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arXiv:cond-mat/9511031 · Physical review. B, Condensed matter

Scaling theory of the Kondo screening cloud

Erik Schwartz Sørensen, Ian K. Affleck

Published Apr 1, 1996• 118 citations
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Abstract

A scaling form for the local susceptibility, derived from renormalization group arguments, is proposed. The scale over which the uniform part of this scaling form varies can be viewed as a definition of the Kondo "screening cloud" $\ensuremath{\sim}{\ensuremath{\xi}}_{K}$. The proposed scaling form interpolates between Ruderman-Kittel-Kasuya-Yosida (RKKY) results in the high-temperature limit $T\ensuremath{\gg}{T}_{K}$ and Fermi liquid results in the low-temperature long-distance limit $T\ensuremath{\ll}{T}_{K}$, $r\ensuremath{\gg}{\ensuremath{\xi}}_{K}$. The predicted form of the Knight shift is longer range at low temperatures, where the screening cloud has formed, than at high temperatures, where it has not. Using weak and strong coupling perturbation theory combined with large scale density matrix renormalization group results we study the validity of the finite size version of the scaling form at $T=0$. We explicitly extract a length scale proportional to the Kondo length scale ${\ensuremath{\xi}}_{K}$. The numerical results are in good agreement with the proposed scaling form and confirm the existence of the Kondo screening cloud.

DOI
10.1103/physrevb.53.9153

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A scaling form for the local susceptibility, derived from renormalization group arguments, is proposed. The scale over which the uniform part of this scaling form varies can be viewed as a definition of the Kondo "screening cloud" \ensuremath∼\ensuremathξK\ensuremath{\sim}{\ensuremath{\xi}}_{K}\ensuremath∼\ensuremathξK​. The proposed scaling form interpolates between Ruderman-Kittel-Kasuya-Yosida (RKKY) results in the high-temperature limit T\ensuremath≫TKT\ensuremath{\gg}{T}_{K}T\ensuremath≫TK​ and Fermi liquid results in the low-temperature long-distance limit T\ensuremath≪TKT\ensuremath{\ll}{T}_{K}T\ensuremath≪TK​, r\ensuremath≫\ensuremathξKr\ensuremath{\gg}{\ensuremath{\xi}}_{K}r\ensuremath≫\ensuremathξK​. The predicted form of the Knight shift is longer range at low temperatures, where the screening cloud has formed, than at high temperatures, where it has not. Using weak and strong coupling perturbation theory combined with large scale density matrix renormalization group results we study the validity of the finite size version of the scaling form at T=0T=0T=0. We explicitly extract a length scale proportional to the Kondo length scale \ensuremathξK{\ensuremath{\xi}}_{K}\ensuremathξK​. The numerical results are in good agreement with the proposed scaling form and confirm the existence of the Kondo screening cloud.
The existence of a length scale ${\ensuremath{\xi}}_{K}\ensuremath{\sim}1/{T}_{K}$ (with TK{T}_{K}TK​ the Kondo temperature) has long been predicted in quantum impurity systems. At low temperatures $T\ensuremath{\ll}{T}_{K}$, the standard interpretation is that a spin-12\frac{1}{2}21​ impurity is screened by a surrounding ``Kondo cloud'' of spatial extent ${\ensuremath{\xi}}_{K}$. We argue that renormalization group (RG) flow between any two fixed points (FPs) results in a characteristic length scale, observed in real space as a crossover between physical behavior typical of each FP. In the simplest example of the Anderson impurity model, three FPs arise, and we show that ``free orbital,'' ``local moment,'' and ``strong coupling'' regions of space can be identified at zero temperature. These regions are separated by two crossover length scales ${\ensuremath{\xi}}_{\text{LM}}$ and ${\ensuremath{\xi}}_{K}$, with the latter diverging as the Kondo effect is destroyed on increasing temperature through TK{T}_{K}TK​. One implication is that moment formation occurs inside the ``Kondo cloud'', while the screening process itself occurs on flowing to the strong coupling FP at distances $\ensuremath{\sim}{\ensuremath{\xi}}_{K}$. Generic aspects of the real-space physics are exemplified by the two-channel Kondo model, where ${\ensuremath{\xi}}_{K}$ now separates local moment and overscreening clouds.
Abstract The screening of an impurity spin in the Kondo model occurs over a characteristic length scale ξ K , that defines the size of the Kondo screening cloud or ‘mist’. The presence of such a length scale in experimental and numerical results is rather subtle. A consistent way to show the presence of the screening cloud is to demonstrate scaling in the spatial correlations in terms of the single variable, r / ξ K instead of the correlations depending on r and ξ K separately. Here we study the paradigmatic one channel Kondo model using a spin chain representation, with an impurity spin at one end of the chain coupled with a strength J K ′ . Using Fermi liquid theory combined with numerical results, we obtain new high precision estimates of the non-universal terms in the entanglement entropy which leads to a verification of the expected non-integer ground-state degeneracy, g. This then allows us to study the impurity contribution to the entanglement in detail. If the impurity coupling J K ′ is varied, a precise determination of ξ K can then be obtained. The length scale, ξ K , is then shown to characterize the scaling of both the uniform and alternating part of a measure of the magnetization of part of an odd length chain.
Measurements of the persistent current in a ring containing a quantum dot would afford a unique opportunity to finally detect the elusive Kondo screening cloud. We present the first large-scale numerical results on this controversial subject using exact diagonalization and density matrix renormalization group (RG). These extremely challenging numerical calculations confirm RG arguments for weak to strong coupling crossover with varying ring length and give results on the universal scaling functions. We also study, analytically and numerically, the important and surprising effects of particle-hole symmetry breaking.
We propose that real-space properties of the two-impurity Kondo model can be obtained from an effective spin model where two single-impurity Kondo spin chains are joined via an Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between the two impurity spins. We then use a density matrix renormalization group approach, valid in all ranges of parameters, to study its features using two complementary quantum-entanglement measures, the negativity and the von Neumann entropy. This nonperturbative approach enables us to uncover the precise dependence of the spatial extent ξ(K) of the Kondo screening cloud with the Kondo and RKKY couplings. Our results reveal an exponential suppression of the Kondo temperature T(K)~1/ξ(K) with the size of the effective impurity spin in the limit of large ferromagnetic RKKY coupling, a striking display of "Kondo resonance narrowing" in the two-impurity Kondo model. We also show how the antiferromagnetic RKKY interaction produces an effective decoupling of the impurities from the bulk already for intermediate strengths of this interaction, and, furthermore, exhibit how the non-Fermi liquid quantum critical point is signaled in the quantum entanglement between various parts of the system.
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.
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.