This work focuses on the nondissipative, parity-odd spin transport of $(2+1)$-dimensional relativistic electrons, generated by torsion, and the torsional Hall viscosity ${\zeta }_{\mathrm{H}}$. We first determine ${\zeta }_{\mathrm{H}}$ for massive Dirac fermions in the presence of a constant electromagnetic field. We predict that the magnetic field induces a contribution to ${\zeta }_{\mathrm{H}}$ competing with the one originating from the Dirac mass. Moreover, we quantify the impact on ${\zeta }_{\mathrm{H}}$ originating from the band structure deformation quadratic in momentum terms that was proposed by Bernevig-Hughes-Zhang (BHZ). We find that the BHZ deformation substantially enhances ${\zeta }_{\mathrm{H}}$ in magnitude as measured in a domain wall configuration, when compared to the free Dirac fermion result. Nevertheless, the torsional Hall viscosity still discriminates between topologically trivial and nontrivial regimes. Our results, hence, pave the way for a deeper understanding of hydrodynamic spin transport and its possible verification in experiments.

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• Chern insulators
• Dirac fermions
• Nonperturbative effects in field theory
• Quantum spin Hall effect
• Two-dimensional electron system
• Functional analytical methods