Quantum dynamics of tunneling between ferromagnets

We study the Josephson-like spin currents between two ferromagnetic metals by deriving the effective action of the junction. A DC spin Josephson current with the full O(3) symmetry is obtained. By the analogy to the superconducting junctions, we have shown that a time-independent uniform magnetic field can serve as the source of the AC spin Josephson effect. That is, the spin current becomes a periodic function of the time with the period proportional to the inverse of the magnitude of the external magnetic field.