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# 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.

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## 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.

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## 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

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.

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

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

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