Why is an optical magnon with k=0 not an eigenenergy state?

I found in a paper the following explanation. Unfortunately, I can't understand it. Can anyone help me on this?

In the limit of equal spins an optical magnon with k=0 gets an acoustical one at the boundary of the Brillouin zone where it is known to have a finite life time.

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Which paper is it? –  Qmechanic Feb 9 at 13:14
@Qmechanic This one. –  huotuichang Feb 10 at 0:36

What comes into my mind is the three-magnon process, in which an optical magnon with $k=0$ and $\omega=\omega_0$ splits into two acoustic magnons, one $k=k_1, \omega=\frac{\omega_0}{2}$, the other $k=-k_1, \omega=\frac{\omega_0}{2}$. Note that in this process energy and momentum are conserved. This 3-magnon process can lead to a finite life time.