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

We predict the presence of minority magnons in monolayer Fe$_3$GeTe$_2$ using first principles calculations. Minority magnons constitute a new type of collective magnetic excitation which increase the magnetic moment -- contrary to ordinary (majority) magnons which lower it -- giving rise to spin-raising poles in the dynamic susceptibility $\chi^{-+}(\omega)$. The presence of such quasi-particles is made possible by the nontrivial ferromagnetic band structure of Fe$_3$GeTe$_2$. We calculate the susceptibility using time-dependent density functional theory and perform a detailed mode analysis, which allows us to identify and investigate individual magnon modes as well as the Stoner excitations that constitute the many-body spectrum. The analysis reveals a plethora of both majority and minority excitations, which in addition to the main magnon branches include both satellite, valley and spin-inversion magnons thanks to the electron itinerancy of the system. Crucially, the analysis allows us to separate peaks of a coherent collective nature from those of mixed magnon/Stoner nature. To this end, we predict that the lowest energy minority magnon mode of monolayer Fe$_3$GeTe$_2$ indeed constitutes a coherent collective quasi-particle at long wavelengths and introduce a simplistic gaussian model for the observed minority magnon enhancement. The underlying physics is in no way restricted to Fe$_3$GeTe$_2$, and minority magnons are thus expected to be observable in other complex ferromagnetic materials as well.