Coupling theory of emergent spin electromagnetic field and electromagnetic field

In ferromagnetic metals, an effective electromagnetic field which couples to conduction electron spin is induced by the sd exchange interaction. We investigate how this effective field, spin electromagnetic field, interacts with the ordinary electromagnetic field by deriving an effective Hamiltonian based on the path-integral formalism. It turns out that the dominant coupling term is the product of the electric field and the spin gauge field. This term descrives the spin-transfer effect, as was pointed out before. The electric field couples also to the spin electric field, but this contribution is smaller than the spin-transfer contribution in the low frequency regime. The magnetic field couples to the spin magnetic field, and this interaction suggests an intriguing intrinsic mechanism of frustration in very weak metallic ferromagnets under a uniform magnetic field. We also propose a voltage generation mechanism due to a non-linear effect of non-monochromatic spin wave excitation.