The quantum spin Hall effect arises due to band inversion in topological insulators, and has the defining characteristic that it hosts helical edge channels at zero magnetic field, leading to a finite spin Hall conductivity. The spin Hall conductivity is understood as the difference of the contributions of two spin states. In the effective four-band Bernevig-Hughes-Zhang model, these two spin states appear as two uncoupled blocks in the Hamiltonian matrix. However, this idea breaks down if additional degrees of freedom are considered. The two blocks cannot be identified by proper spin $S_z$ or total angular momentum $J_z$, both not conserved quantum numbers. In this work, we discuss a notion of block structure for the more general $\mathbf{k}·\mathbf{p}$ model, defined by a conserved quantum number that we call isoparity, a combination of parity $z\to -z$ and spin. Isoparity remains a conserved quantity under a wide range of conditions, in particular in presence of a perpendicular external magnetic field. From point-group considerations, isoparity is fundamentally defined as the action of $z\to -z$ on the spatial and spinorial degrees of freedom. Since time reversal anticommutes with isoparity, the two blocks act as Kramers partners. The combination of conductivity and isoparity defines spin conductivity. This generalized notion of spin Hall conductivity uncovers the meaning of ‘spin’: It is not the proper spin $S_z$, but a crystal symmetry that is realized by a spinorial representation.

中文翻译：

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奇偶对称性是量子自旋霍尔效应中“自旋”的起源

量子自旋霍尔效应是由于拓扑绝缘体中的能带反转而产生的，并且具有定义特征，即它在零磁场下拥有螺旋边缘通道，从而导致有限的自旋霍尔电导率。自旋霍尔电导率被理解为两种自旋态贡献的差异。在有效的四波段 Bernevig-Hughes-Zhang 模型中，这两个自旋态在哈密顿矩阵中表现为两个非耦合块。然而，如果考虑额外的自由度，这个想法就会失效。正确旋转无法识别这两个块${秒}_z$ 或总角动量 $J_z$，两者都不是守恒的量子数。在这项工作中，我们讨论了更一般的块结构的概念$\mathbf{克}·\mathbf{磷}$模型，由一个守恒的量子数定义，我们称之为isoparity，奇偶性的组合$z\to -z$和旋转。在各种条件下，尤其是在存在垂直外部磁场的情况下，等比性仍然是一个守恒量。从点群的角度考虑，等价性从根本上被定义为$z\to -z$关于空间和旋向自由度。由于时间反转具有等价性，因此这两个块充当 Kramers 伙伴。电导率和等价性的组合定义了自旋电导率。自旋霍尔电导率的这种广义概念揭示了“自旋”的含义：这不是正确的自旋${秒}_z$，而是通过旋光表示实现的晶体对称性。