Kondo effect in Dirac systems

We investigate the Kondo effect in Dirac systems, where Dirac electrons interact with the localized spin via the s-d exchange coupling. The Dirac electron in solid state has the linear dispersion and is described typically by the Hamiltonian such as $H_k=v\mathbf{k}\cdot \sigma$ for the wave number ${\bf k}$where$\sigma_j$ are Pauli matrices. We derived the formula of the Kondo temperature $T_{\mathrm{K}}$ by means of the Green’s function theory for small $J$. The $T_{\mathrm{K}}$ is determined from a singularity of Green’s functions in the form $T_{\mathrm{K}}\simeq \bar{D}\exp (-\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{.}\mathrm{/}\rho \mathrm{\mid }J\mathrm{\mid })$ when the exchange coupling $\mathrm{\mid }J\mathrm{\mid }$ is small where $\bar{D}=D\mathrm{/}\sqrt{1+D^2\mathrm{/}(2\mu {)}^2}$ for a cutoff $D$ and $\rho$ is the density of states at the Fermi surface. When $|\mu|\ll D$,$T_{\rm K}$isproportionalto$|\mu|$:$T_{\rm K}\simeq |\mu|\exp(-{\rm const.}/\rho |J|)$. The Kondo screening will, however, disappear when the Fermi surface shrinks to a point called the Dirac point, that is, $T_{\mathrm{K}}$ vanishes when the chemical potential $\mu$ is just at the Dirac point. The resistivity and the specific heat exhibit a log-$T$ singularity in the range $T_{\mathrm{K}}<T\ll \mathrm{\mid }\mu \mathrm{\mid }\mathrm{/}{k}_{\mathrm{B}}$. Instead, for $T\sim O(\mathrm{\mid }\mu \mathrm{\mid })$ or $T>\mathrm{\mid }\mu \mathrm{\mid }$, they never show log-$T$.