Torsional Hall Viscosity of Massive Chern Insulators: Magnetic Field and Momentum Deformations

TL;DR

This work analyzes the torsional Hall viscosity $\zeta_{\rm H}$ for massive Dirac fermions in 2+1D under non-perturbative deformations. It derives a closed-form, nonperturbative expression for $\zeta_{\rm H}$ in a constant magnetic field, reveals a magnetoviscosity with a leading negative ${\cal O}(B^2)$ correction and a zero crossing at finite $B$, and shows that a momentum-quadratic BHZ deformation qualitatively alters $\zeta_{\rm H}$, introducing topological versus trivial phases and enhancing domain-wall jumps. A nonperturbative electric-field analysis indicates an imaginary part beyond the Schwinger field, signaling vacuum instability and non-equilibrium effects. A complementary Berry-curvature computation corroborates the BHZ results, linking the torsional response to geometric phases. Collectively, the results provide experimentally testable predictions for spin transport in 2D Chern insulators and guide hydrodynamic modeling of torsional spin responses in realistic materials such as HgTe quantum wells.

Abstract

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.

Figures (5)

• Figure 1: $\zeta_{\rm H}$ in units of the electron rest mass as a function of the dimensionless ratio $B/B_c$, where $B_c$ is the Schwinger limit. In making this plot, we have assumed $m>0$.
• Figure 2: $\zeta_{\rm H}$ in units of the magnetic length $l_B$ as a function of the dimensionless mass $\tilde{m}=ml_B$.
• Figure 3: Contour used for the evaluation of $I(x)$ in Eq. \ref{['Eq:AuxIntegral']}. Taking $R\to\infty$ we pick up the rest of the poles of the integrand.
• Figure 4: Plot of the real (blue/continuous curve) and imaginary (orange/dashed curve) parts of $\zeta_{\rm H}$ in units of the electron rest mass as a function of the dimensionless combination $|E|/E_c$.
• Figure 5: The energy $E$ of the BHZ model as a function of the spatial momentum $p$ for different cases of $m$ and $b$.