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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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Fig. 1: Delocalization of a TZM by the NHSE.
Fig. 2: Experimental observation of the delocalization effect.
Fig. 3: Morphing of the TEM in a 2D stacked topological lattice.
Fig. 4: Morphing of a TCM by HO-NHSE.

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.

References

  1. Haldane, F. D. M. Nobel Lecture: Topological quantum matter. Rev. Mod. Phys. 89, 040502 (2017).

    MathSciNet  Article  ADS  Google Scholar 

  2. Kosterlitz, J. M. Nobel Lecture: Topological defects and phase transitions. Rev. Mod. Phys. 89, 040501 (2017).

    MathSciNet  Article  ADS  Google Scholar 

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

    CAS  PubMed  Article  ADS  Google Scholar 

  4. 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 

  5. 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).

    CAS  PubMed  Article  ADS  Google Scholar 

  6. 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 (2018).

    CAS  Article  ADS  Google Scholar 

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

    MathSciNet  CAS  Article  ADS  Google Scholar 

  8. Ma, G., Xiao, M. & Chan, C. T. Topological phases in acoustic and mechanical systems. Nat. Rev. Phys. 1, 281–294 (2019).

    Article  Google Scholar 

  9. Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nat. Phys. 7, 907–912 (2011).

    CAS  Article  Google Scholar 

  10. Xu, C., Chen, Z.-G., Zhang, G., Ma, G. & Wu, Y. Multi-dimensional wave steering with higher-order topological phononic crystal. Sci. Bull. 66, 1740–1745 (2021).

    CAS  Article  Google Scholar 

  11. Hu, B. et al. Non-Hermitian topological whispering gallery. Nature 597, 655–659 (2021).

    CAS  PubMed  Article  ADS  Google Scholar 

  12. St-Jean, P. et al. Lasing in topological edge states of a one-dimensional lattice. Nat. Photonics 11, 651–656 (2017).

    CAS  Article  ADS  Google Scholar 

  13. Harari, G. et al. Topological insulator laser: theory. Science 359, eaar4003 (2018).

    PubMed  Article  CAS  Google Scholar 

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

    PubMed  Article  CAS  Google Scholar 

  15. Zeng, Y. et al. Electrically pumped topological laser with valley edge modes. Nature 578, 246–250 (2020).

    CAS  PubMed  Article  ADS  Google Scholar 

  16. Barik, S. et al. A topological quantum optics interface. Science 359, 666–668 (2018).

    MathSciNet  CAS  PubMed  MATH  Article  ADS  Google Scholar 

  17. Blanco-Redondo, A., Bell, B., Oren, D., Eggleton, B. J. & Segev, M. Topological protection of biphoton states. Science 362, 568–571 (2018).

    MathSciNet  CAS  PubMed  MATH  Article  ADS  Google Scholar 

  18. Mittal, S., Goldschmidt, E. A. & Hafezi, M. A topological source of quantum light. Nature 561, 502–506 (2018).

    CAS  PubMed  Article  ADS  Google Scholar 

  19. Bender, C. M. Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007).

    MathSciNet  Article  ADS  Google Scholar 

  20. Özdemir, Ş. K., Rotter, S., Nori, F. & Yang, L. Parity–time symmetry and exceptional points in photonics. Nat. Mater. 18, 783–798 (2019).

    PubMed  Article  ADS  CAS  Google Scholar 

  21. Miri, M.-A. & Alù, A. Exceptional points in optics and photonics. Science 363, eaar7709 (2019).

    MathSciNet  CAS  PubMed  MATH  Article  Google Scholar 

  22. Wang, K. et al. Generating arbitrary topological windings of a non-Hermitian band. Science 371, 1240–1245 (2021).

    CAS  PubMed  Article  ADS  Google Scholar 

  23. Wang, K., Dutt, A., Wojcik, C. C. & Fan, S. Topological complex-energy braiding of non-Hermitian bands. Nature 598, 59–64 (2021).

    CAS  PubMed  Article  ADS  Google Scholar 

  24. Gao, T. et al. Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard. Nature 526, 554–558 (2015).

    CAS  PubMed  Article  ADS  Google Scholar 

  25. Zhong, Q., Khajavikhan, M., Christodoulides, D. N. & El-Ganainy, R. Winding around non-Hermitian singularities. Nat. Commun. 9, 4808 (2018).

    PubMed  Article  ADS  CAS  PubMed Central  Google Scholar 

  26. Tang, W. et al. Exceptional nexus with a hybrid topological invariant. Science 370, 1077–1080 (2020).

    MathSciNet  CAS  PubMed  MATH  Article  ADS  Google Scholar 

  27. Tang, W., Ding, K. & Ma, G. Direct measurement of topological properties of an exceptional parabola. Phys. Rev. Lett. 127, 034301 (2021).

    CAS  PubMed  Article  ADS  Google Scholar 

  28. Tang, W., Ding, K. & Ma, G. Experimental realization of non-Abelian permutations in a three-state non-Hermitian system. Preprint at https://doi.org/10.48550/arXiv.2112.00982 (2022).

  29. Ghatak, A., Brandenbourger, M., van Wezel, J. & Coulais, C. Observation of non-Hermitian topology and its bulk–edge correspondence in an active mechanical metamaterial. Proc. Natl Acad. Sci. USA 117, 29561–29568 (2020).

