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.

# Topological Chern vectors in three-dimensional photonic crystals

## Abstract

The paradigmatic example of a topological phase of matter, the two-dimensional Chern insulator1,2,3,4,5, is characterized by a topological invariant consisting of a single integer, the scalar Chern number. Extending the Chern insulator phase from two to three dimensions requires generalization of the Chern number to a three-vector6,7, similar to the three-dimensional (3D) quantum Hall effect8,9,10,11,12,13. Such Chern vectors for 3D Chern insulators have never been explored experimentally. Here we use magnetically tunable 3D photonic crystals to achieve the experimental demonstration of Chern vectors and their topological surface states. We demonstrate Chern vector magnitudes of up to six, higher than all scalar Chern numbers previously realized in topological materials. The isofrequency contours formed by the topological surface states in the surface Brillouin zone form torus knots or links, whose characteristic integers are determined by the Chern vectors. We demonstrate a sample with surface states forming a (2, 2) torus link or Hopf link in the surface Brillouin zone, which is topologically distinct from the surface states of other 3D topological phases. These results establish the Chern vector as an intrinsic bulk topological invariant in 3D topological materials, with surface states possessing unique topological characteristics.

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

## Access options

Get just this article for as long as you need it

\$39.95

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

## Data availability

The data in this study are available from the Digital Repository of NTU at https://doi.org/10.21979/N9/QTBDH7.

## References

1. Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988).

2. Chang, C. Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).

3. Deng, Y. et al. Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi2Te4. Science 367, 895–900 (2020).

4. Zhao, Y. F. et al. Tuning the Chern number in quantum anomalous Hall insulators. Nature 588, 419–423 (2020).

5. Serlin, M. et al. Intrinsic quantized anomalous Hall effect in a moiré heterostructure. Science 367, 900–903 (2020).

6. Haldane, F. D. M. Berry curvature on the Fermi surface: anomalous Hall effect as a topological Fermi-liquid property. Phys. Rev. Lett. 93, 206602 (2004).

7. Vanderbilt, D. Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators (Cambridge Univ. Press, 2018).

8. Halperin, B. I. Possible states for a three-dimensional electron gas in a strong magnetic field. Jpn. J. Appl. Phys. 26, S3-3 (1987).

9. Kohmoto, M., Halperin, B. I. & Wu, Y. S. Diophantine equation for the three-dimensional quantum Hall effect. Phys. Rev. B 45, 13488 (1992).

10. Balents, L. & Fisher, M. P. Chiral surface states in the bulk quantum Hall effect. Phys. Rev. Lett. 76, 2782 (1996).

11. Druist, D. P., Turley, P. J., Maranowski, K. D., Gwinn, E. G. & Gossard, A. C. Observation of chiral surface states in the integer quantum Hall effect. Phys. Rev. Lett. 80, 365–368 (1998).

12. Bernevig, B. A., Hughes, T. L., Raghu, S. & Arovas, D. P. Theory of the three-dimensional quantum Hall effect in graphite. Phys. Rev. Lett. 99, 146804 (2007).

13. Tang, F. et al. Three-dimensional quantum Hall effect and metal–insulator transition in ZrTe5. Nature 569, 537–541 (2019).

14. von Klitzing, K. et al. 40 years of the quantum Hall effect. Nat. Rev. Phys. 2, 397–401 (2020).

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

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

17. 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).

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

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

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

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

22. Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).

23. Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).

24. Belopolski, I. et al. Discovery of topological Weyl fermion lines and drumhead surface states in a room temperature magnet. Science 365, 1278–1281 (2019).

25. Liu, D. et al. Magnetic Weyl semimetal phase in a Kagomé crystal. Science 365, 1282–1285 (2019).

26. Morali, N. et al. Fermi-arc diversity on surface terminations of the magnetic Weyl semimetal Co3Sn2S2. Science 365, 1286–1291 (2019).

27. Burkov, A. A. & Balents, L. Weyl semimetal in a topological insulator multilayer. Phys. Rev. Lett. 107, 127205 (2011).

28. Chen, C. Z. et al. Disorder and metal-insulator transitions in Weyl semimetals. Phys. Rev. Lett. 115, 246603 (2015).

29. Liu, S., Ohtsuki, T. & Shindou, R. Effect of disorder in a three-dimensional layered Chern insulator. Phys. Rev. Lett. 116, 066401 (2016).

30. Xiao, J. & Yan, B. First-principles calculations for topological quantum materials. Nat. Rev. Phys. 3, 283–297 (2021).

