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Entanglement and Correlation in Kondo Impurity Systems

Sam Young Cho

Published Jan 1, 2006• 0 citations
Publisher

Abstract

Nanotechnology has made it possible to fabricate electronic devices in which it is possible to manipulate and tune quantum states. The paper studies quantum entanglement in Kondo impurity systems. Experimental realization occurs in pairs of quantum dots and in endohedral fullerenes inside carbon nanotubes. Competition of many-body effects plays a very important role in determining quantum entanglement of two localized impurity spins. The reduced density matrix shows that the two spins are in a Werner state with probability coefficients being a function of spin-spin correlation. It is found that to be entangled the two spins need a minimum nonzero anti-ferromagnetic correlation. How the entanglement in the whole system is shared was discussed.

DOI
10.1109/iconn.2006.340613

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Nanotechnology has made it possible to fabricate electronic devices in which it is possible to manipulate and tune quantum states. The paper studies quantum entanglement in Kondo impurity systems. Experimental realization occurs in pairs of quantum dots and in endohedral fullerenes inside carbon nanotubes. Competition of many-body effects plays a very important role in determining quantum entanglement of two localized impurity spins. The reduced density matrix shows that the two spins are in a Werner state with probability coefficients being a function of spin-spin correlation. It is found that to be entangled the two spins need a minimum nonzero anti-ferromagnetic correlation. How the entanglement in the whole system is shared was discussed.
We study the ground-state entanglement properties of nanostructured Kondo systems consisting of a pair of impurity spins coupled to a background of confined electrons. The competition between the Ruderman-Kittel-Kasuya-Yosida-like coupling and the Kondo effect determines the development of quantum correlations between the different parts of the system. A key element is the electronic filling due to confinement. An even electronic filling leads to results similar to those found previously for extended systems, where the properties of the reduced impurity spin subsystem are uniquely determined by the spin-correlation function defining a one-dimensional phase space. An odd filling, instead, breaks spin rotation symmetry unfolding a two-dimensional phase space showing rich entanglement characteristics as, e.g., the requirement of a larger amount of entanglement for the development of nonlocal correlations between impurity spins. We illustrate these results by numerical simulations of elliptic quantum corrals with magnetic impurities at the foci as a case study.
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.
Entanglement is a defining feature of quantum mechanics, and its relation to quantum criticality is of considerable current interest. Here we show that measurable spin and charge fluctuations provide an entanglement witness of quantum criticality in the two-impurity Kondo model, which has experimental realizations in terms of coupled quantum dots and magnetic impurities added to surfaces. We use density-matrix renormalization group and numerical renormalization group calculations to resolve the non-Fermi-liquid critical point separating two independently Kondo-screened impurities from an inter-impurity singlet and interpret it as a change in the dominant entanglement partner of each local moment: from entanglement of the local moment with an extended set of conduction-electron degrees of freedom to entanglement with the other impurity. This reorganization is accompanied by a singular response of the impurity-bath entanglement and the inter-impurity susceptibility. We show how the same structure is encoded in the quantum Fisher information of collective spin and charge operators at zero and finite temperatures, connecting the entanglement picture to experimentally accessible dynamical response functions. Our results establish impurity systems as controlled settings in which quantum-critical entanglement can be detected through measurable correlations, and provide further insight into the possibility of understanding heavy-fermion physics in terms of entanglement.
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.
Quantum dots comprise a type of quantum impurity system. The entanglement and coherence of quantum states are significantly influenced by the strong electron-electron interactions among impurities and their dissipative coupling with the surrounding environment. Competition between many-body effects and transfer couplings plays an important role in determining the entanglement among localized impurity spins. In this work, we employ the hierarchical-equations-of-motion approach to explore the entanglement of a strongly correlated double quantum dots system. The relation between the total system entropy and those of subsystems is also investigated.
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.