Transport Properties of Dirac Ferromagnet

We propose a model ferromagnet based on the Dirac Hamiltonian in three spatial dimensions, and study its transport properties which include anisotropic magnetoresistance (AMR) and anomalous Hall (AH) effect. This relativistic extension allows two kinds of ferromagnetic order parameters, denoted by $\bm{M}$ and $\bm{S}$, which are distinguished by the relative sign between the positive- and negative-energy states (at zero momentum) and become degenerate in the non-relativistic limit. Because of the relativistic coupling between the spin and the orbital motion, both $\bm{M}$ and $\bm{S}$ induce anisotropic deformations of the energy dispersion (and the Fermi surfaces) but in mutually opposite ways. The AMR is determined primarily by the anisotropy of the Fermi surface (group velocity), and secondarily by the anisotropy of the damping; the latter becomes important for ${\bm M}=\pm{\bm S}$, where the Fermi surfaces are isotropic. Even when the chemical potential lies in the gap, the AH conductivity is found to take a finite non-quantized value, $\sigma_{ij} = -(\alpha /3\pi^2 \hbar) \epsilon_{ijk} S_k $, where $\alpha$ is the (effective) fine structure constant. This offers an example of Hall insulator in three spatial dimensions.