Microscopic Theory of the Acoustic Faraday Effect in Spin-Orbit Coupled Magnets

The acoustic Faraday effect probes phonon Hall viscosity through the rotation of transverse acoustic polarization, conventionally interpreted in terms of circular birefringence. We show that magnetic order breaking rotational symmetry generates anisotropic phonon responses and elliptically polarized eigenmodes, so that Faraday rotation cannot be attributed to Hall viscosity alone. To elucidate their microscopic origin and consequences, we derive spin-phonon couplings from a generic $J$-$K$-$\mathrm{\Gamma }$-${\mathrm{\Gamma }}^{\mathrm{\prime }}$ model with interlayer coupling, identifying transverse strain couplings for propagation along the $c$-axis arising from the interplay of Kitaev $(K)$ and $\mathrm{\Gamma }$ interactions, along with strain-induced $g$-tensor contributions. Incorporating these couplings into our theory for the Kitaev spin-liquid candidate $\alpha$-RuCl${}_3$, we find that symmetry breaking modifies the Faraday response, revealing a Hall viscosity approximately twice the value inferred assuming rotational symmetry. Our results establish a microscopic framework beyond circular birefringence, refining the connection between acoustic Faraday rotation and phonon Hall viscosity in symmetry-broken spin-orbit-coupled magnets.