An Exactly Solvable Kondo Problem for Interacting One-Dimensional Fermions.

The single impurity Kondo problem in the one-dimensional (1D) δ-potential Fermi gas is exactly solved for two sets of special coupling constants via Bethe ansatz. It is found that ferromagnetic Kondo screening does occur in one case which confirms the Furusaki-Nagaosa conjecture while in the other case it does not, which we explain in a simple physical picture. The surface energy, the low temperature specific heat and the Pauli susceptibility induced by the impurity and thereby the Kondo temperature are derived explicitly. Typeset using REVTEX 1 With the development of nanofabrication techniques for quantum wires and the prediction of edge states in the quantum Hall effect, the interest in 1D electron systems has been renewed in the recent years. Such systems differ fundamentally from those in three dimensions, where the low energy properties can be described very well by the Landau Fermi liquid theory. In the presence of repulsive interaction, quasiparticle excitations are replaced in one dimension by collective excitations. [1,2] The low temperature properties of such systems are then described by the so-called Luttinger liquid theory. Considerable attention has been focused on the response of a Luttinger liquid to localized perturbations. An example is the exchange interaction between such a non-Fermi liquid and a localized magnetic impurity (Kondo problem in a Luttinger liquid). This problem was first considered by Lee and Toner, [3] who used abelian bosonization to map the problem onto a kink-gas action. The dependence of the Kondo temperature TK on the bare exchange J generically has a power-law dependence on the Kondo coupling J determined by the effective Luttinger coupling Kρ, and crosses over to the familiar exponential form for J large or Kρ → 1. Subsequently, a poor man’s scaling treatment on this problem was performed by Furusaki and Nagaosa. [4] They proposed the interesting conjecture that a Luttinger liquid supports a Kondo effect even if the bare exchange interaction is ferromagnetic. Moreover, they showed that the excess specific heat and Pauli susceptibility due to the Kondo impurity are different from those of a local Fermi liquid. There are a few exact results on the Kondo effect in a Luttinger liquid but no exact solution has been proposed to date. In the absence of electron-electron interaction (Luttinger gas), the problem can be mapped onto a standard 2-channel Kondo problem [5,6] but it is not clear, how these results are changed for a Luttinger liquid. A somewhat artificial model, where the propagation direction of electrons is correlated with their spin direction, does show both ferromagnetic and antiferromagnetic Kondo effects but its thermodynamics is Fermiliquid-like and apparently differs from the Furusaki-Nagaosa solution. [7] Finally, boundary conformal field theory has allowed the classification of all possibilities of critical behavior for a Luttinger liquid coupled to a Kondo impurity. [8] It turns out that there are just two possibilities, a local Fermi liquid with standard low-temperature thermodynamics or a nonFermi liquid with the anomalous scaling observed by Furusaki and Nagaosa. In this letter, we propose an exact solution to two cases of the Kondo problem in a Luttinger liquid. In one case, we find a ferromagnetic Kondo effect similar to Furusaki and Nagaosa [4] but with the thermodynamics of a local Fermi-liquid while in the other case, we do not find complete Kondo screening. The proper starting point to consider the Kondo problem in a fermion chain is the Hamiltonian H = H0 +Himp, H0 = −t N−1