Intrinsic Bulk Topological Invariants in Three-Dimensional Chern Insulators: Chern Vectors, Torus Knots and Hopf Links
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. The three-dimensional quantum Hall effect, similar to that of the two-dimensional phase, is needed to extend the Chern insulator phase from two to three dimensions. Such Chern vectors for 3D Chern insulators have never been explored experimentally. We used 3D optical crystals to demonstrate the behavior of the kelvin genes and their 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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