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Journal of Physics A Mathematical and Theoretical

Entanglement in the scattering process by the spin impurity

Dong Yang, Shi-Jian Gu, Haibin Li

Published Nov 21, 2007• 9 citations
Publisher

Abstract

We study entanglement in the scattering processes by the spin-fixed impurity and the Kondo impurity, respectively. The spin-fixed impurity is employed as a filter to concentrate entanglement. One Kondo impurity can create entanglement between two noninteracting particles and one scattered particle can do so between two noninteracting Kondo impurities.

DOI
10.1088/1751-8113/40/49/018

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arXiv:cond-mat/0703293•Physical Review B•Jun 15, 2007•51 citations
We study entanglement in the scattering processes by the spin-fixed impurity and the Kondo impurity, respectively. The spin-fixed impurity is employed as a filter to concentrate entanglement. One Kondo impurity can create entanglement between two noninteracting particles and one scattered particle can do so between two noninteracting Kondo impurities.
In order to quantify quantum entanglement in two-impurity Kondo systems, we calculate the concurrence, negativity, and von Neumann entropy. The entanglement of the two Kondo impurities is shown to be determined by two competing many-body effects, namely the Kondo effect and the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, III. Due to the spin-rotational invariance of the ground state, the concurrence and negativity are uniquely determined by the spin-spin correlation between the impurities. It is found that there exists a critical minimum value of the antiferromagnetic correlation between the impurity spins which is necessary for entanglement of the two impurity spins. The critical value is discussed in relation with the unstable fixed point in the two-impurity Kondo problem. Specifically, at the fixed point there is no entanglement between the impurity spins. Entanglement will only be created [and quantum information processing (QIP) will only be possible] if the RKKY interaction exchange energy, III, is at least several times larger than the Kondo temperature, TK{T}_{K}TK​. Quantitative criteria for QIP are given in terms of the impurity spin-spin correlation.
Based on Yosida's ground state of the single-impurity Kondo Hamiltonian, we study three kinds of entanglement between an impurity and conduction electron spins. First, it is shown that the impurity spin is maximally entangled with all the conduction electrons. Second, a two-spin density matrix of the impurity spin and one conduction electron spin is given by a Werner state. We find that the impurity spin is not entangled with one conduction electron spin even within the Kondo screening length ${\ensuremath{\xi}}_{K}$, although there is the spin-spin correlation between them. Third, we show the density matrix of two conduction electron spins is nearly the same as that of a free electron gas. The single impurity does not change the entanglement structure of the conduction electrons in contrast to the dramatic change in electrical resistance.
Competition between the Kondo effect and Ruderman-Kittel-Kasuya-Yosida interaction in the two-impurity Kondo problem can be phenomenologically described by the Rasul-Schlottmann spin model. We revisit this model from the quantum entanglement perspective by calculating both the inter-impurity entanglement and the local Kondo entanglement, the latter being the entanglement between a local magnetic impurity and its spatially nearby conduction electron. A groundstate phase diagram is derived and a discontinuous breakdown of the local Kondo entanglement is found at the singular point, associated concomitantly with a jump in the inter-impurity entanglement. An entanglement monogamy holds in the whole phase diagram. Our results identify the important role of the frustrated cross-coupling and demonstrate the local characteristic of the quantum phase transition in the two-impurity Kondo problem. The implications of these results for Kondo lattices and quantum information processing are also briefly discussed.
We use the framework setup recently to compute nonperturbatively inelastic scattering from quantum impurities [G. Zar\'and et al., Phys. Rev. Lett. 93, 107204 (2004)] to study the energy dependence of the single-particle SSS matrix and the inelastic scattering cross section for a number of quantum impurity models. We study the case of the spin S=1∕2S=1∕2S=1∕2 two-channel Kondo model, the Anderson model, and the usual S=1∕2S=1∕2S=1∕2 single-channel Kondo model. We discuss the difference between non-Fermi-liquid and Fermi-liquid models and study how a crossover between the non-Fermi-liquid and Fermi-liquid regimes appears in the case of channel anisotropy for the S=1∕2S=1∕2S=1∕2 two-channel Kondo model. We show that for the most elementary non-Fermi-liquid system, the two-channel Kondo model, half of the scattering remains inelastic even at the Fermi energy. Details of the derivation of the reduction formulas and a simple path integral approach to connect the TTT matrix to local correlation functions are also presented.