  While nondissipative hydrodynamics in two-dimensional electron systems has
been extensively studied, the role of nondissipative viscosity in
three-dimensional transport has remained elusive. In this work, we address this
question by studying the nondissipative viscoelastic response of three
dimensional crystals. We show that for systems with tetrahedral symmetries,
there exist new, intrinsically three-dimensional Hall viscosity coefficients
that cannot be obtained via a reduction to a quasi-two-dimensional system. To
study these coefficients, we specialize to a theoretically and experimentally
motivated tight binding model for a chiral magentic metal in (magnetic) space
group [(M)SG] $P2_13$ (No.~198$.$9), a nonpolar group of recent experimental
interest which hosts both chiral magnets and topological semimetals. Using the
Kubo formula for viscosity, we compute the nondissipative Hall viscosity for
the spin-1 fermion in two ways. First we use an electron-phonon coupling ansatz
to derive the "phonon" strain coupling and associated phonon Hall viscosity.
Second we use a momentum continuity equation to derive the viscosity
corresponding to the conserved momentum density. We conclude by discussing the
implication of our results for hydrodynamic transport in three-dimensional
magnetic metals, and discuss some candidate materials in which these effects
may be observed.
