Spin wave interactions in the pyrochlore Heisenberg antiferromagnet with Dzyaloshinskii-Moriya interactions

We study the effect of magnon interactions on the spin wave spectra of the all-in-all-out phase of the pyrochlore nearest neighbour antiferromagnet with a Dzyaloshinskii-Moriya interaction ($D$). The leading order corrections to spin wave energies indicate a significant renormalisation for commonly encountered strengths of the Dzyaloshinskii-Moriya term. For low values of $D$ we find a potential instability of the phase itself, indicated by the renormalisation of magnon frequencies to negative values. We have also studied the renormalized spectra in the presence of magnetic fields along three high symmetry directions of the lattice, namely the $[111]$, $[100]$ and $[110]$ directions. Generically, we find that for a fixed value of the Dzyaloshinskii-Moriya interaction renormalized spectra for the lowest band decrease with an increasing strength of the field. We have also analyzed the limits of the two magnon continuum and probed the possibility of magnon decay. For a range of $D$ and the field strength we identify possible parameter regimes where the decay of the higher bands of the system are kinematically allowed.