We consider a 2-dimensional Bloch–Landau–Pauli Hamiltonian for a spinful electron in a constant magnetic field subject to a periodic background potential. Assuming that the [Formula: see text]-component of the spin operator is conserved, we compute the linear response of the associated spin density of states to a small change in the magnetic field, and identify it with the spin Hall conductivity. This response is in the form of a spin Chern marker, which is in general quantized to a half-integer, and to an integer under the further assumption of time-reversal symmetry. Our result is thus a generalization to the context of the quantum spin Hall effect of the well-known formula by Středa, which is formulated instead for charge transport.

中文翻译：

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电荷和自旋电流的 Středa 公式

我们考虑二维 Bloch-Landau-Pauli 哈密顿量，用于在受周期性背景电势影响的恒定磁场中的自旋电子。假设自旋算子的 [公式：见正文] 分量是守恒的，我们计算状态的相关自旋密度对磁场的微小变化的线性响应，并用自旋霍尔电导率识别它。该响应采用自旋陈标记的形式，通常量化为半整数，并在进一步假设时间反演对称的情况下量化为整数。因此，我们的结果是对 Středa 著名公式的量子自旋霍尔效应背景的推广，该公式是为电荷传输而制定的。