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.

中文翻译：

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电子流体中涡度引起的异常霍尔效应

我们为表现出反常霍尔效应的电子系统开发了流体动力学理论，并表明在边界附近产生的涡度会引起额外的反常霍尔效应。我们使用线性响应理论计算与电场成正比的动量通量和力。流体动力学方程是通过用电流代替局部电场获得的，重点放在比平均自由路径足够大的尺度上。结果表明，电流的涡流与磁化强度之间存在耦合，磁化强度由不均匀的涡流产生压力。考虑霍尔粘度和松弛力，计算边界附近的非均匀流动和附加霍尔力。额外的异常霍尔力与传统的异常霍尔力相反，导致薄系统中的符号反转。由于速度异常，侧跳过程会产生反对称粘度。