  Motivated by experimental observations, Samajdar et al. [Nature Physics 15,
1290 (2019)] have proposed that the insulating Neel state in the parent
compounds of the cuprates is proximate to a quantum phase transition to a state
in which Neel order coexists with semion topological order. We study the manner
in which proximity to this transition can make the phonons chiral, by inducing
a significant phonon Hall viscosity. We describe the spinon-phonon coupling in
a lattice spinon model coupled to a strain field, and also using a general
continuum theory constrained only by symmetry. We find a nonanalytic Hall
viscosity across the transition, with a divergent second derivative at zero
temperature.
