Non-equilibrium correlation effects in spin transport through the 2D ferromagnet Fe$_4$GeTe$_2$

Understanding non-equilibrium spin transport through 2D ferromagnets is a theoretical challenge, as correlations produce a complex electronic structure with coexisting itinerant and localized electrons. We have developed a fully non-equilibrium ab initio method, combining density functional theory, dynamical mean-field theory, and non-equilibrium Green's functions to investigate the transport in Fe$_4$GeTe$_2$, a prototypical high-temperature 2D ferromagnet. We show that, while spin transport remains essentially single-particle under moderate bias, inelastic spin-dependent scattering of carriers with particle-hole excitations drives a distinctive hot-correlated electron regime beyond a critical voltage. This regime is marked by incoherent features in both the electronic spectrum and the conductance, which are experimentally accessible. Our results demonstrates that material-specific many-body non-equilibrium methods are essential for a complete understanding of spin transport in 2D ferromagnets.