    CAS  PubMed  Article  ADS  PubMed Central  Google Scholar 

  30. Zhang, L. et al. Acoustic non-Hermitian skin effect from twisted winding topology. Nat. Commun. 12, 6297 (2021).

    CAS  PubMed  Article  ADS  PubMed Central  Google Scholar 

  31. Weidemann, S. et al. Topological funneling of light. Science 368, 311–314 (2020).

    MathSciNet  CAS  PubMed  MATH  Article  ADS  Google Scholar 

  32. Xiao, L. et al. Non-Hermitian bulk–boundary correspondence in quantum dynamics. Nat. Phys. 16, 761–766 (2020).

    CAS  Article  Google Scholar 

  33. Song, F., Yao, S. & Wang, Z. Non-Hermitian topological invariants in real space. Phys. Rev. Lett. 123, 246801 (2019).

    MathSciNet  CAS  PubMed  Article  ADS  Google Scholar 

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

    CAS  Article  Google Scholar 

  35. Gao, P., Willatzen, M. & Christensen, J. Anomalous topological edge states in non-Hermitian piezophononic media. Phys. Rev. Lett. 125, 206402 (2020).

    CAS  PubMed  Article  ADS  Google Scholar 

  36. Zhu, W., Teo, W. X., Li, L. & Gong, J. Delocalization of topological edge states. Phys. Rev. B 103, 195414 (2021).

    CAS  Article  ADS  Google Scholar 

  37. Brandenbourger, M., Locsin, X., Lerner, E. & Coulais, C. Non-reciprocal robotic metamaterials. Nat. Commun. 10, 4608 (2019).

    PubMed  Article  ADS  CAS  PubMed Central  Google Scholar 

  38. Hsu, C. W., Zhen, B., Stone, A. D., Joannopoulos, J. D. & Soljačić, M. Bound states in the continuum. Nat. Rev. Mater. 1, 16048 (2016).

    CAS  Article  ADS  Google Scholar 

  39. Okuma, N., Kawabata, K., Shiozaki, K. & Sato, M. Topological origin of non-Hermitian skin effects. Phys. Rev. Lett. 124, 086801 (2020).

    MathSciNet  CAS  PubMed  Article  ADS  Google Scholar 

  40. Zhang, K., Yang, Z. & Fang, C. Correspondence between winding numbers and skin modes in non-Hermitian systems. Phys. Rev. Lett. 125, 126402 (2020).

    MathSciNet  CAS  PubMed  Article  ADS  Google Scholar 

  41. Xiao, L. et al. Observation of non-Bloch parity-time symmetry and exceptional points. Phys. Rev. Lett. 126, 230402 (2021).

    CAS  PubMed  Article  ADS  Google Scholar 

  42. Sounas, D. L. & Alù, A. Non-reciprocal photonics based on time modulation. Nat. Photonics 11, 774–783 (2017).

    CAS  Article  ADS  Google Scholar 

  43. Nassar, H. et al. Nonreciprocity in acoustic and elastic materials. Nat. Rev. Mater. 5, 667–685 (2020).

    CAS  Article  ADS  Google Scholar 

  44. Painter, O. et al. Two-dimensional photonic band-gap defect mode laser. Science 284, 1819–1821 (1999).

    CAS  PubMed  Article  Google Scholar 

  45. Hirose, K. et al. Watt-class high-power, high-beam-quality photonic-crystal lasers. Nat. Photonics 8, 406–411 (2014).

    CAS  Article  ADS  Google Scholar 

  46. Zhang, W. et al. Low-threshold topological nanolasers based on the second-order corner state. Light: Sci. Appl. 9, 109 (2020).

    CAS  Article  ADS  Google Scholar 

  47. Teo, W. X., Zhu, W. & Gong, J. Tunable two-dimensional laser arrays with zero-phase locking. Phys. Rev. B 105, L201402 (2022).

    CAS  Article  ADS  Google Scholar 

  48. Zhao, H. et al. Topological hybrid silicon microlasers. Nat. Commun. 9, 981 (2018).

    PubMed  Article  ADS  CAS  PubMed Central  Google Scholar 

  49. Kim, H.-R. et al. Multipolar lasing modes from topological corner states. Nat. Commun. 11, 5758 (2020).

    CAS  PubMed  Article  ADS  PubMed Central  Google Scholar 

  50. Shao, Z.-K. et al. A high-performance topological bulk laser based on band-inversion-induced reflection. Nat. Nanotechnol. 15, 67–72 (2020).

    CAS  PubMed  Article  ADS  Google Scholar 

  51. Fan, S., Suh, W. & Joannopoulos, J. D. Temporal coupled-mode theory for the Fano resonance in optical resonators. J. Opt. Soc. Am. A 20, 569 (2003).

    Article  ADS  Google Scholar 

  52. Chong, Y. D., Ge, L., Cao, H. & Stone, A. D. Coherent perfect absorbers: time-reversed lasers. Phys. Rev. Lett. 105, 053901 (2010).

    CAS  PubMed  Article  ADS  Google Scholar 

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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 and Affiliations

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.

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The authors declare no competing interests.

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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.

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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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