Title: Minority magnons and mode branching in monolayer Fe$_3$GeTe$_2$

Abstract: In this letter, we predict the existence of minority magnons in a monolayer of Fe$_3$GeTe$_2$ using first principles calculations. Minority magnons constitute a new type of collective magnetic excitations which increase the magnetic moment rather than lower it, contrary to ordinary (majority) magnons. The presence of such quasi-particles is made possible by the nontrivial ferromagnetic band structure of Fe$_3$GeTe$_2$ originating from its nonequivalent Fe sublattices. The result is a strong peak in the dynamic spin-raising susceptibility $\chi^{-+}(\omega)$ in the long wavelength limit, which is the hallmark of minority magnon physics. We calculate the susceptibility using time-dependent density functional theory and perform a detailed mode analysis, which allows us to separate and investigate the individual magnon modes as well as the Stoner excitations that constitute the many-body spectrum. For minority as well as majority magnons, the analysis reveals a plethora of magnetic excitations, which in addition to the standard main magnon branches include both satellite, valley and spin-inversion magnons, originating from the itinerancy of the ferromagnetic order. The physics underlying this analysis is in no way restricted to Fe$_3$GeTe$_2$, and minority magnons are expected to be observable in many complex ferromagnetic materials.