Dirac fermions and Kondo effect

In this study, we investigate the Kondo effect induced by the s-d interaction where the conduction bands are occupied by Dirac fermions. The Dirac fermion has the linear dispersion and is described typically by the Hamiltonian such as $H_k= v{\bf k}\cdot {\sigma}+m \sigma_0$ for the wave number ${\bf k}$ where $\sigma_j$ are Pauli matrices and $\sigma_0$ is the unit matrix. We derived the formula of the Kondo temperature $T_K$ by means of the Green's function theory for Green's functions including Dirac fermions and the localized spin. The $T_K$ was determined from a singularity of Green's functions in the form $T_K\propto \exp(-{\rm const}/\rho|J|)$ when $\rho|J|$ is small. The Kondo effect will disappear when the Fermi surface is point like because $T_K$ vanishes as the chemical potential $\mu$ approaches the Dirac point.


Introduction
Recently, the Dirac electron in solid state has been investigated intensively [1][2][3][4][5][6]. It is interesting to examine how the Kondo effect occurs in a system of Dirac electrons with magnetic impurities. It is not so trivial whether the Kondo effect indeed appears there. We expect significant and interesting behaviors when the localized spin interacts with the Dirac electron through the sd interaction. The Dirac Hamiltonian resembles the s-d model with the spin-orbit coupling of Rashba type [7]. In this paper we investigate the s-d Hamiltonian with Dirac electrons by means of the Green's function theory and evaluate the Kondo temperature T K .
The Hamiltonian is given by v and v z are velocities of conduction electrons, µ is the chemical potential and m is the mass of the Dirac fermion. N indicates the number of sites. We set m = 0 so that the chemical potential µ includes m. c kσ and c † kσ are annihilation and creation operators, respectively. S + , S − and S z denote the operators of the localized spin. The term H sd indicates the s-d interaction between the conduction electrons and the localized spin, with the coupling constant J [8,9]. J is negative, as adopted in this paper, for the antiferromagnetic interaction.

Green's Functions
We define thermal Green's functions of the conduction electrons where T τ is the time ordering operator. We note that the spin operators satisfy the following relations: We also define Green's functions which include the localized spins as well as the conduction electron operators. They are for example, following the notation of Zubarev [10], The Fourier transforms are defined as usual: · · · · · · .
Using the commutation relations the equation of motion for G kk ′ ↑ (iω n ) reads Here we have defined The equation of motion for F kk ′ = c k↓ ; c † k ′ ↑ iωn is also obtained in a similar way. By using the decoupling procedure for Green's functions [11][12][13], we can obtain a closed solution for a set of Green's functions. The Green's function G kk ′ ↑ (iω n ) is obtained as where the analytic continuation iω n → ω is carried out. We defined the following functions: with m k = 3 q c † q↑ c k↓ S − and n k = q c † q↑ c k↑ .

Kondo Temperature
From the Green's function G kk ′ ↑ (iω n ), the Kondo temperature T K is determined from a zero of the denominator in this formula. We consider 1 + JG(ω) = 0 (24) in the limit ω → 0 by neglecting higher-order term being proportional to (J/2) 2 . Let us adopt for simplicity that v z = v and then the dispersion is ξ k = ±v k 2 x + k 2 y + k 2 z − µ in three dimensions. We neglect the term of the order of J in n k = c † k↑ c k↑ . The equation 1 + JG(0) = 0 results in the Kondo temperature T K given as where D is a cutoff and γ is Euler's constant. We assumed that |µ| ≫ k B T K to derive the above formula. The result shows that the Kondo effect indeed occurs in a Dirac system. We can consider the Kondo effect in two dimensions by setting v z = 0, and also the d-dimensional case in general. In d dimensions T K reads where Ω d is the solid angle in d-dimensional space, namely, the area of the (d − 1)-sphere S d−1 .
In the limit |µ| → 0, the equation 1 + JG(0) = 0 has no solution. Hence, when |µ| is small, T K is reduced and vanishes. This measn that, when the Fermi surface is point like, the Kondo effect never appears because of the weak scattering by the localized spin. T K shows an algebraic behavior T K ∝ |µ| α with a constant α. In three dimensions, T K is proportional to µ 2 for small |µ|: where ζ(3) is the Riemann zeta function at argument 3. In d-dimensional space, this is generalized as