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Physical review. B, Condensed matter

Kondo effect in a host with fractional statistics: Absence of Kondo logarithms

Yupeng Wang, P. Schlottmann

Published Sep 10, 2001• 1 citation
Publisher

Abstract

By constructing the reflection Dunkl operator we derive several integrable models consisting of a boundary impurity coupled to an electron gas with interactions of the Calogero-Sutherland type. Some of these models were constucted previously using Lax-pair operators. The necessary condition of integrability imposes that the impurity potential has a form similar to that of the bulk interactions. Based on these results we conjecture that a Kondo impurity coupled to the host with long-range interactions of the ${1/r}^{2}$ type is also integrable. Using the asymptotic Bethe Ansatz we show that there are no Kondo logarithms, and depending on the coupling of the impurity to the host, the impurity spin can either be totally screened, partially screened, or unscreened. On the other hand, for a $1/{\mathrm{sinh}}^{2}(r)$ interaction potential a Kondo effect with logarithms is obtained.

DOI
10.1103/physrevb.64.125113

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By constructing the reflection Dunkl operator we derive several integrable models consisting of a boundary impurity coupled to an electron gas with interactions of the Calogero-Sutherland type. Some of these models were constucted previously using Lax-pair operators. The necessary condition of integrability imposes that the impurity potential has a form similar to that of the bulk interactions. Based on these results we conjecture that a Kondo impurity coupled to the host with long-range interactions of the 1/r2{1/r}^{2}1/r2 type is also integrable. Using the asymptotic Bethe Ansatz we show that there are no Kondo logarithms, and depending on the coupling of the impurity to the host, the impurity spin can either be totally screened, partially screened, or unscreened. On the other hand, for a 1/sinh2(r)1/{\mathrm{sinh}}^{2}(r)1/sinh2(r) interaction potential a Kondo effect with logarithms is obtained.
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 study the low-temperature behavior of a boundary magnetic impurity ${S}^{\ensuremath{'}}=1/2$ in an open ferromagnetic Takhatajian-Babujian spin-SSS chain. For antiferromagnetic Kondo coupling, it is shown via the Bethe ansatz solution that the impurity spin is always locked into the critical behavior of the bulk. At low temperatures, a local composite of spin-$(S\ensuremath{-}1/2)$ forms near the impurity site and its contribution to specific heat is of simple power law T1/2.{T}^{1/2}.T1/2. The absence of the overscreened Kondo effect is due to the large correlation length of host spins which is divergent near the quantum critical point.
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.
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.
In a metal, a magnetic impurity is fully screened by the conduction electrons at low temperature. In contrast, impurity moments coupled to spin-1 bulk bosons, such as triplet excitations in paramagnets, are only partially screened, even at the bulk quantum critical point. We argue that this difference is not due to the quantum statistics of the host particles but instead related to the structure of the impurity-host coupling, by demonstrating that frustrated magnets with bosonic spinon excitations can display a bosonic version of the Kondo effect. However, the Bose statistics of the bulk implies distinct behavior, such as a weak-coupling impurity quantum phase transition, and perfect screening for a range of impurity spin values. We discuss implications of our results for the compound Cs2CuCl4, as well as possible extensions to multicomponent bosonic gases.
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.