Dynamical spin susceptibility of $d$-wave Hatsugai-Kohmoto altermagnet

We investigate the interplay between altermagnetic band structures and electronic correlations by focusing on the $d_{x^2-y^2}$ altermagnetic generalization of the Hatsugai-Kohmoto model. We find that with increasing interaction, a many-body Lifshitz transition takes place when doubly occupied regions disappear from the Fermi surface and each momentum state becomes fully spin polarized. The spin susceptibility is directly evaluated from the Kubo formula in terms of many-body occupation probabilities. We find that the dynamical susceptibility, which possesses only transverse non-zero components for small wavevectors, develops a gap proportional to the interaction strength, and displays a sharp peak at a frequency increasing with the interaction. %with increasing frequency. Above the Lifshitz transition, this peak moves to the lower gap edge and becomes log-divergent. The signal intensity increases with the interaction up until the Lifshitz transition and saturates afterwards. The static susceptibility remains unaffected by the correlations and altermagnetism reduces the static transverse response.