Density Wave Mediated Dzyaloshinskii-Moriya Interactions*

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 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 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 anisotropy; upon calculating the non-Abelian Berry curvature for this magnon branch we show explicitly that the magnon contribution to the thermal Hall conductivity is zero. Finally, we calculate corrections to the ground state energy, and the spin-wave dispersion due to the 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.