supplementary from Phase-space methods for the spin dynamics in condensed matter systems

Using the phase-space formulation of quantum mechanics, we derive a four-component Wigner equation for a system composed of spin-½ fermions (typically, electrons) including the Zeeman effect and the spin–orbit coupling. This Wigner equation is coupled to the appropriate Maxwell equations to form a self-consistent mean-field model. A set of semiclassical Vlasov equations with spin effects is obtained by expanding the full quantum model to first order in the Planck constant. The corresponding hydrodynamic equations are derived by taking velocity moments of the phase-space distribution function. A simple closure relation is proposed to obtain a closed set of hydrodynamic equations.

本研究采用量子力学相空间表述，针对由自旋1/2费米子（典型如电子）构成的体系，推导得到包含塞曼效应与自旋轨道耦合的四分量维格纳方程（Wigner equation）。将该维格纳方程与恰当的麦克斯韦方程组相耦合，构建自洽平均场模型。通过将全量子模型按普朗克常数展开至一阶近似，得到含自旋效应的半经典弗拉索夫方程组（Vlasov equations）。通过对相空间分布函数取速度矩，推导得到对应的流体力学方程组；随后提出简单的封闭关系，以获得闭合的流体力学方程组集合。