Itinerant spin-flip particle-hole continuum

Spin reversal in exchange-split bands creates an electron-hole excitation with energy ${ϵ}_{k+q}-{ϵ}_k+2M$ for a majority-to-minority flip in the simple uniform model with $M>0$. Its allowed continuum depends on occupied initial and empty final states. At $q=0$ all such allowed transitions have exchange energy $2M$; finite momentum changes the thresholds. A collective ferromagnetic magnon can remain gapless below this continuum when spin rotation is exact. The continuum is not the same excitation as a longitudinal amplitude mode.