Equilibrium spin currents and the magnetoelectric effect in magnetic nanostructures

We discuss the problem of equilibrium spin currents in ferromagnets with inhomogeneous magnetization. Using simple microscopic models we explain the physical origin of equilibrium spin currents. Next we derive the equilibrium spin current from the Hamiltonian with a gauge field associated with local rotations in the spin space. Several examples of magnetic systems are studied in detail, and the persistent spin current is found to exist in the ground state of these systems. We also demonstrate the possibility of measuring the equilibrium spin current using the magnetoelectrically induced electric field near the ring.