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arXiv:cond-mat/9506104 · Physical Review Letters

Exact Perturbative Solution of the Kondo Problem

Paul Fendley, Hubert Saleur

Published Dec 11, 1995• 34 citations
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Abstract

We explicitly evaluate the infinite series of integrals that appear in the "Anderson-Yuval" reformulation of the anisotropic Kondo model in terms of a one-dimensional Coulomb gas. We do this by developing a general approach relating the anisotropic Kondo model of arbitrary spin with the boundary sine-Gordon model, which describes an impurity in a Luttinger liquid and tunneling in the fractional quantum Hall effect. The Kondo solution then follows from the exact perturbative solution of the latter model in terms of Jack polynomials.

DOI
10.1103/physrevlett.75.4492

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We explicitly evaluate the infinite series of integrals that appear in the "Anderson-Yuval" reformulation of the anisotropic Kondo model in terms of a one-dimensional Coulomb gas. We do this by developing a general approach relating the anisotropic Kondo model of arbitrary spin with the boundary sine-Gordon model, which describes an impurity in a Luttinger liquid and tunneling in the fractional quantum Hall effect. The Kondo solution then follows from the exact perturbative solution of the latter model in terms of Jack polynomials.
We develop a unified theoretical framework for the anisotropic Kondo model and the boundary sine-Gordon model. They are both boundary integrable quantum field theories with a quantum-group spin at the boundary which takes values, respectively, in standard or cyclic representations of the quantum groupSU(2) q. This unification is powerful, and allows us to find new results for both models. For the anisotropic Kondo problem, we find exact expressions (in the presence of a magnetic field) for all the coefficients in the Anderson-Yuval perturbative expansion. Our expressions hold initially in the very anisotropic regime, but we show how to continue them beyond the Toulouse point all the way to the isotropic point using an analog of dimensional regularization. The analytic structure is transparent, involving only simple poles which we determine exactly, together with their residues. For the boundary sine-Gordon model, which describes an impurity in a Luttinger liquid, we find the nonequilibrium conductance for all values of the Luttinger coupling. This is an intricate computation because the voltage operator and the boundary scattering do not commute with each other.
We show that the problem of impurity tunneling in a Luttinger liquid of electrons with spin is solvable in the spin isotropic case ${(g}_{\ensuremath{\sigma}}=2$, ${g}_{\ensuremath{\rho}}$ arbitrary). The resulting integrable model is similar to a two-channel anisotropic Kondo model, but with the impurity spin in a ``cyclic representation'' of the quantum algebra su(2)q\mathrm{su}{(2)}_{q}su(2)q​ associated with the anisotropy. Using exact, nonperturbative techniques we study the renormalization-group flow, and compute the dc conductance. As expected from the analysis of Kane and Fisher we find that the IR fixed point corresponds to two separate leads. We also prove an exact duality between the UV and IR expansions of the current at vanishing temperature.
To study the impurity potential effect to the Kondo problem in a Luttinger liquid, we introduce a model of ferromagnetic single-impurity Kondo problem in a multi-band one-dimensional δ-potential Fermi gas, which is exactly solvable for a special Kondo coupling constant via Bethe ansatz. It is found that the boundary potential modifies the local moment seriously due to the charge-spin coupling and the strong backward scattering off the impurity, and thus strongly affects the fixed point of the system. Generally, their is a residual entropy for the ground state which depends on the strength of the boundary potential. In addition, the tunneling effect through the impurity and the breakdown of the open boundary induced by the impurity are also discussed.
We propose a series of models with inverse-square interactions, for which the ground states have an exact closed form, that describe one localized spin-1/21/21/2 Kondo impurity coupled to Luttinger liquids of itinerant bosons or fermions in the continuum or on a lattice. The model Hamiltonians contain two parts: a two-component Calogero-Sutherland model with an open boundary in the continuum or a long-range $t\text{\ensuremath{-}}J$-type model on the lattice, and long-range Kondo couplings between the impurity and the particles. The wave functions of the ground states in a certain basis have Jastrow product forms with powers depending on the particle statistics. It is shown that the wave functions can be expressed as chiral correlators of certain conformal field theories, which reveals that the particles are fractionalized to fully screen the Kondo impurity. This insight would be very useful in future studies on the very difficult problem of Kondo physics with interacting baths.
We analyze the structure of the perturbation expansion of the general multichannel Kondo model with channel anisotropic exchange couplings and in the presence of an external magnetic field, generalizing to this case the Anderson-Yuval technique. For two channels, we are able to map the Kondo model onto a generalized resonant-level model. Limiting cases in which the equivalent resonant-level model is solvable are identified. The solution correctly captures the properties of the two-channel Kondo model, and also allows an analytic description of the crossover from the non-Fermi-liquid to the Fermi-liquid behavior caused by the channel anisotropy.
We propose an integrable Kondo problem in a one-dimensional $t\ensuremath{-}J$ model. With the open boundary condition of the wave functions at the impurity sites, the model can be exactly solved via Bethe ansatz for a set of JL,R{J}_{L,R}JL,R​ (Kondo coupling constants) and VL,R{V}_{L,R}VL,R​ (impurity potentials) parametrized by a single parameter ccc. The integrable value of JL,R{J}_{L,R}JL,R​ runs from negative infinity to positive infinity, which allows us to study both the ferromagnetic Kondo problem and the antiferromagnetic Kondo problem in a strongly correlated electron system. Generally, there is a residual entropy for the ground state, which indicates a typical non-Fermi liquid behavior.