907.11619 (2019).

  48. 48.

    Brody, D. C. Biorthogonal quantum mechanics. J. Phys. A Math. Theor. 47, 035305 (2014).

    ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work has been supported by the National Natural Science Foundation of China (grant nos. 11674056, 11674189, U1930402 and 11974331) and a start-up fund from the Beijing Computational Science Research Center. W.Y. acknowledges support from the National Key Research and Development Program of China (grant nos. 2016YFA0301700 and 2017YFA0304100).

Author information

Affiliations

Authors

Contributions

L.X. performed the experiments, with contributions from K.W. and G.Z.; W.Y., T.D. and Z.W. developed the theoretical aspects and performed the theoretical analysis; P.X. designed the experiments and analysed the results; P.X., W.Y. and Z.W. wrote the paper, with input from all authors.

Corresponding authors

Correspondence to Zhong Wang or Wei Yi or Peng Xue.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Andrea Alu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Information and Figs. 1–5.

Source data

Source Data Fig. 3

Source data for Figure 3.

Source Data Fig. 4

Source data for Figure 4.

Source Data Fig. 5

Source data for Figure 5.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Xiao, L., Deng, T., Wang, K. et al. Non-Hermitian bulk–boundary correspondence in quantum dynamics. Nat. Phys. 16, 761–766 (2020). https://doi.org/10.1038/s41567-020-0836-6

Download citation

Further reading

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing