Cubic Hall viscosity in three-dimensional topological semimetals

The nondissipative (Hall) viscosity is known to play an interesting role in two-dimensional (2D) topological states of matter, in the hydrodynamic regime of correlated materials, and in classical active fluids with broken time-reversal symmetry (TRS). However, generalizations of these effects to 3D have remained elusive. In this work, we address this question by studying the Hall viscoelastic response of 3D crystals. We show that for systems with tetrahedral symmetries, there exist new, intrinsically 3D Hall viscosity coefficients that cannot be obtained via a reduction to a quasi-2D system. To study these coefficients, we specialize to a theoretically and experimentally motivated tight-binding model for a chiral magnetic metal in (magnetic) space group [(M)SG] $P{2}_{1}3$ (No. 198.9), a nonpolar group of recent experimental interest that hosts both chiral magnets and topological semimetals (TSMs). Using the Kubo formula for viscosity, we compute two forms of the Hall viscosity, phonon and “momentum” (conventional) and show that for the tight-binding model we consider, both forms realize the novel cubic Hall viscosity. We conclude by discussing the implication of our results for transport in 2D magnetic metals and discuss some candidate materials in which these effects may be observed.

Figure 1

Schematic of ${\eta }_1$ and ${\eta }_2$. Dynamic strains (yellow) and viscosity give rise to stresses (orange). In response to a dynamic strain that elongates the length and width of a cubic parcel of fluid while compressing the depth, ${\eta }_1$ produces a diagonal shear stress. In response to a dynamic rotation of the parcel, ${\eta }_2$ produces an off-diagonal shear stress

Figure 2

(a) Band structure of the full tight-binding model Eq. (15). The gap at the $\mathrm{\Gamma }$ point scales with the hopping strength $t$. (b) Effective three band description of the $\mathrm{\Gamma }$ point for $\mu$ near the threefold degeneracy, with $\mathrm{\Lambda }$ a momentum cutoff chosen to regulate the Hall viscosity, which is plotted in (c) for the PHV “(p)” and MHV “(c)” methods (in the plot ${\overline{a}}^3=\frac{a^3}{8{\pi }^3}$).