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# Non-Hermitian morphing of topological modes

## Abstract

Topological modes (TMs) are usually localized at defects or boundaries of a much larger topological lattice1,2. Recent studies of non-Hermitian band theories unveiled the non-Hermitian skin effect (NHSE), by which the bulk states collapse to the boundary as skin modes3,4,5,6. Here we explore the NHSE to reshape the wavefunctions of TMs by delocalizing them from the boundary. At a critical non-Hermitian parameter, the in-gap TMs even become completely extended in the entire bulk lattice, forming an ‘extended mode outside of a continuum’. These extended modes are still protected by bulk-band topology, making them robust against local disorders. The morphing of TM wavefunction is experimentally realized in active mechanical lattices in both one-dimensional and two-dimensional topological lattices, as well as in a higher-order topological lattice. Furthermore, by the judicious engineering of the non-Hermiticity distribution, the TMs can deform into a diversity of shapes. Our findings not only broaden and deepen the current understanding of the TMs and the NHSE but also open new grounds for topological applications.

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## Data availability

The data represented in Figs. 2c–e and 4c 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

We thank C. T. Chan, Z.-Q. Zhang, K. Ding and R.-Y. Zhang for discussions, and Q. Wang for assisting with the experiments. This work was supported by the National Natural Science Foundation of China (grant no. 11922416) and the Hong Kong Research Grants Council (grant nos. 12302420, 12300419, and C6013-18G).

## Author information

Authors

### Contributions

W.W. developed the theory and performed numerical calculations. X.W. developed the experimental platform. W.W. and G.M. designed the experimental systems. W.W. and X.W. carried out the measurements. All authors analysed the results. W.W. and G.M. wrote the manuscript. G.M. led the research

### Corresponding author

Correspondence to Guancong Ma.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature thanks Corentin Coulais, Evelyn Tang and Zhong Wang for their contribution to the peer review of this work. Peer reviewer reports are available.

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## Extended data figures and tables

### Extended Data Fig. 1 Lasing effect of the extended TZM.

a, b, The real (a) and imaginary (b) part of the energy spectra of a nine-site NH-SSH chain (Supplementary Fig. 2a). We set $$\gamma =0.01$$, $$g=0.018$$. c, The evolution of the instantaneous total intensity $${I}_{{\rm{t}}{\rm{o}}{\rm{t}}}(t)$$ in the chain with $${\delta }_{x}={\delta }_{{xc}}$$ and $${\delta }_{x}=0$$. In the calculation, we set $${I}_{{\rm{s}}{\rm{a}}{\rm{t}}}=10$$. An initial field with random complex amplitudes of 0.01$$\left(m+{ni}\right)$$ with $$m,n\in \left(-\mathrm{1,1}\right)$$ at each site is applied. d, The steady-state intensity distribution. e, $${I}_{{\rm{t}}{\rm{o}}{\rm{t}}}^{{\rm{s}}}$$ as a function of $$g$$. f, $${I}_{{\rm{t}}{\rm{o}}{\rm{t}}}^{{\rm{s}}}$$ as a function of the lattice size (total site number $$N$$).

### Extended Data Fig. 2 Scattering coefficients of the TZM in a nine-site NH-SSH chain.

a, b, The scattering coefficients $$\left|{S}_{i1}\right|$$ with $${\delta }_{x}={\delta }_{{xc}}$$ (a) and $${\delta }_{x}=0$$ (b), pumped at site 1, as a function of $$\Delta f=f-{f}_{{TZM}}$$, where $${f}_{{\rm{T}}{\rm{Z}}{\rm{M}}}$$ is the TZM’s eigenfrequency and $$f$$ is the pumping frequency.

### Extended Data Fig. 3 Scattering coefficients of a non-Hermitian topological quadrupole insulator.

The scattering coefficients $$\left|{S}_{1}^{{ij}}\right|$$ [with $$\left(i,{j}\right)$$ indexing all the lattice sites] of a non-Hermitian topological quadrupole insulator, pumped at the left-most corner, as a function of $$\Delta f=f-{f}_{{\rm{T}}{\rm{C}}{\rm{M}}}$$, where $${f}_{{\rm{T}}{\rm{Z}}{\rm{M}}}$$ is the TCM’s eigenfrequency, and $$f$$ is the pumping frequency.

### Extended Data Fig. 4 A coherent beam splitter based on the delocalized TMs.

a, A coherent topological beam splitter. All red (blue) ports can send out coherent waves. b, The schematic model of a topological interface (similar to the one shown in Fig. 1d) stacked along $$y$$ direction. Here, the right part is also non-Hermitian with a biased hopping $${\delta }_{x{\rm{R}}}(y)$$ toward the right end. c, The real part of the response field (middle panel) with $${\delta }_{x{\rm{L}}}$$ and $${\delta }_{x{\rm{R}}}$$ taking the profiles in the left and right panels, respectively. Clearly, all the red (blue) sites are in-phase with each other. The increase in the amplitudes at the outputs is also due to the NHSE, which injects energy into the system. d, The real part of the response field with $${\delta }_{x{\rm{L}}}={\delta }_{x{\rm{R}}}=0$$.

## Supplementary information

### Supplementary Information

This Supplementary Information file includes 12 sections, 19 figures and 19 references.

### Supplementary Video 1

The non-Hermitian skin effect in a one-dimensional topological interface system.

### Supplementary Video 2

Morphing of topological edge modes in two-dimensional non-Hermitian topological lattices.

### Supplementary Video 3

Delocalization of a topological corner mode by the non-Hermitian skin effect.

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Wang, W., Wang, X. & Ma, G. Non-Hermitian morphing of topological modes. Nature 608, 50–55 (2022). https://doi.org/10.1038/s41586-022-04929-1

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• DOI: https://doi.org/10.1038/s41586-022-04929-1