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 $H_k= v{\bf k}\cdot σ$ for the wave number ${\bf k}$ where $σ_j$ are Pauli matrices. We derived the formula of the Kondo temperature $T_{\rm K}$ by means of the Green's function theory for small $J$. The $T_{\rm K}$ is determined from a singularity of Green's functions in the form $T_{\rm K}\simeq \bar{D}\exp(-{\rm const.}/ρ|J|)$ when the exchange coupling $|J|$ is small where $\bar{D}=D/\sqrt{1+D^2/(2μ)^2}$ for a cutoff $D$ and $ρ$ is the density of states at the Fermi surface. When $|μ|\ll D$, $T_{\rm K}$ is proportional to $|μ|$: $T_{\rm K}\simeq |μ|\exp(-{\rm const.}/ρ|J|)$. The Kondo screening will, however, disappear when the Fermi surface shrinks to a point called the Dirac point, that is, $T_{\rm K}$ vanishes when the chemical potential $μ$ is just at the Dirac point. The resistivity and the specific heat exhibit a log-$T$ singularity in the range $T_{\rm K} < T\ll |μ|/k_{\rm B}$. Instead, for $T\sim O(|μ|)$ or $T>|μ|$, they never show log-$T$.

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