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 Středa 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 Středa 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 $\mu$ of the orbital magnetization at half-filling $\mu =0$ for all $U$ and $h$.

Figure 1

(a) Density of states for noninteracting (dashed blue line) and interacting (continuous red line) Weyl semimetals. $U=12$ corresponds to the bandwidth of the model (2). Sharp peaks are due to the finite number of bath sites of the DMFT impurity solver. We use a Lorentzian broadening $\eta =0.02$ for both densities of states.

Figure 2

(a) Anomalous Hall conductivity as a function of $U$ at half-filling. (b) Anomalous Hall conductivity as a function of the renormalized Zeeman term $\tilde{h}=h+{\mathrm{\Sigma }}^z$. The continuous black line is the theoretical value found by replacing $h$ by $\tilde{h}$ in Eq. (10).

Figure 3

(a) Noninteracting orbital partial magnetization for $h=0$ (continuous lines) and $h=0.2$ (dashed lines) as a function of the chemical potential $\mu$. Note that the partial orbital magnetization is basis dependent and we have chosen the spin basis to represent it. (b) Value of the positive partial orbital magnetization as a function of Hubbard interaction $U$ for $h=0$ and $\mu =0$. Interaction renormalizes orbital magnetization via $Z$ as shown in the inset where the positive band orbital magnetization is plotted as a function of the spectral weight. (c) Noninteracting orbital magnetization as a function of chemical potential $\mu$ for several values of $h$. Two regimes are present: the cyan background delimitates the region where bands are mainly linear and white backgrounds where bands have quadratic behavior. The inset is a zoom of the same figure in the linear regime. Marks represent the numerical value from Eq. (12) and dashed lines are the functions $\mu {\sigma }_{xy}\left(h\right)$ with ${\sigma }_{xy}\left(h\right)$ the Hall conductivity in the absence of interaction but with a Zeeman term $h$. (d) Interacting orbital magnetization as a function of the electronic density near half-filling for $h=0.05$. The inset is the same plot as a function of chemical potential.

Figure 4

Comparison between numerical results for the the orbital magnetization and the modified Středa formula (14) where the derivative of ${\mathrm{\Sigma }}^I$ is obtained using a finite-difference method on the self-energy computed to second order in perturbation theory.