Charge-Hall effect driven by spin force: reciprocal of the spin-Hall\n effect

A new kind of charge-Hall effect is shown. Unlike in the usual Hall effect,\nthe driving force in the longitudinal direction is a spin force, which may\noriginate from the gradient of a Zeeman field or a spin-dependent chemical\npotential. The transverse force is provided by a Berry curvature in a mixed\nposition-momentum space. We can establish an Onsager relation between this\neffect and the spin-Hall effect provided the spin current in the latter is\nmodified by a torque dipole contribution. This remarkable relation leads to new\nways for experimental detection of spin accumulation predicted by the spin Hall\neffect.\n