Generalized Spin Electro-Motive Force in Spatially Dependent Systems

The coupling of an electron to an electromagnetic field is usually described by U(1) gauge fields in the Hamiltonian, whose gradient and curvature yield the electric and magnetic fields, respectively. We apply this gauge field formalism for two cases: i) electrons traveling through a medium with a spatial moment texture, and adiabatically relaxing along the local moment orientation, and ii) in the presence of Rashba spin-orbit interaction. Having obtained the Abelian U(1) and non-Abelian SU(2) gauge fields corresponding to the two cases, we apply the generalized Maxwell´s relation to derive the resulting spin electromotive force (e.m.f.). Finally, we analyze the spin e.m.f. effect in an exemplary domain-wall system in a ferromagnetic nanostrip.