Magnon interactions in the frustrated pyrochlore ferromagnet Yb$_2$Ti$_2$O$_7$

The frustrated rare-earth pyrochlore Yb$_2$Ti$_2$O$_7$ is remarkable among magnetic materials: despite a ferromagnetically ordered ground state it exhibits a broad, nearly gapless, continuum of excitations. This broad continuum connects smoothly to the sharp one-magnon excitations expected, and indeed observed, at high magnetic fields, raising the question: how does this picture of sharp magnons break down as the field is lowered? In this paper, we consider the effects of magnon interactions in Yb$_2$Ti$_2$O$_7$, showing that their inclusion greatly extends the reach of spin-wave theory. First, we show that magnon interactions shift the phase boundary between the (splayed) ferromagnet (SFM) and the antiferromagnetic $\Gamma_5$ phase so that Yb$_2$Ti$_2$O$_7$ lies very close to it. Next, we show how the high-field limit connects to lower fields; this includes corrections to the critical fields for the $[111]$ and $[1\bar{1}0]$ directions, bringing them closer to the observed experimental values, as well as accounting for the departures from linear spin wave theory that appear in $[001]$ applied fields below 3 T [Thompson et al., Phys. Rev. Lett. 119, 057203 (2017)]. Turning to low-fields, though the extent of the experimentally observed broadening is not quite reproduced, we find a rough correspondence between non-linear spin-wave theory and inelastic neutron scattering data on both a single-crystal sample, as well as on a powder sample [Pe\c{c}anha-Antonio et al., Phys. Rev. B, 96, 214415 (2017)]. We conclude with an outlook on implications for future experimental and theoretical work on Yb$_2$Ti$_2$O$_7$ and related materials, highlighting the importance of proximity to the splayed ferromagnet-$\Gamma_5$ phase boundary and its potential role in intrinsic or extrinsic explanations of the low-field physics of Yb$_2$Ti$_2$O$_7$.