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Non-Hermitian topological whispering gallery


In 1878, Lord Rayleigh observed the highly celebrated phenomenon of sound waves that creep around the curved gallery of St Paul’s Cathedral in London1,2. These whispering-gallery waves scatter efficiently with little diffraction around an enclosure and have since found applications in ultrasonic fatigue and crack testing, and in the optical sensing of nanoparticles or molecules using silica microscale toroids. Recently, intense research efforts have focused on exploring non-Hermitian systems with cleverly matched gain and loss, facilitating unidirectional invisibility and exotic characteristics of exceptional points3,4. Likewise, the surge in physics using topological insulators comprising non-trivial symmetry-protected phases has laid the groundwork in reshaping highly unconventional avenues for robust and reflection-free guiding and steering of both sound and light5,6. Here we construct a topological gallery insulator using sonic crystals made of thermoplastic rods that are decorated with carbon nanotube films, which act as a sonic gain medium by virtue of electro-thermoacoustic coupling. By engineering specific non-Hermiticity textures to the activated rods, we are able to break the chiral symmetry of the whispering-gallery modes, which enables the out-coupling of topological ‘audio lasing’ modes with the desired handedness. We foresee that these findings will stimulate progress in non-destructive testing and acoustic sensing.

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Fig. 1: Complex band diagram of a sonic topological insulator with acoustic gain.
Fig. 2: Assembly and non-Hermitian phase engineering.
Fig. 3: Topological WG mode splitting.
Fig. 4: Routing of amplified topological WG modes.

Data availability

The data that support the findings of this study are available from the corresponding authors on reasonable request.

Code availability

The code used to calculate the results for this work is available from the corresponding authors on reasonable request.


  1. 1.

    Lord Rayleigh The Theory of Sound Vol. II, 1st edn (MacMillan, 1878).

  2. 2.

    Lord Rayleigh CXII. The problem of the whispering gallery. Phil. Mag. 20, 1001–1004 (1910).

    MATH  Article  Google Scholar 

  3. 3.

    Fleury, R., Sounas, D. L. & Alù, A. Parity–time symmetry in acoustics: theory, devices, and potential applications. IEEE J. Sel. Top. Quantum Electron. 22, 121–129 (2016).

    ADS  Article  Google Scholar 

  4. 4.

    Gupta, S. K. et al. Parity–time symmetry in non-Hermitian complex optical media. Adv. Mater. 32, 1903639 (2020).

    CAS  Google Scholar 

  5. 5.

    Zhang, X., Xiao, M., Cheng, Y., Lu, M.-H. & Christensen, J. Topological sound. Commun. Phys. 1, 97 (2018).

    Article  Google Scholar 

  6. 6.

    Khanikaev, A. B. et al. Photonic topological insulators. Nat. Mater. 12, 233–239 (2013).

    ADS  CAS  PubMed  Article  Google Scholar 

  7. 7.

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

    CAS  Google Scholar 

  8. 8.

    Foa Torres, L. E. F. Perspective on topological states of non-Hermitian lattices. J. Phys. Mater. 3, 014002 (2019).

    Article  CAS  Google Scholar 

  9. 9.

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

    ADS  PubMed  Article  CAS  Google Scholar 

  10. 10.

    Wang, M., Ye, L., Christensen, J. & Liu, Z. Valley physics in non-Hermitian artificial acoustic boron nitride. Phys. Rev. Lett. 120, 246601 (2018).

    ADS  CAS  PubMed  Article  Google Scholar 

  11. 11.

    Zhang, Z., López, M. R., Cheng, Y., Liu, X. & Christensen, J. Non-Hermitian sonic second-order topological insulator. Phys. Rev. Lett. 122, 195501 (2019).

    ADS  CAS  PubMed  Article  Google Scholar 

  12. 12.

    Zhao, H. et al. Non-Hermitian topological light steering. Science 365, 1163–1166 (2019).

    ADS  CAS  PubMed  Article  Google Scholar 

  13. 13.

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

    ADS  CAS  PubMed  Article  Google Scholar 

  14. 14.

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

    ADS  MathSciNet  CAS  PubMed  Article  Google Scholar 

  15. 15.

