Skip to main content

Thank you for visiting nature.com. 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:

Higher-order topological states in photonic kagome crystals with long-range interactions

Nature Photonics volume 14, pages 89–94 (2020)Cite this article

Subjects

Abstract

Photonic topological insulators enable topological boundary modes that are resilient to defects and disorder, irrespective of manufacturing precision. This property is known as topological protection. Although originally limited to dimensionality of modes one lower than that of topological insulators, the recently discovered higher-order topological insulators (HOTIs) offer topological protection over an extended range of dimensionalities. Here, we introduce a photonic HOTI with kagome lattice that exhibits topological bulk polarization, leading to the emergence of one-dimensional edge states, as well as higher-order zero-dimensional states confined to the corners of the structure. Interestingly, in addition to the corner states due to nearest-neighbour interactions, we discover a new class of topological corner states induced by long-range interactions and specific to photonic systems. Our findings demonstrate that photonic HOTIs possess richer physics than their condensed-matter counterparts, offering opportunities for engineering novel designer electromagnetic states with unique topological robustness.

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

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

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

Fig. 1: Kagome lattice and corresponding bulk and edge band spectra.
Fig. 2: Simulation results of topological edge states and type I and type II corner states.
Fig. 3: Localization mechanism of type II corner states.
Fig. 4: Experimental observation of HOT type I and type II corner and edge states.

Similar content being viewed by others

Data availability

Data that are not already included in the paper and/or in the Supplementary Information are available on request from the authors.

References

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  3. Bernevig, B. A. & Hughes, T. L. Topological Insulators and Topological Superconductors (Princeton University Press, 2013).

  4. Raghu, S. & Haldane, F. D. M. Analogs of quantum-Hall-effect edge states in photonic crystals. Phys. Rev. A 78, 033834 (2008).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  7. 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).

    Article  ADS  Google Scholar 

  8. Fang, K. & Fan, S. Controlling the flow of light using the inhomogeneous effective gauge field that emerges from dynamic modulation. Phys. Rev. Lett. 111, 203901 (2013).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  11. Plotnik, Y. et al. Observation of unconventional edge states in ‘photonic graphene’. Nat. Mater. 13, 57–62 (2014).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  13. Cheng, X. et al. Robust reconfigurable electromagnetic pathways within a photonic topological insulator. Nat. Mater. 15, 542–548 (2016).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Kitaev, A. Unpaired Majorana fermions in quantum wires. Phys.-Usp. 44, 131–136 (2001).

    Article  ADS  Google Scholar 

  20. Bahari, B. et al. Nonreciprocal lasing in topological cavities of arbitrary geometries. Science 358, 636–639 (2017).

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  26. Ni, X. et al. Spin- and valley-polarized one-way Klein tunneling in photonic topological insulators. Sci. Adv. 4, eaap8802 (2018).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Song, Z., Fang, Z. & Fang, C. (d − 2)-dimensional edge states of rotation symmetry protected 437 topological states. Phys. Rev. Lett. 119, 246402 (2017).

  29. Schindler, F. et al. Higher-order topological insulators. Sci. Adv. 4, eaat0346 (2018).

    Article  ADS  Google Scholar 

  30. Ezawa, M. Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices. Phys. Rev. Lett. 120, 026801 (2018).

    Article  ADS  Google Scholar 

  31. Geier, M., Trifunovic, L., Hoskam, M. & Brouwer, P. W. Second-order topological insulators and superconductors with an order-two crystalline symmetry. Phys. Rev. B 97, 205135 (2018).

    Article  ADS  Google Scholar 

  32. Khalaf, E. Higher-order topological insulators and superconductors protected by inversion symmetry. Phys. Rev. B 97, 205136 (2018).

    Article  ADS  Google Scholar 

  33. Liu, T., He, J. J. & Nori, F. Majorana corner states in a two-dimensional magnetic topological insulator on a high-temperature superconductor. Phys. Rev. B 98, 245413 (2018).

    Article  ADS  Google Scholar 

  34. Noh, J. et al. Topological protection of photonic mid-gap defect modes. Nat. Photon. 12, 408–415 (2018).

    Article  ADS  Google Scholar 

  35. Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators. Phys. Rev. B 96, 245115 (2017).

    Article  ADS  Google Scholar 

  36. Serra-Garcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342–345 (2018).

    Article  ADS  Google Scholar 

  37. Peterson, C. W., Benalcazar, W. A., Hughes, T. L. & Bahl, G. A quantized microwave quadrupole insulator with topologically protected corner states. Nature 555, 346–350 (2018).

    Article  ADS  Google Scholar 

  38. Mittal, S. et al. Photonic quadrupole topological phases. Nat. Photon. 13, 692–696 (2019).

    Article  ADS  Google Scholar 

  39. Imhof, S. et al. Topolectrical-circuit realization of topological corner modes. Nat. Phys. 14, 925–929 (2018).

    Article  Google Scholar 

  40. Benalcazar, W. A., Li, T. & Hughes, T. L. Quantization of fractional corner charge in C n-symmetric topological crystalline insulators. Phys. Rev. B 99, 245151 (2019).

    Article  ADS  Google Scholar 

  41. Ni, X., Weiner, M., Alù, A. & Khanikaev, A. B. Observation of higher-order topological acoustic states protected by generalized chiral symmetry. Nat. Mater. 18, 113–120 (2019).

    Article  ADS  Google Scholar 

  42. Weiner, M., Ni, X., Li, M., Alu, A. & Khanikaev, A. B. Demonstration of a 3rd order hierarchy of topological states in a three-dimensional acoustic metamaterial. Preprint at https://arxiv.org/abs/1903.00428 (2019).

  43. El Hassan, A. et al. Corner states of light in photonic waveguides. Nat. Photon. 13, 697–700 (2019).

    Article  ADS  Google Scholar 

  44. Teo, J. C. Y. & Kane, C. L. Topological defects and gapless modes in insulators and superconductors. Phys. Rev. B 82, 115120 (2010).

    Article  ADS  Google Scholar 

  45. Seradjeh, B., Weeks, C. & Franz, M. Fractionalization in a square-lattice model with time-reversal symmetry. Phys. Rev. B 77, 033104 (2008).

    Article  ADS  Google Scholar 

  46. Leykam, D., Mittal, S., Hafezi, M. & Chong, Y. Reconfigurable topological phases in next-nearest-neighbor coupled resonator lattices. Phys. Rev. Lett. 121, 023901 (2018).

    Article  ADS  Google Scholar 

  47. Poli, C., Schomerus, H., Bellec, M., Kuhl, U. & Mortessagne, F. Partial chiral symmetry-breaking as a route to spectrally isolated topological defect states in two-dimensional artificial materials. 2D Mater. 4, 025008 (2017).

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  49. Bliokh, K. Y., Leykam, D., Lein, M. & Nori, F. Topological non-Hermitian origin of surface Maxwell waves. Nat. Commun. 10, 580 (2019).

    Article  ADS  Google Scholar 

  50. Liu, T. et al. Second-order topological phases in non-Hermitian systems. Phys. Rev. Lett. 122, 076801 (2019).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was supported by the Defense Advanced Research Projects Agency under the Nascent programme with grant no. HR00111820040 and by the National Science Foundation with grant nos EFRI-1641069 and DMR-1809915. Experimental investigation of corner states was partially supported by the Russian Science Foundation (grant no. 16-19-10538). Theoretical analysis of type II corner states was partially supported by the Russian Foundation for Basic Research (grant no. 18-32-20065).

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Grove School of Engineering, City College of New York, New York, NY, USA

    Mengyao Li, Dmitry Zhirihin, Xiang Ni, Andrea Alù & Alexander B. Khanikaev

  2. Physics Program, Graduate Center of the City University of New York, New York, NY, USA

    Mengyao Li, Xiang Ni, Andrea Alù & Alexander B. Khanikaev

  3. Department of Nanophotonics and Metamaterials, ITMO University, Saint Petersburg, Russia

    Dmitry Zhirihin, Maxim Gorlach & Alexey Slobozhanyuk

  4. Center for Photonics and 2D Materials, Moscow Institute of Physics and Technology, Dolgoprudny, Russia

    Dmitry Filonov

  5. Photonics Initiative, Advanced Science Research Center, City University of New York, New York, NY, USA

    Andrea Alù

Authors
  1. Mengyao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitry Zhirihin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Maxim Gorlach
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiang Ni
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Dmitry Filonov
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Alexey Slobozhanyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Andrea Alù
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Alexander B. Khanikaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

A.B.K. initiated the research. M.L., X.N. and M.G. derived the theoretical results. M.L. and D.Z. performed numerical simulations and designed the structure. D.Z., D.F. and A.S. designed and carried out the experiment. M.L. and D.Z. performed the data analysis. A.B.K. and A.A. supervised the research and wrote the main text. All authors discussed the results and contributed to the final version of the manuscript.

Corresponding author

Correspondence to Alexander B. Khanikaev.

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 Figs. 1–8 and refs. 1–4.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, M., Zhirihin, D., Gorlach, M. et al. Higher-order topological states in photonic kagome crystals with long-range interactions. Nat. Photonics 14, 89–94 (2020). https://doi.org/10.1038/s41566-019-0561-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41566-019-0561-9

This article is cited by

Search

Advanced search

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