  We investigate the effect that density wave states have on the localized
spins of a square lattice. We find that topologically nontrivial density wave
states can induce stable Dzyaloshinskii-Moriya (DM) interactions among the
localized spins of the lattice in the presence of an external magnetic field,
and we study the resulting spin models for both antiferromagnetic and
ferromagnetic backgrounds. While the density wave state itself can contribute
to the the thermal Hall effect, as shown by Li & Lee (arXiv:1905.04248v3),
symmetry considerations preclude the resulting spin excitations from inducing a
further thermal Hall effect. We utilize a Holstein-Primakoff (HP) substitution
about the classical mean-field ground state to calculate the magnon dispersion
for LSCO and find that the density wave induces a weak $d_{x^2-y^2}$
anisotropy; upon calculating the non-Abelian Berry curvature for this magnon
branch we show explicitly that the magnon contribution to $\kappa_{xy}$ is
zero. Finally, we calculate corrections to the magnetic ground state energy,
spin canting angles, and the spin-wave dispersion due to the topological
density wave for ferromagnetic backgrounds. We find that terms linear in the HP
bosons can affect the critical behavior, a point previously overlooked in the
literature.
