Vorticity-induced anomalous Hall effect in an electron fluid

We develop a hydrodynamic theory for an electron system exhibiting the anomalous Hall effect, and show that an additional anomalous Hall effect is induced by a vorticity generated near boundaries. We calculate the momentum flux and force proportional to the electric field using linear response theory. The hydrodynamic equation is obtained by replacing the local electric field with the electric current, focusing on a scale that is sufficiently larger than the mean-free path. It is demonstrated that there is a coupling between a vorticity of an electric current and a magnetization which generates a pressure from nonuniform vorticity. Taking into account Hall viscosity and relaxation forces, a nonuniform flow near a boundary and an additional Hall force are calculated. The additional anomalous Hall force is opposite to a conventional anomalous Hall force, resulting in a sign reversal in thin systems. An antisymmetric viscosity turns out to arise from the side-jump process due to the anomalous velocity.