Parity symmetry as the origin of ‘spin’ in the quantum spin Hall effect

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 $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ model, defined by a conserved quantum number that we call isoparity, a combination of parity $z\ensuremath{\rightarrow}\ensuremath{-}z$ 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 $z\ensuremath{\rightarrow}\ensuremath{-}z$ 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.