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Topology and broken Hermiticity


Topology and symmetry have emerged as compelling guiding principles to predict and harness the propagation of waves in natural and artificial materials. Be it for quantum particles (such as electrons) or classical waves (such as light, sound or mechanical motion), these concepts have so far been mostly developed in idealized scenarios, in which the wave amplitude is neither attenuated nor amplified, and time evolution is unitary. In recent years, however, there has been a considerable push to explore the consequences of topology and symmetries in non-conservative, non-equilibrium or non-Hermitian systems. A plethora of driven artificial materials has been reported, blurring the lines between a wide variety of fields in physics and engineering, including condensed matter, photonics, phononics, optomechanics, as well as electromagnetic and mechanical metamaterials. Here we discuss the latest advances, emerging opportunities and open challenges for combining these exciting research endeavours into the new pluridisciplinary field of non-Hermitian topological systems.

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Fig. 1: Classification of topological systems and examples of strategies to engineer symmetries and/or break Hermiticity.
Fig. 2: Emergence of topology and broken symmetry in non-Hermitian materials.


  1. 1.

    Raman, A. & Fan, S. Photonic band structure of dispersive metamaterials formulated as a Hermitian eigenvalue problem. Phys. Rev. Lett. 104, 087401 (2010).

    ADS  Article  Google Scholar 

  2. 2.

    El-Ganainy, R. et al. Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11–19 (2018).

    Article  Google Scholar 

  3. 3.

    Bender, C. M. & Boettcher, S. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Feng, L. et al. Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical †frequencies. Nat. Mater. 12, 108–113 (2013).

    ADS  Article  Google Scholar 

  5. 5.

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

    ADS  Article  Google Scholar 

  6. 6.

    Doppler, J. et al. Dynamically encircling an exceptional point for asymmetric mode switching. Nature 537, 76–79 (2016).

    ADS  Article  Google Scholar 

  7. 7.

    Yoon, J. W. et al. Time-asymmetric loop around an exceptional point over the full optical communications band. Nature 562, 86–90 (2018).

    ADS  Article  Google Scholar 

  8. 8.

    Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    ADS  Google Scholar 

  9. 9.

    Lu, L., Joannopoulos, J. D. & Soljačić, M. Topological photonics. Nat. Photon. 8, 821–829 (2014).

    ADS  Article  Google Scholar 

  10. 10.

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

    ADS  MathSciNet  Article  Google Scholar 

  11. 11.

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

    Article  Google Scholar 

  12. 12.

    Huber, S. D. Topological mechanics. Nat. Phys. 12, 621–623 (2016).

    Article  Google Scholar 

  13. 13.

    Bandres, M. A., Rechtsman, M. C. & Segev, M. Topological photonic quasicrystals: fractal topological spectrum and protected transport. Phys. Rev. X 6, 011016 (2016).

    Google Scholar 

  14. 14.

    Fremling, M., van Hooft, M., Smith, C. M. & Fritz, L. Existence of robust edge currents in Sierpiński fractals. Phys. Rev. Res. 2, 013044 (2020).

    Article  Google Scholar 

  15. 15.

    Souslov, A., van Zuiden, B. C., Bartolo, D. & Vitelli, V. Topological sound in active-liquid metamaterials. Nat. Phys. 13, 1091–1094 (2017).

    Article  Google Scholar 

  16. 16.

    Mitchell, N. P., Nash, L. M., Hexner, D., Turner, A. M. & Irvine, W. T. M. Amorphous topological insulators constructed from random point sets. Nat. Phys. 14, 380–385 (2018).

    Article  Google Scholar 

  17. 17.

    Delplace, P., Marston, J. B. & Venaille, A. Topological origin of equatorial waves. Science 358, 1075–1077 (2017).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Zangeneh-Nejad, F. & Fleury, R. Topological fano resonances. Phys. Rev. Lett. 122, 014301 (2019).

    ADS  Article  Google Scholar 

  19. 19.

    Zangeneh-Nejad, F. & Fleury, R. Topological analog signal processing. Nat. Commun. 10, 2058 (2019).

    ADS  Article  Google Scholar 

  20. 20.

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

    ADS  Article  Google Scholar 

  21. 21.

    Yin, X., Jin, J., Soljačić, M., Peng, C. & Zhen, B. Observation of topologically enabled unidirectional guided resonances. Nature 580, 467–471 (2020).

    ADS  Article  Google Scholar 

  22. 22.

    Lohse, M., Schweizer, C., Zilberberg, O., Aidelsburger, M. & Bloch, I. A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice. Nat. Phys. 12, 350–354 (2016).

    Article  Google Scholar 

  23. 23.

    Aidelsburger, M. et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nat. Phys. 11, 162–166 (2015).

    Article  Google Scholar 

  24. 24.

    Fleury, R., Khanikaev, A. & Alù, A. Floquet topological insulators for sound. Nat. Commun. 7, 11744 (2016).