31. Devescovi, C. et al. Cubic 3D Chern photonic insulators with orientable large Chern vectors. Nat. Commun. 12, 7330 (2021).

32. Wang, Z. Y. et al. Realization of an ideal Weyl semimetal band in a quantum gas with 3D spin-orbit coupling. Science 372, 271–276 (2021).

33. Manturov, V. Knot Theory (CRC Press, 2018).

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

35. Belopolski, I. et al. Observation of a linked-loop quantum state in a topological magnet. Nature 604, 647–652 (2022).

36. Slobozhanyuk, A. et al. Three-dimensional all-dielectric photonic topological insulator. Nat. Photon. 11, 130–136 (2017).

37. Yang, Y. et al. Realization of a three-dimensional photonic topological insulator. Nature 565, 622–626 (2019).

## Acknowledgements

We acknowledge funding from the Singapore National Research Foundation Competitive Research Program (grant no. NRF-CRP23-2019-0007) and Singapore Ministry of Education Academic Research Fund Tier 3 (grant no. MOE2016-T3-1-006). P.Z., Z.G. and Y.Y. acknowledge funding from the National Natural Science Foundation of China (grant nos. 52022018, 52021001, 12104211, 6101020101 and 62175215) and Chinese Academy of Engineering (grant no. 2022-XY-127).

## Author information

Authors

### Contributions

Y.Y. and G.-G.L. initiated the project. Q.W. performed the tight-binding calculation. G.-G.L. and X.X. performed the simulation. G.-G.L., Z.G., Y.Y. and B.Z. designed experiments. Z.G. and P.Z. fabricated samples. P.Z., Y.-H.H., M.W. and C.L. carried out measurements. G.-G.L., Y.Y., P.Z., Z.G., Q.W., X.L., X.X., L.D., S.A.Y., Y.C. and B.Z. analysed the results and wrote the manuscript. B.Z., Y.C., Y.Y. and P.Z. supervised the project.

### Corresponding authors

Correspondence to Peiheng Zhou, Yihao Yang, Yidong Chong or Baile Zhang.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature thanks Alexander Khanikaev and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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 Permeability tensor of the gyromagnetic material.

a,b, Frequency-dependent elements μr and κ of the permeability tensor of the gyromagnetic material for B = 0.20 T and 0.45 T, respectively. The grey dashed line in a indicates the Weyl frequency in Fig. 2, and the grey rectangle in b indicates the bandgap of the photonic crystal in Fig. 2. c, Dispersionless μr and κ adopted in all simulations as a function of biasing magnetic fields.

### Extended Data Fig. 2 Chern-number component Cz of photonic crystals.

ac, Chern-number component Cz calculated in the 2D BZ plane at a fixed kz along the purple dashed line in Fig. 1b. The magnetic field B = 0 T, 0.20 T, and 0.45 T, corresponds to the band diagrams in Fig. 1d, e, and f, respectively. dh, Chern-number component Cz calculated in the 2D BZ plane at a fixed kz along the purple dashed line in Fig. 3b. The magnetic field B = 0 T, 0.20 T, 0.35 T, 0.42 T, and 0.50 T, corresponds to the simulated and measured surface intensities in Fig. 3e.

### Extended Data Fig. 3 Qualitative tight-binding models.

a, Qualitative tight-binding model for the photonic crystal designed in Fig. 1. b, Phase diagram of the model in a. It exhibits a trivial 3D insulating phase and a 3D Chern insulator phase, characterized by Chern vectors of (0, 0, 0) and (0, 0, ±1), respectively. c, Qualitative tight-binding model with two layers in a unit cell. d, Phase diagram of the model in c. Each gapped phase has been labelled by a Chern vector. The brown and the grey regions represent the gapless Weyl semimetal phases hosting four and two WPs in the first BZ, respectively. Here, t2 = 1.2, and the interlayer couplings represented by pink, blue, brown, and green dashed lines are 3, 0.5, 2, and 1.5, respectively.

### Extended Data Fig. 4 Phase transition by rotation of magnetic fields.

The photonic crystal designed in Fig. 1 is considered in the simulation as a typical example. a, Magnetic field rotatable in the x-z plane. b, Phase diagram of the photonic crystal by tuning the radius difference between two coupling holes R = r2r1 and the angle between the x-axis and the magnetic field α (B = 0.45 T). Grey regions: Weyl semimetal phases hosting two WPs. c, Bulk BZ. The blue and red dots are the two ideal WPs with opposite topological charges, when α = π/6. The red and blue arrows indicate the moving directions of the WPs with the enhancement of the α. df, Band diagrams of the photonic crystal with R = −0.7 mm and α of 0, π/6, and π/2, respectively. The green rectangle in d represents a complete 3D trivial bandgap from 19.1 to 19.7 GHz. The blue dot in e denotes a WP. The red rectangle in f represents a complete 3D topological bandgap from 19.3 to 19.7 GHz.

