Current response of a topological insulator to a static Zeeman field

We study the magnetoelectric coupling at the surface of a topological insulator. We are particularly interested in the surface current induced by a static Zeeman/exchange field. This surface current can be related to the orbital magnetization of the system. For an insulator with zero Chern number, the orbital magnetization is independent of the details at the boundary. With the appearance of surface states in the topological insulator, it is not immediately obvious if the response is affected by the conditions at the surface. We investigate this question using exact diagonalization to a lattice model. By applying a time-reversal symmetry-breaking term near the boundary, even if the surface states are gapped out, we still find no change in the surface current. This arises from cancelations between Pauli and Van Vleck contributions between surface and bulk scattering states. We also show that the surface current response is independent of the chemical potential when it is within the bulk gap. Our results are consistent with the claim that orbital magnetization is a bulk property.