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

Abstract

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 Formula Presented Chern insulator Formula Presented with a Formula Presented Chern insulator Formula Presented leads to a Formula Presented Chern insulator Formula Presented, where in the brackets, Formula Presented is the Formula Presented Hamiltonian with Formula Presented even, Formula Presented is the corresponding Formula Presented Chern number, and Formula Presented 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 “Formula Presented” 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 Formula Presented. (ii) Fascinatingly, the boundary states feature flat nodal surfaces with nontrivial Chern charges. For the constructed Formula Presented Chern insulator, a boundary perpendicular to a direction of Formula Presented generically hosts Formula Presented nodal surfaces, each of which has topological charge Formula Presented. 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.