Orbital magnetization and anomalous Hall effect in interacting Weyl semimetals

Ferromagnetic Weyl semimetals exhibit an anomalous Hall effect, a consequence of their topological properties. In the noninteracting case, the derivative of the orbital magnetization with respect to chemical potential is proportional to this anomalous Hall effect, the St\ifmmode \check{r}\else \v{r}\fi{}eda formula. Motivated by compounds such as ${\text{Mn}}_{3}\text{Sn}$, here we investigate how interactions modeled by a Hubbard $U$ impact on both quantities when the Fermi energy is either aligned with the Weyl nodes or away from them. Using dynamical mean-field theory, we first find interaction-induced Mott- and band-insulating transitions. In the Weyl semimetal regime, away from insulators, interactions lead to an increase in the imbalance between the densities of spin species induced by a Zeeman term $h$. This increased imbalance leads to an increase of the anomalous Hall effect that can also be understood from the displacement of the Weyl nodes and topological arguments. This interaction-induced spin imbalance also compensates the reduction in orbital magnetization of each spin species that comes from smaller quasiparticle weight. In the small interaction regime, the combined effects lead to an orbital magnetization that depends weakly on interaction and still changes linearly with chemical potential at small doping. In the intermediate and strong correlation regimes, the localization due to interaction affects strongly the orbital magnetization, which becomes small. A quasiparticle picture explains the anomalous Hall effect but does not suffice for the orbital magnetization. We propose a modified St\ifmmode \check{r}\else \v{r}\fi{}eda formula to relate anomalous Hall effect and orbital magnetization in the weak correlation limit. We also identify mirror and particle-hole symmetries of the lattice model that explain, respectively, the vanishing of the anomalous Hall effect at $h=0$ for all $U$ and $\ensuremath{\mu}$ of the orbital magnetization at half-filling $\ensuremath{\mu}=0$ for all $U$ and $h$.

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