    ADS  Article  Google Scholar 

  25. 25.

    Salerno, G., Ozawa, T., Price, H. M. & Carusotto, I. Floquet topological system based on frequency-modulated classical coupled harmonic oscillators. Phys. Rev. B 93, 085105 (2016).

    ADS  Article  Google Scholar 

  26. 26.

    Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nat. Photon. 6, 782–787 (2012).

    ADS  Article  Google Scholar 

  27. 27.

    Dutt, A. et al. A single photonic cavity with two independent physical synthetic dimensions. Science 367, 59–64 (2020).

    ADS  Article  Google Scholar 

  28. 28.

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

    Google Scholar 

  29. 29.

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

    ADS  Article  Google Scholar 

  30. 30.

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

    ADS  Article  Google Scholar 

  31. 31.

    Kodigala, A. et al. Lasing action from photonic bound states in continuum. Nature 541, 196–199 (2017).

    ADS  Article  Google Scholar 

  32. 32.

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

    Article  Google Scholar 

  33. 33.

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

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Rivet, E. et al. Constant-pressure sound waves in non-Hermitian disordered media. Nat. Phys. 14, 942–947 (2018).

    Article  Google Scholar 

  35. 35.

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

    ADS  Article  Google Scholar 

  36. 36.

    Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Bacot, V., Durey, G., Eddi, A., Fink, M. & Fort, E. Phase-conjugate mirror for water waves driven by the faraday instability. Proc. Natl Acad. Sci. USA 116, 8809–8814 (2019).

    ADS  Article  Google Scholar 

  38. 38.

    Svidzinsky, A. A., Yuan, L. & Scully, M. O. Quantum amplification by superradiant emission of radiation. Phys. Rev. X 3, 041001 (2013).

    Google Scholar 

  39. 39.

    Koutserimpas, T. T. & Fleury, R. Nonreciprocal gain in non-Hermitian time-floquet systems. Phys. Rev. Lett. 120, 087401 (2018).

    ADS  Article  Google Scholar 

  40. 40.

    Haldane, F. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).

    ADS  Article  Google Scholar 

  41. 41.

    Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljacić, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772–775 (2009).

    ADS  Article  Google Scholar 

  42. 42.

    Fleury, R., Sounas, D. L., Sieck, C. F., Haberman, M. R. & Alù, A. Sound isolation and giant linear nonreciprocity in a compact acoustic circulator. Science 343, 516–519 (2014).

    ADS  Article  Google Scholar 

  43. 43.

    Khanikaev, A. B., Fleury, R., Mousavi, S. H. & Alu, A. Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice. Nat. Commun. 6, 8260 (2015).

    ADS  Article  Google Scholar 

  44. 44.

    Nash, L. M. et al. Topological mechanics of gyroscopic metamaterials. Proc. Natl Acad. Sci. USA 112, 14495–14500 (2015).

    ADS  Article  Google Scholar 

  45. 45.

    Ruesink, F., Miri, M. A., Alu, A. & Verhagen, E. Nonreciprocity and magnetic-free isolation based on optomechanical interactions. Nat. Commun. 7, 13662 (2016).

    ADS  Article  Google Scholar 

  46. 46.

    Kane, C. L. & Mele, E. J. Quantum spin hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).

    ADS  Article  Google Scholar 

  47. 47.

    Ma, J., Zhou, D., Sun, K., Mao, X. & Gonella, S. Edge modes and asymmetric wave transport in topological lattices: experimental characterization at finite frequencies. Phys. Rev. Lett. 121, 094301 (2018).

    ADS  Article  Google Scholar 

  48. 48.

    Wu, L.-H. & Hu, X. Scheme for achieving a topological photonic crystal by using dielectric material. Phys. Rev. Lett. 114, 223901 (2015).

    ADS  Article  Google Scholar 

  49. 49.

    He, C. et al. Acoustic topological insulator and robust one-way sound transport. Nat. Phys. 12, 1124–1129 (2016).

    Article  Google Scholar 

  50. 50.

    Miniaci, M., Pal, R. K., Morvan, B. & Ruzzene, M. Experimental observation of topologically protected helical edge modes in patterned elastic plates. Phys. Rev. X 8, 031074 (2018).

    Google Scholar 

  51. 51.

    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(R) (2018).

    ADS  Article  Google Scholar 

  52. 52.

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

    ADS  Article  Google Scholar 

  53. 53.

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

    Article  Google Scholar 

  54. 54.

    Bergholtz, E. J., Budich, J. C. & Kunst, F. K. Exceptional topology of non-Hermitian systems. Preprint at (2019).

  55. 55.

    Ghatak, A., Brandenbourger, M., Wezel, J. v. & Coulais, C. Observation of non-hermitian topology and its bulk–edge correspondence. (2019).

  56. 56.

    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).

    ADS  Article  Google Scholar 

  57. 57.

