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Physical review. B, Condensed matter

Absence of significant conduction-electron polarization in the Kondo effect

J. H. Lowenstein

Published Apr 1, 1984• 12 citations
Publisher

Abstract

Within the Bethe-ansatz formulation of the $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}$ Kondo model, it is found that the low-temperature suppression of the impurity's contribution to the magnetization is due entirely to many-body correlations, with zero net polarization of the conduction electrons. This result agrees with the prediction of conventional perturbation theory (including use of the renormalization-group equation) in the extreme scaling regime where $O(\frac{1}{\mathrm{ln}\mathcal{D}})$ terms ($\mathcal{D}$ is the ultraviolet cutoff) are neglected. The Kondo model with arbitrary impurity magnetic moment ${\ensuremath{\mu}}_{0}$ is discussed using both perturbative and nonperturbative methods. It is found that in the infinite-cutoff limit, the impurity's contribution to thermodynamic quantities is independent of the electron's magnetic moment $\ensuremath{\mu}$, and so one can make use of the known results for $\ensuremath{\mu}={\ensuremath{\mu}}_{0}$.

DOI
10.1103/physrevb.29.4120

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Within the Bethe-ansatz formulation of the spin\ensuremath−12\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}spin\ensuremath−21​ Kondo model, it is found that the low-temperature suppression of the impurity's contribution to the magnetization is due entirely to many-body correlations, with zero net polarization of the conduction electrons. This result agrees with the prediction of conventional perturbation theory (including use of the renormalization-group equation) in the extreme scaling regime where O(1lnD)O(\frac{1}{\mathrm{ln}\mathcal{D}})O(lnD1​) terms (D\mathcal{D}D is the ultraviolet cutoff) are neglected. The Kondo model with arbitrary impurity magnetic moment \ensuremathμ0{\ensuremath{\mu}}_{0}\ensuremathμ0​ is discussed using both perturbative and nonperturbative methods. It is found that in the infinite-cutoff limit, the impurity's contribution to thermodynamic quantities is independent of the electron's magnetic moment \ensuremathμ\ensuremath{\mu}\ensuremathμ, and so one can make use of the known results for \ensuremathμ=\ensuremathμ0\ensuremath{\mu}={\ensuremath{\mu}}_{0}\ensuremathμ=\ensuremathμ0​.
The bosonized Kondo Hamiltonian is obtained by comparing the long-time limit of the reduced grand partition function for the impurity spin. We analyze this bosonized Kondo Hamiltonian at zero temperature and derive an effective Hamiltonian describing the effective interaction between the conduction electrons via an impurity-spin scattering. The infrared divergences encountered in the conventional perturbation theory are caused by this growth effective coupling at the low-energy limit. With the Bogoliubov transformation, we have developed a perturbation approach for this effective Hamiltonian and investigated the nontrivial ferromagnetic-antiferromagnetic crossover of the impurity spin. The critical condition derived here is in agreement with the renormalization-group numerical result. In particular, the ground-state wave function and its excitation spectrum of the conduction electrons are also obtained. Moreover, this effective Hamiltonian may be mapped onto a modified quantum sine-Gordon model. Paralleling the renormalization-group theory of the quantum sine-Gordon model, we straightforwardly reproduce previous results and derive the higher-order terms and a new universal correction in the flow equations.
The Kondo problem, which describes the interaction of a spin sss magnetic impurity with a free Fermi gas, is a classic example of strongly coupled physics. Historically, the problem has been solved by Wilson's numerical renormalization group and later by Bethe ansatz. In this paper, we present an alternate analytic solution of the Kondo problem that combines an expansion in 1/s1/s1/s with the renormalization group. We study both the case of an impurity interacting with a single channel K=1K=1K=1 of fermions in the s→∞s \rightarrow \inftys→∞ limit and the case with KKK channels in the double-scaling limit K→∞K \rightarrow \inftyK→∞, s→∞s \to \inftys→∞, K/sK/sK/s fixed. Our approach allows us to describe analytically intermediate scales of the Kondo problem at large sss and compute thermodynamic observables such as the impurity entropy and susceptibility. We find these observables to agree with the Bethe ansatz results. We also compute the impurity spectral function, finite temperature resistivity and the Kondo screening cloud profile, properties that are not easily accessible from Bethe ansatz. Notably, in the regime K>2sK > 2sK>2s we access the "non-Fermi-liquid" overscreened fixed point of the multichannel Kondo problem.
A magnetic moment in a metal or in a quantum dot is, at low temperatures, screened by the conduction electrons through the mechanism of the Kondo effect. This gives rise to spin-spin correlations between the magnetic moment and the conduction electrons, which can have a substantial spatial extension. We study this phenomenon, the so-called Kondo cloud, by means of the density matrix renormalization group method for the case of the single-impurity Anderson model. We focus on the question whether the Kondo screening length, typically assumed to be proportional to the inverse Kondo temperature, can be extracted from the spin-spin correlations. For several mechanisms -- the gate potential and a magnetic field -- that destroy the Kondo effect, we investigate the behavior of the screening cloud induced by these perturbations.
A magnetic moment in a metal or in a quantum dot is, at low temperatures, screened by the conduction electrons through the mechanism of the Kondo effect. This gives rise to spin-spin correlations between the magnetic moment and the conduction electrons, which can have a substantial spatial extension. We study this phenomenon, the so-called Kondo cloud, by means of the density matrix renormalization group method for the case of the single-impurity Anderson model. We focus on the question whether the Kondo screening length, typically assumed to be proportional to the inverse Kondo temperature, can be extracted from the spin-spin correlations. For several mechanisms---the gate potential and a magnetic field---which destroy the Kondo effect, we investigate the behavior of the screening cloud induced by these perturbations.
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.