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arXiv:cond-mat/9809093 · Physical Review Letters

Hidden Kondo Effect in a Correlated Electron Chain

Andrei A. Zvyagin, Henrik Johannesson

Published Sep 28, 1998• 24 citations
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Abstract

We develop a general Bethe ansatz formalism for diagonalizing an integrable model of a magnetic impurity of arbitrary spin coupled ferro- or antiferromagnetically to a chain of interacting electrons. The method is applied to an open chain, with the exact solution revealing the existence of a ``hidden'' Kondo effect driven by forward electron scattering off the impurity. We argue that the so-called ``operator reflection matrices'' proposed in recent Bethe ansatz studies of related models emulate only forward electron-impurity scattering, which may explain the absence of complete Kondo screening for certain values of the impurity-electron coupling in these models.

DOI
10.1103/physrevlett.81.2751

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We develop a general Bethe ansatz formalism for diagonalizing an integrable model of a magnetic impurity of arbitrary spin coupled ferro- or antiferromagnetically to a chain of interacting electrons. The method is applied to an open chain, with the exact solution revealing the existence of a hidden'' Kondo effect driven by forward electron scattering off the impurity. We argue that the so-called operator reflection matrices'' proposed in recent Bethe ansatz studies of related models emulate only forward electron-impurity scattering, which may explain the absence of complete Kondo screening for certain values of the impurity-electron coupling in these models.
It is well known that the free electron Kondo problem can be described by a one-dimensional (1D) model because only the s-wave part of the electronic wavefunction is affected by the Kondo coupling. Moreover, since only the spin degrees of freedom are involved in the Kondo interaction, and due to spin–charge separation in 1D, the universal low energy long distance physics of the Kondo model also arises when a magnetic impurity is coupled to the end of a gapless antiferromagnetic J 1 –J 2 spin- chain, where J 1 ( J 2 ) is the (next) nearest neighbor coupling. Experimental realizations of such spin chain models are possible and, using various analytical and numerical techniques, we present a detailed and quantitative comparison between the usual free electron Kondo model and such spin chain versions of the Kondo problem. For the gapless J 1 –J 2 spin chain two cases are studied, with zero next nearest neighbor coupling, J 2 = 0, and with a critical second-neighbor coupling, J 2 = J 2 c . We first focus on the spin chain impurity model with a critical second-neighbor antiferromagnetic exchange where a bulk marginal coupling present in the spin chain model for J 2 < J 2 c vanishes . There, the usual Kondo physics is recovered in the spin chain model in the low energy regime (up to negligible corrections, dropping as powers of inverse length or energy). At J 2 c the spin chain model is not exactly solvable and we demonstrate the equivalence to the Kondo problem by comparing density matrix renormalization group calculations on the frustrated spin chain model with exact Bethe ansatz calculations of the electronic Kondo problem. We then analyze the nearest neighbor model ( J 2 = 0) where a new kind of Kondo effect occurs due to the presence of the bulk marginal coupling. This marginal coupling slightly alters the β function for the Kondo coupling, leading to a slower variation of the Kondo temperature, T K , with the bare Kondo coupling. In their exact Bethe ansatz solution of this spin chain impurity model ( J 2 = 0), Frahm and Zvyagin noted this relation as well as the connection to the Kondo problem. Here, by numerically solving the Bethe ansatz equations we provide further evidence for the connection to Kondo physics and in addition we present low temperature quantum Monte Carlo results for the impurity susceptibility that further support this connection.
We propose an integrable model of the spin-1/2 Heisenberg chain coupled to two impurity moments. With the open boundary conditions at the impurity sites, the model can be exactly solved for arbitrary impurity spin and arbitrary exchange constants between the bulk and the impurities. The absence of redundant terms in the Hamiltonian makes the model very reasonable. The Hamiltonian is diagonalized via algebraic Bethe ansatz. It is found that the impurity spins can only be screened (partially for S>1/2S>1/2S>1/2) for antiferromagnetic coupling between the impurity and the bulk. Otherwise the impurity spins cannot be screened. The residual entropy of the ground state and the Kondo temperature are also derived explicitly based on the Bethe ansatz and the local Fermi-liquid theory.
To specify the difference between the one-dimensional Kondo problem and the three-dimensional Kondo problem, we consider a model which describes the interaction between one-dimensional free electrons and a magnetic impurity. Bethe ansatz exact results and scaling analysis show that if J F > -| J B |, the Kondo screening occurs. There is no ferromagnetic Kondo screening in such a system but the long range antiferromagnetic Kondo coupling is relevant. As soon as the Coulomb interaction is introduced, the crossovers between the scaling trajectories in the parameter region J F < -| J B | and the trajectories in other parameter regions arise and ferromagnetic Kondo screening may occur.
Integrable Kondo impurities in the one-dimensional supersymmetric U model of strongly correlated electrons are studied by means of the boundary graded quantum inverse scattering method. The boundary K -matrices depending on the local magnetic moments of the impurities are presented as non-trivial realizations of the reflection equation algebras in an impurity Hilbert space. Furthermore, the model Hamiltonian is diagonalized and the Bethe ansatz equations are derived. It is interesting to note that our model exhibits a free parameter in the bulk Hamiltonian but no free parameter exists on the boundaries. This is in sharp contrast to the impurity models arising from the supersymmetric t-J and extended Hubbard models where there is no free parameter in the bulk but there is a free parameter on each boundary.
We study the problem of a boundary magnetic impurity coupled with the solvable $t\ensuremath{-}J$ chain. This model provides a good starting point to understand the Kondo problem in a Luttinger liquid as well as in a strongly correlated host. As the Kondo coupling constant Ji{J}_{i}Ji​ may take arbitrary values without breaking the integrability, we can study the ferromagnetic and antiferromagnetic Kondo problems simultaneously. It is shown that the boundary coupling generally splits the impurity spin into two effective spins and induces the interaction-dependent residual entropy. The absence of Kondo screening in an antiferromagnetic Kondo coupling regime is found, which indicates a genuine competing effect between the Kondo coupling Ji{J}_{i}Ji​ and the impurity potential Vi.{V}_{i}.Vi​. A local Landau-Luttinger liquid description is proposed to calculate the specific heat of the impurity.