Torsional Hall viscosity of massive Chern insulators: Magnetic field and momentum deformations

This work focuses on the non-dissipative, parity-odd spin transport of $(2+1)$-dimensional relativistic electrons, generated by torsion, and the torsional Hall viscosity $ζ_{\rm H}$. We first determine $ζ_{\rm H}$ for massive Dirac fermions in the presence of a constant electromagnetic field. We predict that the magnetic field induces a contribution to $ζ_{\rm H}$ competing with the one originating from the Dirac mass. Moreover, we quantify the impact on $ζ_{\rm H}$ 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 $ζ_{\rm H}$ 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 non-trivial regimes. Our results, hence, pave the way for a deeper understanding of hydrodynamic spin transport and its possible verification in experiments.