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Topological Chern vectors in three-dimensional photonic crystals


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.

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Fig. 1: Photonic design of a 3D Chern insulator.
Fig. 2: Observation of topological bandgap opening with WP pair annihilation.
Fig. 3: Photonic 3D Chern insulators with large Chern vectors.
Fig. 4: Hopf link surface states formed by perpendicular Chern vectors.

Data availability

The data in this study are available from the Digital Repository of NTU at


  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).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

  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).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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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).

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Authors and Affiliations



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.

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The authors declare no competing interests.

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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.

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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.

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Liu, GG., Gao, Z., Wang, Q. et al. Topological Chern vectors in three-dimensional photonic crystals. Nature 609, 925–930 (2022).

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