Splitting of the electronic spectrum in the paramagnetic phase of itinerant ferromagnets and altermagnets

We study self-energy effects induced by strong magnetic fluctuations in the paramagnetic phase of strongly-correlated itinerant magnets within the density functional theory combined with the dynamical mean field theory (DFT+DMFT approach) and its non-local extension. We show that both local and non-local magnetic correlations yield a splitting of the electronic spectrum in the paramagnetic phase, such that it closely resembles the DFT band structure in the ordered phase. We demonstrate these effects on $α$-iron, half-metal CrO$_2$, van der Waals material CrTe$_2$, and altermagnet CrSb. Although the obtained split bands do not possess a certain spin projection, their splitting suppresses spectral weight at the Fermi level. Even when originating from local magnetic correlations, the splitting is strongly momentum dependent as a consequence of the orbital selectivity of non-quasiparticle states. The relative importance of non-local vs. local correlations depends on the proximity to half filling of $d$ states: closer to half filling, the role of local correlations increases.