Spontaneous magnon decays in planar ferromagnet

We predict that spin-waves in an easy-plane ferromagnet have a finite lifetime at zero temperature due to spontaneous decays. In zero field the damping is determined by three-magnon decay processes, whereas decays in the two-particle channel dominate in a transverse magnetic field. Explicit calculations of the magnon damping are performed in the framework of the spin-wave theory for the $XXZ$ square-lattice ferromagnet with an anisotropy parameter $\lambda <1$. In zero magnetic field the decays occur for ${\lambda }^{\ast }<\lambda <1$ with ${\lambda }^{\ast }\approx 1\mathrm{/}7$. We also discuss possibility of experimental observation of the predicted effect in a number of ferromagnetic insulators.