    Lee, C. H. & Thomale, R. Anatomy of skin modes and topology in non-Hermitian systems. Phys. Rev. B 99, 201103(R) (2019).

    ADS  Article  Google Scholar 

  58. 58.

    McDonald, A., Pereg-Barnea, T. & Clerk, A. Phase-dependent chiral transport and effective non-Hermitian dynamics in a bosonic Kitaev–Majorana chain. Phys. Rev. X 8, 041031 (2018).

    Google Scholar 

  59. 59.

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

    Article  Google Scholar 

  60. 60.

    Scheibner, C., Irvine, I. W. T. M. & Vitelli, V. Non-Hermitian band topology and skin modes in active elastic media. Phys. Rev. Lett. 125, 118001 (2020).

    ADS  MathSciNet  Article  Google Scholar 

  61. 61.

    Shen, H., Zhen, B. & Fu, L. Topological band theory for non-Hermitian Hamiltonians. Phys. Rev. Lett. 120, 146402 (2018).

    ADS  MathSciNet  Article  Google Scholar 

  62. 62.

    Shen, H., Zhen, B. & Fu, L. Topological band theory for non-hermitian hamiltonians. Phys. Rev. Lett. 120, 146402 (2018).

    ADS  MathSciNet  Article  Google Scholar 

  63. 63.

    Rudner, M. S., Lindner, N. H., Berg, E. & Levin, M. Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X 3, 031005 (2013).

    Google Scholar 

  64. 64.

    Reimer, V., Pedersen, K. G. L., Tanger, N., Pletyukhov, M. & Gritsev, V. Nonadiabatic effects in periodically driven dissipative open quantum systems. Phys. Rev. A 97, 043851 (2018).

    ADS  Article  Google Scholar 

  65. 65.

    Nassar, H., Chen, H., Norris, A. N. & Huang, G. L. Quantization of band tilting in modulated phononic crystals. Phys. Rev. B 97, 014305 (2018).

    ADS  Article  Google Scholar 

  66. 66.

    Yang, Z. et al. Mode-locked topological insulator laser utilizing synthetic dimensions. Phys. Rev. X 10, 011059 (2020).

    Google Scholar 

  67. 67.

    Milton, G. W. & Mattei, O. Field patterns: a new mathematical object. Proc. Math. Phys. Eng. Sci. 473, 20160819 (2017).

    MathSciNet  MATH  Google Scholar 

  68. 68.

    Li, M., Ni, X., Weiner, M., Alù, A. & Khanikaev, A. B. Topological phases and nonreciprocal edge states in non-hermitian floquet insulators. Phys. Rev. B 100, 045423 (2019).

    ADS  Article  Google Scholar 

  69. 69.

    Li, L., Lee, C. H. & Gong, J. Topological switch for non-Hermitian skin effect in cold-atom systems with loss. Phys. Rev. Lett. 124, 250402 (2020).

    ADS  Article  Google Scholar 

  70. 70.

    Kruthoff, J., de Boer, J., van Wezel, J., Kane, C. L. & Slager, R.-J. Topological classification of crystalline insulators through band structure combinatorics. Phys. Rev. X 7, 041069 (2017).

    Google Scholar 

  71. 71.

    Po, H., Vishwanath, A. & Watanabe, H. Symmetry-based indicators of band topology in the 230 space groups. Nat. Commun. 8, 50 (2017).

    ADS  Article  Google Scholar 

  72. 72.

    Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298–305 (2017).

    ADS  Article  Google Scholar 

  73. 73.

    Kruthoff, J., de Boer, J. & van Wezel, J. Topology in time-reversal symmetric crystals. Phys. Rev. B 100, 075116 (2019).

    ADS  Article  Google Scholar 

  74. 74.

    Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  75. 75.

    Cha, J., Kim, K. W. & Daraio, C. Experimental realization of on-chip topological nanoelectromechanical metamaterials. Nature 564, 229–233 (2018).

    ADS  Article  Google Scholar 

  76. 76.

    Assawaworrarit, S., Yu, X. & Fan, S. Robust wireless power transfer using a nonlinear parity–time-symmetric circuit. Nature 546, 387–390 (2017).

    ADS  Article  Google Scholar 

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We thank the Dutch Institute for Emergent Phenomena for sponsoring the workshop ‘Topology and broken symmetries: from driven quantum matter to active metamaterials’, which took place in Utrecht, the Netherlands from 1 to 3 July 2019, and the workshop participants for insightful discussions that inspired this Perspective. We acknowledge funding by ERC-StG-Coulais-852587-Extr3Me (C.C.) and by the Swiss National Science Foundation under SNSF grant number 172487 and the SNSF Eccellenza award number 181232 (R.F.).

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Correspondence to Corentin Coulais, Romain Fleury or Jasper van Wezel.

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Coulais, C., Fleury, R. & van Wezel, J. Topology and broken Hermiticity. Nat. Phys. 17, 9–13 (2021).

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