Topological defects and boundary states in four-dimensional topological insulator

We show the connection between the second Chern number and topological defects, in a (4+1) -dimensional time -reversal invariant Dirac lattice model. It is discovered that two types of topological defects, the five -dimensional (5D) and four-dimensional (4D) point defects arise from the singular points of wave functions together with the geometric meaning of the second Chern number. We demonstrated that the 5D point defects appear at the band crossing positions with a topological transition, leading to a jump of the second Chern number. The 4D point defects exist in an insulating bulk, whose topological charges can give the evaluations of the second Chern number of energy bands. Finally, we discussed the possible structures of the boundary states in the light of the realization way of the 4D model. Our theory provides not only a new perspective to grasp the second Chern number, but also a simple approach to derive its values without calculating any integrals.