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.
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Data availability
The data in this study are available from the Digital Repository of NTU at https://doi.org/10.21979/N9/QTBDH7.
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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).
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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.
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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.
a–c, 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. d–h, 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 = r2−r1 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. d–f, 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. 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). https://doi.org/10.1038/s41586-022-05077-2
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DOI: https://doi.org/10.1038/s41586-022-05077-2
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