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

    ADS  MathSciNet  CAS  PubMed  Article  Google Scholar 

  16. 16.

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

    CAS  Article  Google Scholar 

  17. 17.

    Helbig, T. et al. Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits. Nat. Phys. 16, 747–750 (2020).

    CAS  Article  Google Scholar 

  18. 18.

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

    ADS  MathSciNet  CAS  PubMed  Article  Google Scholar 

  19. 19.

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

    ADS  CAS  PubMed  Article  Google Scholar 

  20. 20.

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

    ADS  CAS  Article  Google Scholar 

  21. 21.

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

    ADS  PubMed  PubMed Central  Article  CAS  Google Scholar 

  22. 22.

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

    PubMed  Article  CAS  Google Scholar 

  23. 23.

    Klembt, S. et al. Exciton-polariton topological insulator. Nature 562, 552–556 (2018).

    ADS  CAS  PubMed  Article  Google Scholar 

  24. 24.

    Bahari, B. et al. Nonreciprocal lasing in topological cavities of arbitrary geometries. Science 358, 636–640 (2017).

    ADS  CAS  PubMed  Article  Google Scholar 

  25. 25.

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

    ADS  CAS  PubMed  Article  Google Scholar 

  26. 26.

    Hutson, A. R., McFee, J. H. & White, D. L. Ultrasonic amplification in CdS. Phys. Rev. Lett. 7, 237–239 (1961).

    ADS  CAS  Article  Google Scholar 

  27. 27.

    Arnold, H. & Crandall, I. The thermophone as a precision source of sound. Phys. Rev. 10, 22–38 (1917).

    ADS  Article  Google Scholar 

  28. 28.

    Xiao, L. et al. Flexible, stretchable, transparent carbon nanotube thin film loudspeakers. Nano Lett. 8, 4539–4545 (2008).

    ADS  CAS  PubMed  Article  Google Scholar 

  29. 29.

    Aliev, A. E., Lima, M. D., Fang, S. & Baughman, R. H. Underwater sound generation using carbon nanotube projectors. Nano Lett. 10, 2374–2380 (2010).

    ADS  CAS  PubMed  Article  Google Scholar 

  30. 30.

    Ma, T. & Shvets, G. All-Si valley-Hall photonic topological insulator. New J. Phys. 18, 025012 (2016).

    ADS  Article  CAS  Google Scholar 

  31. 31.

    Ye, L. et al. Observation of acoustic valley vortex states and valley-chirality locked beam splitting. Phys. Rev. B 95, 174106 (2017).

    ADS  Article  Google Scholar 

  32. 32.

    Lu, J. et al. Observation of topological valley transport of sound in sonic crystals. Nat. Phys. 13, 369–374 (2017).

    CAS  Article  Google Scholar 

  33. 33.

    Ni, X., Gorlach, M. A., Alù, A. & Khanikaev, A. B. Topological edge states in acoustic kagome lattices. New J. Phys. 19, 055002 (2017).

    ADS  MathSciNet  Article  CAS  Google Scholar 

  34. 34.

    Zhang, Z. et al. Directional acoustic antennas based on valley-Hall topological insulators. Adv. Mater. 30, 1803229 (2018).

    Article  CAS  Google Scholar 

  35. 35.

    Mei, J., Wu, Y., Chan, C. T. & Zhang, Z.-Q. First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals. Phys. Rev. B 86, 035141 (2012).

    ADS  Article  CAS  Google Scholar 

  36. 36.

    Makwana, M. P. & Craster, R. V. Geometrically navigating topological plate modes around gentle and sharp bends. Phys. Rev. B 98, 184105 (2018).

    ADS  CAS  Article  Google Scholar 

  37. 37.

    Ochiai, T. Photonic realization of the (2+1)-dimensional parity anomaly. Phys. Rev. B 86, 075152 (2012).

    ADS  Article  CAS  Google Scholar 

  38. 38.

    Vesterinen, V., Niskanen, A. O., Hassel, J. & Helisto, P. Fundamental efficiency of nanothermophones: modeling and experiments. Nano Lett. 10, 5020–5024 (2010).

    ADS  CAS  PubMed  Article  Google Scholar 

  39. 39.

    Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958).

    ADS  CAS  Article  Google Scholar 

  40. 40.

    Stützer, S. et al. Photonic topological Anderson insulators. Nature 560, 461–465 (2018).

    ADS  PubMed  Article  CAS  Google Scholar 

  41. 41.

    Liu, G.-G. et al. Topological Anderson insulator in disordered photonic crystals. Phys. Rev. Lett. 125, 133603 (2020).

    ADS  CAS  PubMed  Article  Google Scholar 

  42. 42.

    Zangeneh-Nejad, F. & Fleury, R. Disorder-induced signal filtering with topological metamaterials. Adv. Mater. 32, 2001034 (2020).

    CAS  Article  Google Scholar 

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This work was supported by the National Basic Research Program of China (2017YFA0303702), NSFC (12074183, 11922407, 11904035, 11834008, 11874215 and 12104226) and the Fundamental Research Funds for the Central Universities (020414380181). Z.Z. acknowledges the support from the China National Postdoctoral Program for Innovative Talents (BX20200165) and the China Postdoctoral Science Foundation (2020M681541). L.Z. acknowledges support from the CONEX-Plus programme funded by Universidad Carlos III de Madrid and the European Union’s Horizon 2020 research and innovation programme under Marie Sklodowska-Curie grant agreement 801538. J.C. acknowledges support from the European Research Council (ERC) through the Starting Grant 714577 PHONOMETA and from the MINECO through a Ramón y Cajal grant (grant number RYC-2015-17156).

Author information




Y.C. initiated the project and conceived the idea. Y.C., X.L. and J.C. guided the research. B.H., Z.Z., H.Z., L.Z. and J.C. carried out the theoretical analyses. Z.Z. and L.Z. developed the Hamiltonian model. B.H., Z.Z. and H.Z. conducted finite-element-method simulations, designed the experimental setup and conducted the measurements. W.X., Z.Y., X.W. and J.X. assisted with sample fabrication. Z.Z., Y.C. and J.C. wrote the manuscript. All the authors contributed to the discussions of the results and the manuscript preparation.

Corresponding authors

Correspondence to Zhiwang Zhang, Ying Cheng, Xiaojun Liu or Johan Christensen.

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Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Chunyin Qiu, Farzad Zangeneh-Nejad 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.

Extended data figures and tables

Extended Data Fig. 1 Experimental setup for modulating the acoustic gain.

The CNT films wrapped around cylinders are connected with the electrical input and thus play the role of acoustic gain thanks to electro-thermoacoustic coupling.

Extended Data Fig. 2 Symmetry analysis and the dispersion relations calculated from the k·p method.

a, Schematic of the unit cell under C3v symmetry. b, c, Band diagrams of the C3v-symmetric sonic crystal with β = 0 (b) and with β = 0.05 (c). Coloured circles and solid curves epitomize the calculated results from the finite-element method and the k·p method, respectively. df, Same as ac, but for the sonic crystal preserved under C3 symmetry with the rotation angle θ = −π/6.

Extended Data Fig. 3 Comparison between amplification through the meta-fluid and thermal-acoustic gain.

a, b, Simulated scattering pressure fields under the inward radiation of coaxial cylindrical waves by the meta-fluid model (a) and the thermal-acoustic gain model (b). c, Enhancement of the scattering pressure fields calculated by the meta-fluid model (purple line) and the thermal-acoustic gain model (orange dots) with different β at f = 9.1 kHz. d, Corresponding frequency dependence at β = 0.05.

Extended Data Fig. 4 Characterizations of the CNT film.

a, Experimental setup for measuring the sound pressure and the surface vibration displacement. b, Acoustic pressure amplitude spectra measured near the CNT film (solid curve) and the loudspeaker (dashed curve). c, Vibration displacements spectra measured by laser vibrometry. The solid and dashed curves represent surface displacement on the CNT film and the traditional paper basin loudspeaker, respectively. d, Photograph of the measurement setup for electrical impedance analysis. Inset: enlarged view of the single sample. e, Experimentally measured amplitude and phase of the impedance curve. f, Experimentally measured directivity pattern.

Extended Data Fig. 5 Physical model of the topological WG insulator.

