The screening of an impurity spin by conduction electrons is associated with the formation of a large Kondo screening cloud, of size ξ K . We study the quantum entanglement between a region of size r surrounding the impurity and the rest of the sample (of total size R ) using density matrix renormalization group and analytic methods. The corresponding ‘impurity entanglement entropy’, S imp , is shown to be a universal scaling function of r /ξ K and r / R . We calculate this universal function using Fermi liquid theory in the regime ξ K ≪ r .The screening of an impurity spin by conduction electrons is associated with the formation of a large Kondo screening cloud, of size ξ K . We study the quantum entanglement between a region of size r surrounding the impurity and the rest of the sample (of total size R ) using density matrix renormalization group and analytic methods. The corresponding ‘impurity entanglement entropy’, S imp , is shown to be a universal scaling function of r /ξ K and r / R . We calculate this universal function using Fermi liquid theory in the regime ξ K ≪ r .

arXiv:cond-mat/0606705Journal of Statistical Mechanics Theory and ExperimentJan 5, 200752 citations

Numerical results in the Kondo problem

P. P. TANS, Ger Vertogen, J.G.J. Ypma +1 more

Physics Letters AApr 1, 19730 citations

Kondo screening cloud and charge staircase in one-dimensional mesoscopic devices

Rodrigo G. Pereira, Nicolas Laflorencie, Ian K. Affleck +1 more

We propose that the finite size of the Kondo screening cloud, ${\ensuremath{\xi}}_{K}$, can be probed by measuring the charge quantization in a one-dimensional system coupled to a small quantum dot. When the chemical potential $\ensuremath{\mu}$ in the system is varied at zero temperature, one should observe charge steps whose locations are at values of $\ensuremath{\mu}$ that are controlled by the Kondo effect when the system size $L$ is comparable to ${\ensuremath{\xi}}_{K}$. We show that, if the standard Kondo model is used, the ratio between the widths of the Coulomb blockade valleys with odd or even number of electrons is a universal scaling function of ${\ensuremath{\xi}}_{K}∕L$. If we take into account electron-electron interactions in a single-channel wire, this ratio also depends on the parameters of the effective Luttinger model; in addition, the scaling is weakly violated by a marginal bulk interaction. For the geometry of a quantum dot embedded in a ring, we show that the dependence of the charge steps on a magnetic flux through the ring is controlled by the size of the Kondo screening cloud.We propose that the finite size of the Kondo screening cloud, ${\ensuremath{\xi}}_{K}$, can be probed by measuring the charge quantization in a one-dimensional system coupled to a small quantum dot. When the chemical potential $\ensuremath{\mu}$ in the system is varied at zero temperature, one should observe charge steps whose locations are at values of $\ensuremath{\mu}$ that are controlled by the Kondo effect when the system size $L$ is comparable to ${\ensuremath{\xi}}_{K}$. We show that, if the standard Kondo model is used, the ratio between the widths of the Coulomb blockade valleys with odd or even number of electrons is a universal scaling function of ${\ensuremath{\xi}}_{K}∕L$. If we take into account electron-electron interactions in a single-channel wire, this ratio also depends on the parameters of the effective Luttinger model; in addition, the scaling is weakly violated by a marginal bulk interaction. For the geometry of a quantum dot embedded in a ring, we show that the dependence of the charge steps on a magnetic flux through the ring is controlled by the size of the Kondo screening cloud.

arXiv:0801.4166Physical Review BMar 19, 200831 citations

Space-Time and Scaling Techniques in the Kondo Problem

Philip Warren Anderson

World Scientific series in 20th century physicsJan 1, 20050 citations

Scaling theory of the Kondo screening cloud

Erik Schwartz Sørensen, Ian K. Affleck

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.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.

arXiv:cond-mat/9511031Physical review. B, Condensed matterApr 1, 1996118 citations

L

is comparable to ${\ensuremath{\xi}}_{K}$. We show that, if the standard Kondo model is used, the ratio between the widths of the Coulomb blockade valleys with odd or even number of electrons is a universal scaling function of ${\ensuremath{\xi}}_{K}∕L$. If we take into account electron-electron interactions in a single-channel wire, this ratio also depends on the parameters of the effective Luttinger model; in addition, the scaling is weakly violated by a marginal bulk interaction. For the geometry of a quantum dot embedded in a ring, we show that the dependence of the charge steps on a magnetic flux through the ring is controlled by the size of the Kondo screening cloud.

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.