Tensor theory for higher-dimensional Chern insulators with large Chern numbers

Recent advances in topological artificial systems open the door to realizing topological states in dimensions higher than the usual three-dimensional space. Here we present a "tensor product" theory, which offers a method to construct Chern insulators with arbitrarily high dimensions and Chern numbers. Particularly, we show that the tensor product of a d(A)D Chern insulator < H-A((kappa A)),C-A > with a d(B)D Chern insulator < H-B((kappa B)),C-B > leads to a (d(A) + d(B))D Chern insulator < H-AB((kappa A star kappa B)),-2C(A)C(B)>, where in the brackets, H-(kappa) is the dD Hamiltonian with d even, C is the corresponding (d/2)th Chern number, and kappa labels the five nonchiral Altland-Zirnbauer symmetry classes A, AI, D, AII, and C. The four real classes AI, D, AII, and C form a Klein four-group under the multiplication "star," with class AI the identity and class A is the zero element. Our theory leads to novel higher-dimensional topological physics. (i) The construction can generate large higher-order Chern numbers, e.g., for some cases the resultant classification is 8Z. (ii) Fascinatingly, the boundary states feature flat nodal surfaces with nontrivial Chern charges. For the constructed (d(A) + d(B))D Chern insulator, a boundary perpendicular to a direction of H-A generically hosts vertical bar C-A vertical bar d(B)D nodal surfaces, each of which has topological charge +/- 2C(B). Under perturbations, each nodal surface bursts into stable unit nodal points with the total Chern charge conserved. Examples are given to demonstrate our theory, which can be experimentally realized in artificial systems such as acoustic crystals, electric-circuit arrays, ultracold atoms, or mechanical networks.