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# Non-Hermitian topology and exceptional-point geometries

## Abstract

Non-Hermitian theory is a theoretical framework used to describe open systems. It offers a powerful tool in the characterization of both the intrinsic degrees of freedom of a system and the interactions with the external environment. The non-Hermitian framework consists of mathematical structures that are fundamentally different from those of Hermitian theories. These structures not only underpin novel approaches for precisely tailoring non-Hermitian systems for applications but also give rise to topologies not found in Hermitian systems. In this Review, we provide an overview of non-Hermitian topology by establishing its relationship with the behaviours of complex eigenvalues and biorthogonal eigenvectors. Special attention is given to exceptional points — branch-point singularities on the complex eigenvalue manifolds that exhibit nontrivial topological properties. We also discuss recent developments in non-Hermitian band topology, such as the non-Hermitian skin effect and non-Hermitian topological classifications.

## Key points

• A non-Hermitian Hamiltonian that describes an open system generically has complex eigenvalues, which must be studied on the complex plane, which leads to the emergence of eigenvalue topology, or spectral topology. This additional ‘layer’ of topology is a unique feature for non-Hermitian systems.

• Spectral topology fundamentally affects the parallel-transport behaviours of eigenvectors of a non-Hermitian Hamiltonian.

• Exceptional points are branch singularities on non-Hermitian eigenvalue manifolds and exhibit exotic topological phenomena associated with the winding of eigenvalues and eigenvectors.

• The confluence of non-Hermiticity and band topology generates new phenomena such as the non-Hermitian skin effect, which is characterized by non-Bloch band theory and the re-establishment of bulk–boundary correspondence. It also ramifies the possible symmetry classes, thereby expanding the classifications of topological bands.

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## Relevant articles

• ### News and views (11&12)

AAPPS Bulletin Open Access 22 December 2022

• ### Transient non-Hermitian skin effect

Nature Communications Open Access 12 December 2022

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## Acknowledgements

K.D. and G.M. thanks Zhen Li, Wei Wang and Mengying Hu for their help with preparing the figures and Ruo-Yang Zhang for discussions. C.F. thanks Zhesen Yang and Kai Zhang for discussions. G.M. is supported by the National Natural Science Foundation of China (11922416) and the Hong Kong Research Grants Council (RFS2223-2S01, 12302420, 12300419, 12301822). K.D. is supported by the National Natural Science Foundation of China (12174072) and the Natural Science Foundation of Shanghai (21ZR1403700). C.F. is supported by the Ministry of Science and Technology of China (2016YFA0302400) and the Chinese Academy of Sciences (XDB33000000).

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## Glossary

Branch cut

A curve across which a multivalued complex function is discontinuous.

Defective matrix

A square matrix that does not have a complete basis of eigenvectors.

Jordan canonical form

A particular form of the upper triangular matrix, which is block-diagonalized filled with Jordan block matrices.

Fibre bundles

A mathematical structure composed of three topological spaces (total space, base space and fibres) and a projection map. The fibres adhere to the base space and their product space further gives the total space by the projection map, a continuous surjection satisfying a local triviality condition. In band theory, wavefunctions are considered as fibres and the momentum space is the base space, such that wavefunctions’ evolution in the momentum space is mathematically described by fibre bundles.

Antilinear symmetry

The symmetry defined by an antilinear operator, a map between two complex vector spaces that are additive but conjugate homogeneous.

Diabolic point

A parametric point at which a matrix has two degenerate eigenvalues and the eigenvectors still span the vector space.

Jordan block form

A matrix with zeroes everywhere except for the diagonal and with the superdiagonal being equal to 1.

Zak phase

The Berry phase defined for 1D Bloch states.

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Ding, K., Fang, C. & Ma, G. Non-Hermitian topology and exceptional-point geometries. Nat Rev Phys 4, 745–760 (2022). https://doi.org/10.1038/s42254-022-00516-5

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