### Extended Data Fig. 5 Experimental setups.

a, Top view of the fabricated sample in Fig. 2a, where the first copper plate on the top is shifted for visualization. b, Copper pillars inserted into the coupling holes to function as metallic obstacles. c, Electromagnet used to produce magnetic fields. d, Triangular samples whose three wall surfaces are identical. e, Setup of two interfaced photonic crystals with perpendicular Chern vectors used in the demonstration in Fig. 4.

### Extended Data Fig. 6 Frequency-dependent surface dispersion and robustness of chiral surface states.

a, c, Measured surface dispersions on the frontal (010) surface of the fabricated sample in Fig. 2 for B = 0.20 T and 0.45 T, respectively. Three values of kz = 0, 0.53π/h, and 1π/h are selected. b, d, Simulated band structure on the frontal (010) surface for B = 0.20 T and 0.45 T, respectively. The white and green curves in a and c indicate the simulated envelopes of the projected bulk dispersions and surface dispersions, respectively. The blue curved surfaces in b and d represent the topological surface states, while the orange sheets indicate the envelopes of the projected bulk dispersions. e, Measured field distribution of chiral surface states in the fabricated sample in Fig. 2. The surface states are excited by a point source (cyan star) oscillating at 19.6 GHz. f, Measured field distribution in the same setup as in e, while copper pillars (yellow rods) are inserted into the sample as metallic obstacles. The frontal (010), left (100), and right (100) surfaces of the sample are covered with copper claddings, and all other surfaces with microwave absorbers. The samples are biased at 0.45 T along +z axis. The chiral surface states can propagate smoothly around the sharp corners and obstacles without scattering. The surface waves are mainly confined at their individual layers when passing around the copper pillars due to the weak dispersion along the z-axis.

### Extended Data Fig. 7 Band diagram simulations of models in Fig. 3.

ae, Band diagrams for the photonic crystal in Fig. 3e at different magnetic fields. fj, Band diagrams for different photonic crystals in Fig. 3f–j under the same magnetic field B = 0.45 T.

### Extended Data Fig. 8 Summary of (m, n)-torus knots/links with different combinations of m and n.

The colors of red, blue, green and yellow represent the first, second, third and fourth loops that wrap around the torus surface without crossing. The links with non-coprime m and n are highlighted with grey background. The simplest link is the (2, 2)-torus link, or the Hopf link on the torus surface.

### Extended Data Fig. 9 Construction of perpendicular Chern vectors.

a, e, Unit cells of the photonic crystals with Chern vectors $${{\bf{C}}}_{1}=2\hat{z}$$ and $${{\bf{C}}}_{2}=2\hat{x}$$, respectively. The two unit cells are identical except for different orientations. The dimensions are r = 1.2 mm, h1 = 4 mm, h2 = 1 mm, r1 = 2.6 mm, and r2 = 1.2 mm. The biasing magnetic field B = 0.5 T is oriented along the direction of $$\hat{x}+\hat{z}$$. b, f, Simulated band diagrams for a, e, respectively, which are identical. The bandgap is highlighted in pink. c, g, Measured surface intensity at 19.6 GHz for frontal and back (010) surfaces, respectively. The green lines indicate the simulated Fermi loop surface states. d, h, Simulated Fermi loop surface states wrap around the surface BZ in a torus geometry.

### Extended Data Fig. 10 Formation of Hopf link surface states.

a, b, Illustrations for surface Fermi loops induced by Chern vectors $${{\bf{C}}}_{1}=2\hat{z}$$ and $${{\bf{C}}}_{2}=2\hat{x}$$, respectively. c, Surface Fermi loops rearranged in the presence of coupling between two photonic crystals with $${{\bf{C}}}_{1}=2\hat{z}$$ and $${{\bf{C}}}_{2}=2\hat{x}$$. Blue and red solid lines depict the resulted two Fermi loops around the BZ. d, Blue and red lines individually form a loop in a torus geometry, and the two loops form a Hopf link.

## Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

Liu, GG., Gao, Z., Wang, Q. et al. Topological Chern vectors in three-dimensional photonic crystals. Nature 609, 925–930 (2022). https://doi.org/10.1038/s41586-022-05077-2

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1038/s41586-022-05077-2