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A magnetic Impurity in a Weyl semimetal

We utilize the variational method to study the Kondo screening of a spin-$1/2$ magnetic impurity in a three-dimensional (3D) Weyl semimetal with two Weyl nodes along the $k_z$-axis. The model reduces to a 3D Dirac semimetal when the separation of the two Weyl nodes vanishes. When the chemical potential lies at the nodal point, $μ=0$, the impurity spin is screened only if the coupling between the impurity and the conduction electron exceeds a critical value. For finite but small $μ$, the impurity spin is weakly bound due to the low density of state, which is proportional to $μ^2$, contrary to that in a 2D Dirac metal such as graphene and 2D helical metal where the density of states is proportional to $|μ|$. The spin-spin correlation function $J_{uv}(\mathbf{r})$ between the spin $v$-component of the magnetic impurity at the origin and the spin $u$-component of a conduction electron at spatial point $\mathbf{r}$, is found to be strongly anisotropic due to the spin-orbit coupling, and it decays in the power-law. The main difference of the Kondo screening in 3D Weyl semimetals and in Dirac semimetals is in the spin $x$- ($y$-) component of the correlation function in the spatial direction of the $z$-axis.