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. We also show that a time-independent uniform magnetic field can serve as the source of the ac spin Josephson effect. That is, the spin current in a uniform magnetic field becomes a periodic function of the time with the period proportional to the inverse of the magnitude of the external magnetic field.