Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Self-induced topological protection in nonlinear circuit arrays


The interplay between topology and many-body physics has been a topic of strong interest in condensed matter physics for several years. For electronic systems, research has so far focused on linear topological phenomena due to the lack of a proper experimental platform and, in classical systems, due to the weak nature of nonlinear phenomena. Recently, it has been shown, however, that nonlinear effects can lead to unique phenomena, including self-induced topological states. Here we report nonlinear circuit arrays that exhibit self-induced topological protection. Our arrays are based on one-dimensional transmission-line circuits that emulate the Su–Schrieffer–Heeger model. We show that these nonlinear circuit arrays can exhibit self-induced topological transitions as a function of the input intensity, leading to topologically robust edge states that are immune to the presence of defects. The emergence of these topological states could lead to the design of electronic, optical and acoustic devices with functionalities that are highly tolerant to fabrication imperfections and unwanted parasitic effects.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Model and geometry.
Fig. 2: Self-induced edge states and the origin of their immunity against defects.
Fig. 3: Experimental demonstration of self-induced edge state with immunity against defects.
Fig. 4: Mode profile at low and high intensities and the emergence of the nonlinear edge state.

Similar content being viewed by others


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

    Article  Google Scholar 

  2. Kim, D. et al. Surface conduction of topological Dirac electrons in bulk insulating Bi2Se3. Nat. Phys. 8, 459–463 (2012).

    Article  Google Scholar 

  3. Bernevig, B. A., Hughes, T. L. & Zhang, S. C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

  4. Fu, L., Kane, C. & Mele, E. Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007).

    Article  Google Scholar 

  5. Hsieh, D. et al. A topological Dirac insulator in a quantum spin Hall phase. Nature 452, 970–974 (2008).

    Article  Google Scholar 

  6. Moore, J. & Balents, L. Topological invariants of time-reversal-invariant band structures. Phys. Rev. B 75, 121306 (2007).

    Article  Google Scholar 

  7. Roy, R. Z 2 classification of quantum spin Hall systems: an approach using time-reversal invariance. Phys. Rev. B 79, 195321 (2009).

    Article  Google Scholar 

  8. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  11. Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nat. Photon. 7, 1001–1005 (2013).

    Article  Google Scholar 

  12. Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).

  13. Schomerus, H. Topologically protected midgap states in complex photonic lattices. Opt. Lett. 38, 1912–1914 (2013).

    Article  Google Scholar 

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

  15. Zeuner, J. M. et al. Observation of topological transition in the bulk of a non-Hermitian system. Phys. Rev. Lett. 115, 040402 (2015).

    Article  Google Scholar 

  16. Mittal, S. et al. Measurements of topological invariants in a 2D photonic system. Nat. Photon. 10, 180–183 (2016).

    Article  Google Scholar 

  17. Peano, V., Brendel, C., Schmidt, M. & Marquardt, F. Topological phases of sound and light. Phys. Rev. X 5, 031011 (2015).

    Google Scholar 

  18. Yang, Z. et al. Topological acoustics. Phys. Rev. Lett. 114, 114301 (2015).

    Article  Google Scholar 

  19. Xiao, M. et al. Geometric phase and band inversion in periodic acoustic system. Nat. Phys. 11, 240–244 (2015).

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

  21. Ningyuan, J., Owens, C., Sommer, A., Schuster, D. & Simon, J. Time- and site-resolved dynamics in a topological circuit. Phys. Rev. X 5, 021031 (2015).

    Google Scholar 

  22. Albert, V. V., Glazman, L. I. & Jiang, L. Topological properties of linear circuit lattices. Phys. Rev. Lett. 114, 173902 (2015).

    Article  MathSciNet  Google Scholar 

  23. Roushan, P. et al. Chiral ground-state currents of interacting photons in a synthetic magnetic field. Nat. Phys. 13, 146–151 (2017).

    Article  Google Scholar 

  24. Hadad, Y., Khanikaev, A. B. & Alù, A. Self-induced topological transitions and edge states supported by nonlinear staggered potential. Phys. Rev. B 93, 155112 (2016).

  25. Hadad, Y., Vitelli, V. & Alù, A. Solitons and propagating domain walls in optical resonator arrays. ACS Photon. 4, 1974–1979 (2017).

  26. Lumer, Y., Plotnik, Y., Rechtsman, M. C. & Segev, M. Self-localized states in photonic topological insulator. Phys. Rev. Lett. 111, 243905 (2013).

    Article  Google Scholar 

  27. Leykam, D. & Chong, Y. D. Edge solitons in nonlinear photonic topological insulator. Phys. Rev. Lett. 117, 143901 (2016).

    Article  Google Scholar 

  28. Heeger, A. J., Kivelson, S., Schrieffer, J. R. & Su, W. P. Solitons in conducting polymers. Rev. Mod. Phys. 60, 781–823 (1988).

    Article  Google Scholar 

  29. Zak, J. Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747–2750 (1989).

    Article  Google Scholar 

  30. Xiao, M., Zhang, Z. Q. & Chan, C. T. Surface impedance and bulk band geometric phases in one dimensional systems. Phys. Rev. X 4, 021017 (2014).

    Google Scholar 

  31. Freeman, R. H. & Karbowiak, A. E. An investigation of nonlinear transmission lines and shock waves. J. Phys. D 10, 633–643 (1977).

    Article  Google Scholar 

  32. Wilson, C. R., Turner, M. M. & Smith, P. W. Pulse sharpening in a uniform LC ladder network containing nonlinear ferroelectric capacitors. IEEE Trans. Electron. Dev. 38, 767–771 (1991).

    Article  Google Scholar 

  33. Hanamura, E. Rapid radiative decay and enhanced optical nonlinearity of exitons in a quantum well. Phys. Rev. B 38, 1228–1234 (1988).

  34. Lee, J. et al. Giant nonlinear response from plasmonic metasurfaces coupled to intraband transitions. Nature 511, 66–69 (2014).

    Google Scholar 

Download references


This work was supported by the National Science Foundation, the Air Force Office of Scientific Research and the Simons Foundation.

Author information

Authors and Affiliations



All authors conceived the idea. Y.H. developed the theory and the basic circuit design. J.C.S. optimized the circuit and designed the circuit layout. Y.H. and J.C.S. designed the experimental set-up and performed the measurements. All authors discussed the results and wrote the paper.

Corresponding author

Correspondence to Andrea Alù.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figures 1–3 and Supplementary Notes 1–8

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hadad, Y., Soric, J.C., Khanikaev, A.B. et al. Self-induced topological protection in nonlinear circuit arrays. Nat Electron 1, 178–182 (2018).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing