# Fermionic time-reversal symmetry in a photonic topological insulator

## Abstract

Much of the recent attention directed towards topological insulators is motivated by their hallmark feature of protected chiral edge states. In electronic (or fermionic) topological insulators, these states originate from time-reversal symmetry and allow carriers with opposite spin-polarization to propagate in opposite directions at the edge of an insulating bulk. By contrast, photonic (or bosonic) systems are generally assumed to be precluded from supporting edge states that are intrinsically protected by time-reversal symmetry. Here, we experimentally demonstrate counter-propagating chiral states at the edge of a time-reversal-symmetric photonic waveguide structure. The pivotal step in our approach is the design of a Floquet driving protocol that incorporates effective fermionic time-reversal symmetry, enabling the realization of the photonic version of an electronic topological insulator. Our findings allow for fermionic properties to be harnessed in bosonic systems, thereby offering alternative opportunities for photonics as well as acoustics, mechanical waves and cold atoms.

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## Data availability

The data represented in Figs. 3d,e,g,h and 4b are provided with the paper as source data. All other data that support results in this Article are available from the corresponding author upon reasonable request.

## Code availability

The numerical codes that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

## References

1. 1.

Kane, C. L. & Mele, E. J. Z 2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).

2. 2.

Bernevig, B. A. & Zhang, S.-C. Quantum spin Hall effect. Phys. Rev. Lett. 96, 106802 (2006).

3. 3.

König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).

4. 4.

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

5. 5.

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

6. 6.

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

7. 7.

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

8. 8.

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

9. 9.

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

10. 10.

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

11. 11.

Süsstrunk, R. & Huber, S. D. Observation of phononic helical edge states in a mechanical topological insulator. Science 349, 47–50 (2015).

12. 12.

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

13. 13.

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

14. 14.

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

15. 15.

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

16. 16.

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

17. 17.

Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the ‘parity anomaly’. Phys. Rev. Lett. 61, 2015 (1988).

18. 18.

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

19. 19.

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).

20. 20.

Maczewsky, L. J., Zeuner, J. M., Nolte, S. & Szameit, A. Observation of photonic anomalous Floquet topological insulators. Nat. Commun. 8, 13756 (2017).

21. 21.

Mukherjee, S. et al. Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice. Nat. Commun. 8, 13918 (2017).

22. 22.

Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).

23. 23.

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).

24. 24.

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

25. 25.

Slobozhanyuk, A. P. et al. Experimental demonstration of topological effects in bianisotropic metamaterials. Sci. Rep. 6, 22270 (2016).

26. 26.

He, C. et al. Photonic topological insulator with broken time-reversal symmetry. Proc. Natl Acad. Sci. USA 113, 4924–4928 (2016).

27. 27.

Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).

28. 28.

Carpentier, D., Delplace, P., Fruchart, M. & Gawędzki, K. Topological index for periodically driven time-reversal invariant 2D systems. Phys. Rev. Lett. 114, 106806 (2015).

29. 29.

Kitagawa, T., Rudner, M. S., Berg, E. & Demler, E. Exploring topological phases with quantum walks. Phys. Rev. A 82, 033429 (2010).

30. 30.

Roy, R. & Harper, F. Periodic table for Floquet topological insulators. Phys. Rev. B 96, 155118 (2017).

31. 31.

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

32. 32.

Nathan, F. & Rudner, M. S. Topological singularities and the general classification of Floquet-Bloch systems. New J. Phys. 17, 125014 (2015).

33. 33.

Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405 (1982).

34. 34.

Fu, L. & Kane, C. L. Time reversal polarization and a Z 2 adiabatic spin pump. Phys. Rev. B 74, 195312 (2006).

35. 35.

Kitagawa, T., Berg, E., Rudner, M. & Demler, E. Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, 235114 (2010).

36. 36.

Höckendorf, B., Alvermann, A. & Fehske, H. Efficient computation of the W 3 topological invariant and application to Floquet-Bloch systems. J. Phys. A 50, 295301 (2017).

37. 37.

Höckendorf, B., Alvermann, A. & Fehske, H. Topological invariants for Floquet-Bloch systems with chiral, time-reversal, or particle-hole symmetry. Phys. Rev. B 97, 045140 (2018).

38. 38.

Szameit, A. & Nolte, S. Discrete optics in femtosecond-laser-written photonic structures. J. Phys. B 43, 163001 (2010).

39. 39.

Rachel, S. Interacting topological insulators: a review. Rep. Prog. Phys. 81, 116501 (2018).

40. 40.

Poli, C., Bellec, M., Kuhl, U., Mortessagne, F. & Schomerus, H. Selective enhancement of topologically induced interface states in a dielectric resonator chain. Nat. Commun. 6, 6710 (2015).

41. 41.

Weimann, S. et al. Topologically protected bound states in photonic parity-time-symmetric crystals. Nat. Mater. 16, 433–438 (2017).

42. 42.

Kremer, M. et al. Demonstration of a two-dimensional $${\cal P}{\cal T}$$-symmetric crystal. Nat. Commun. 10, 435 (2019).

43. 43.

Heinrich, M. et al. Supersymmetric mode converters. Nat. Commun. 5, 3698 (2014).

44. 44.

Queraltó, G. et al. Topological state enineering via supersymmetric transformations. Commun. Phys. 3, 49 (2020).

## Acknowledgements

A.S. gratefully acknowledges financial support from the Deutsche Forschungsgemeinschaft (grants SZ 276/9-1, SZ 276/19-1, SZ 276/20-1, BL 574/13-1) and the Alfried Krupp von Bohlen und Halbach Foundation. We thank C. Otto for preparing the high-quality fused silica samples used in all experiments presented here.

## Author information

Authors

### Contributions

The theory was established by B.H., A.A. and H.F. The sample design and lattice implementation were developed by L.M., M.H. and A.S. The characterization of the lattice structure was carried out by L.M., M.K. and T.B. The project was supervised by H.F. and A.S. All authors discussed the results and co-wrote the paper.

### Corresponding authors

Correspondence to Andreas Alvermann or Alexander Szameit.

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### Competing interests

The authors declare no competing interests.

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## Supplementary information

### Supplementary Information

Detailed experimental techniques, additional measurements, extended theory, stability checking

## Source data

### Source Data Fig. 3

Image data of Fig. 3

### Source Data Fig. 4

Experimental data points of Fig. 4b

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