Effect of conduction-electron interactions on Anderson impurities.

The effect of conduction electron interactions for an Anderson impurity is investigated in one dimension using a scaling approach. The flow diagrams are obtained by solving the renormalization group equations numerically. It is found that the Anderson impurity case is different from its counterpart – the Kondo impurity case even in the local moment region. The Kondo temperature for an Anderson impurity shows nonmonotonous behavior, increasing for weak interactions but decreasing for strong interactions. The implication of the study to other related impurity models is also discussed. PACS Numbers: 72.10.Fk, 75.20.Hr, 71.28+d, 72.15.Nj Typeset using REVTEX 1 Recently there has been much interest in magnetic impuirties interacting with a onedimensional (1D) correlated fermion system [1–5]. On the one hand, the progress in the nanofabrication technology could make the question accessible experimentally. On the other hand, it would shed some light on the impurity as well as lattice systems in D > 1 in the presence of conduction electron correlations. It has been found that localized electrons interact with strongly correlated conduction electrons in many materials, particularly in high-Tc superconductors and the heavy-fermion-like compound Nd1.8Ce0.2CuO4 [6]. It is known that electrons in 1D systems are in a Luttinger liquid state [8]. A Kondo impurity in a Luttinger liquid was studied recently by Lee and Toner [1], and Furusaki and Nagaosa [2]. They found that the Kondo temperature TK has an algebraic dependence on the Kondo coupling rather than the exponential dependence of the usual Kondo impurity, and always rises as the strength of conduction electron interactions increases. In effect, there is no competition between electron correlations and the Kondo effect. The usual Anderson impurity Hamiltonian with a free conduction bath can be transformed into the Kondo Hamiltonian with an irrelevant potential scattering [9] using the Schrieffer-Wolff transformation in the local moment regime. However, when the SchriefferWolff transformation is applied in the presence of conduction electron interactions, the effective Kondo coupling strongly depends on the interactions. Moreover, the impurity spin will interact not only with conduction electron spins at the impurity site but also with spins at neighboring sites [10]. The potential scattering is also relevant in a Luttinger liquid [11,5]. One would expect that an Anderson impurity behave differently from a Kondo impurity when conduction interactions are included even in the local moment region. In this paper, we mainly report results for an Anderson impurity in a Luttinger liquid. Using the scaling approach of Anderson-Yuval-Hamanna (AYH) [12] and Cardy [13], we obtain the renormalization group (RG) equations, which are solved exactly by numerical methods. We find that there is a strong interplay between the Kondo effect and the electron interactions for the Anderson impurity, unlike the Kondo impurity case. The Kondo temperature is enhanced for weak electron interactions but reduced for the strong interacting 2 case. The zeroth-order approximation which is widely used in the literature for solving the RG equations is checked against our numerical results. It suggests that the zeroth-order approximation could be very misleading. The Anderson impurity in a Luttinger liquid is described by the Hamiltonian H = HL +Hf +Hc−f , HL = vF ∑ k,σ k(c†k,1,σck,1,σ − c†k,2,σck,2,σ) + g2 N ∑ k1,k2,p,σ1,σ2 c†k1,1,σ1c † k2,2,σ2 ck2+p,2,σ2ck1−p,1,σ, Hf = E 0 f ∑ σ nfσ + Unf↑nf↓, (1) Hc−f = t √ N ∑ k,i,σ (c†k,i,σfσ +H.c.), Where ck,1,σ (ck,2,σ) is the annihilation operator of right-moving (left-moving) electrons with spin σ and momentum k, fσ is the annihilation operator for localized electrons, nfσ = f † σfσ, and N is the number of lattice sites. HL is the so-called Tomonaga-Luttinger Hamiltonian, where the g2 term represents forward scattering. HL generally describes 1D fermion systems away from half-filling and with repulsive interactions, for which umklapp and backward scattering can be ignored. Hf and Hc−f are the local and mixing terms, respectively. In the case of g2 = 0, the total Hamiltonian H reduces to the usual Anderson Hamiltonian. We will derive the partition function using bosonization technique [7] and then obtain the renormalization group equations applying the scaling approach of AndersonYuval-Hamann [12] and Cardy [13]. The partition function of the system is Z =