Green's function of the magnetic topological insulator in a gradient expansion approach

We study the Keldysh Green's function of the Weyl-fermion surface state of the three-dimensional topological insulator coupled with a space-time dependent magnetization in the gradient expansion. Based on it we analyze the electric charge and current densities as well as the energy density and current induced by spatially and temporally slowly-varying magnetization fields. We show that all the above quantities except the energy current are generated by the emergent electromagnetic fields. The energy current emerges as the circular current reflecting the spatial modulation of an induced gap of the Weyl fermion.