The proposed topological domain wall can be regarded as a triangular acoustic waveguide, and the phase along each edge is labelled as αj, with = 1, 2 and 3.

Extended Data Fig. 6 The influence of loss on the topological WG insulator.

ac, Spectrally resolved amplification factors through simulations considering the inherent loss with three different gain-phase textures: ϕ = 0 (a), ϕ = π (b) and ϕ = 2π (c). df, Pressure-field distributions and their chiralities of the three resonances at three different gain-phase textures corresponding to the frequencies f = 8,889 Hz (d), f0 = 8,999 Hz (e) and f+ = 9,103 Hz (f).

Extended Data Fig. 7 Valley chirality-selective sound emissions of the topological WG insulator.

a, Left: illustration of the designed device for the out-coupling of the chiral WG modes. Right: enlarged view of the router. The insets in the right panel show photographs of the cylinder trimers wrapped without or with CNT films. b, c, Momentum space analysis of the out-coupled K valley-projected topological WG mode of CW chirality at frequency f = 8,889 Hz (b) and K′ valley-WG mode of CCW chirality at frequency f+ = 9,106 Hz (c). The white solid hexagon represents the first Brillouin zone and the white dashed circle shows the equi-frequency contour in air. Ambient thermal colour represents the corresponding simulated sound energy fields.

Extended Data Fig. 8 Thermogram of the device and influence of temperature.

a, Photograph of the sample. b, c, Corresponding thermogram of the sample without (passive; b) and with (active; c) applied electric control. In ac, the left column shows the entire sample and the right column shows the enlarged view of the partial sample outlined by the dashed frame. d, Temperature evolutions of air near the CNT film (blue curve) and in the background (orange curve) with time during the measurements. The shaded area corresponds to the temperature range of 21–22 °C in the experiments. e, The frequency shifts of the peaks corresponding to f, f0 and f+ under the variation of the temperature. Lines and dots represent the theoretical and simulated results.

Extended Data Fig. 9 Other types of active topological gallery and robustness against disturbances.

a, Schematic of the WG with a snowflake-shaped domain wall. b, Energy distributions of the CCW WG mode with ϕ = 2π. c, Introducing phase disturbances. The solid curves in orange, light blue and purple represent the undistorted gain signals, and the dashed curves represent the distorted gain signals. d, Amplification spectrum including phase inhomogeneities with ϕ = 2π. e, f, Same as c, d, but amplitude disturbances are introduced instead of phase disturbances.

Extended Data Fig. 10 Robustness of the topological WG insulator against the geometric defects.

a, b, Numerical defect analysis comprising one defective unit cell at each corner or side of the structure. At a gain-phase texture of ϕ = 2π, we simulate the pressure fields of the system including defective units, that is, gainless, displaced or expanded cylinders (a) together with their corresponding spectral amplification factors (b). c, Schematic of the sample where the red and blue highlighted units label the perturbed rods located at the sides and corners, respectively. d, In the experiments, we chose three sets of perturbation displacements Δd = 0.04a−0.10a with ϕ = 2π, whose measured amplification factors include both corner (top) and side (bottom) defects.

Supplementary information

Video 1

Distributions of the phase emanating the edges. Time evolution for the simulated distributions of the phase emanating the edges at f = 8.80 kHz within the topological band gap. The left panel shows the result when three gain assisted rods emit sound with zero delay (ϕ = 0), while the right one shows the case when the phase increment acquires 2π/3 to assume a full gain cycle of ϕ = 2π.

Video 2

Topological whispering gallery modes splitting. Time evolution for the simulated pressure field distributions of the achiral/chiral topological whispering gallery modes. The middle panel shows the result at f0 with the phase texture ϕ = 0, behaving as the achiral topological whispering gallery mode. The left and right panels represent the cases with the phase texture ϕ = 2π at f- and f+, respectively, which behave as the CW and CCW chiral modes, respectively.

Video 3

| Chiral topological whispering gallery mode along a snowflake-shaped domain wall. Time evolution for the simulated pressure field distributions along a complicated snowflake-shaped domain wall, where the CCW chiral propagations of sound waves tightly confined along the domain wall can be clearly observed.

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

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