Hybridization and anisotropy in the exchange interaction in three-dimensional Dirac semimetals

We study the Ruderman-Kittel-Kasuya-Yosida interaction in three-dimensional Dirac semimetals. Using retarded Green's functions in real space, we obtain and analyze asymptotic expressions for the interaction, with magnetic impurities at different distances and relative angle with respect to high symmetry directions on the lattice. We show that the Fermi velocity anisotropy in these materials produces a strong renormalization of the magnitude of the interaction, as well as a correction to the frequency of oscillation in real space. Hybridization of the impurities to different conduction electron orbitals are shown to result in interesting anisotropic spin-spin interactions which can generate spiral spin structures in doped samples.

Figure 1

Lattice structure of ${\mathrm{Na}}_3\mathrm{Bi}$ in the $xy$ plane (left) with dashed lines denoting different symmetry directions. ${\mathrm{Na}}_3\mathrm{Bi}$ unit cell [24] with a magnetic impurity in its center as a possible location (right).

Figure 2

Effective impurity interaction as a function of their separation $R$ in the $xy$ plane. The anisotropy in the Fermi velocity, characterized by $\lambda =0.25$, has a big impact on the strength of the interaction with respect to the isotropic case $\left(\lambda =1\right)$. It also introduces a correction in the period of the oscillation.