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International Journal of Modern Physics B

ENTANGLEMENT IN THE ANISOTROPIC KONDO NECKLACE MODEL

Juan José Mendoza-Arenas, Roberto Franco, Jereson Silva-Valencia

Published Dec 20, 2010• 1 citation
Publisher

Abstract

We study the entanglement in the one-dimensional Kondo necklace model with exact diagonalization, calculating the concurrence as a function of the Kondo coupling J and an anisotropy η in the interaction between conduction spins, and we review some results previously obtained in the limiting cases η = 0 and 1. We observe that as J increases, localized and conduction spins get more entangled, while neighboring conduction spins diminish their concurrence; localized spins require a minimum concurrence between conduction spins to be entangled. The anisotropy η diminishes the entanglement for neighboring spins when it increases, driving the system to the Ising limit η = 1 where conduction spins are not entangled. We observe that the concurrence does not give information about the quantum phase transition in the anisotropic Kondo necklace model (between a Kondo singlet and an antiferromagnetic state), but calculating the von Neumann block entropy with the density matrix renormalization group in a chain of 100 sites for the Ising limit indicates that this quantity is useful for locating the quantum critical point.

DOI
10.1142/s0217979210055433

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We study the entanglement in the one-dimensional Kondo necklace model with exact diagonalization, calculating the concurrence as a function of the Kondo coupling J and an anisotropy η in the interaction between conduction spins, and we review some results previously obtained in the limiting cases η = 0 and 1. We observe that as J increases, localized and conduction spins get more entangled, while neighboring conduction spins diminish their concurrence; localized spins require a minimum concurrence between conduction spins to be entangled. The anisotropy η diminishes the entanglement for neighboring spins when it increases, driving the system to the Ising limit η = 1 where conduction spins are not entangled. We observe that the concurrence does not give information about the quantum phase transition in the anisotropic Kondo necklace model (between a Kondo singlet and an antiferromagnetic state), but calculating the von Neumann block entropy with the density matrix renormalization group in a chain of 100 sites for the Ising limit indicates that this quantity is useful for locating the quantum critical point.
"We use the density matrix renormalization group to calculate the von Neumann block entropy in the Kondo necklace model. We have included two anisotropies: η in the exchange interaction between conduction spins, and ∆ in the coupling between localized and conduction spins. Calculating the energy gap for the one-dimensional model at zero temperature, it was obtained that when η > 0, a quantum phase transition between Kondo singlet and antiferomagnetic states for finite couplings occurs, and as ∆ increases the Kondo singlet phase is favoured. Now we observe that the quantum critical points of the model can be identified by a maximum of the von Neumann block entropy, so entanglement measures can be useful for determining the phase diagram of the anisotropic Kondo necklace model."
We study the von Neumann block entropy in the Kondo necklace model for different anisotropies $\ensuremath{\eta}$ in the XY\mathit{XY}XY interaction between conduction spins using the density matrix renormalization group method. It was found that the block entropy presents a maximum for each $\ensuremath{\eta}$ considered, and, comparing it with the results of the quantum criticality of the model based on the behavior of the energy gap, we observe that the maximum block entropy occurs at the quantum critical point between an antiferromagnetic and a Kondo singlet state, so this measure of entanglement is useful for giving information about where a quantum phase transition occurs in this model. We observe that the block entropy also presents a maximum at the quantum critical points that are obtained when an anisotropy $\ensuremath{\Delta}$ is included in the Kondo exchange between localized and conduction spins; when $\ensuremath{\Delta}$ diminishes for a fixed value of $\ensuremath{\eta}$, the critical point increases, favoring the antiferromagnetic phase.
We analyze the one-dimensional Kondo necklace model, at zero temperature, with an anisotropy parameter $\ensuremath{\eta}$ in the interaction of the conduction chain, by means of the density matrix renormalization group. We calculate the energy gap and estimate the quantum critical points that separate a Kondo singlet state from an antiferromagnetic state, assuming a Kosterlitz-Thouless tendency. We also observe the correlation functions and the structure factors that support our critical points. The resulting phase diagram is presented and compared to that reported previously using Lanczos calculations. It is shown that the quantum critical points vary very slowly with $\ensuremath{\eta}$, but when $\ensuremath{\eta}$ approaches zero, they drop abruptly.
The Kondo-necklace model can describe magnetic low-energy limit of strongly correlated heavy fermion materials. There exist multiple energy scales in this model corresponding to each phase of the system. Here, we study quantum phase transition between the Kondo-singlet phase and the antiferromagnetic long-range ordered phase, and show the effect of anisotropies in terms of quantum information properties and vanishing energy gap. We employ the perturbative continuous unitary transformations approach to calculate the energy gap and spin-spin correlations for the model in the thermodynamic limit of one, two, and three spatial dimensions as well as for spin ladders. In particular, we show that the method, although being perturbative, can predict the expected quantum critical point, where the gap of low-energy spectrum vanishes, which is in good agreement with results of other numerical and Green's function analyses. In addition, we employ concurrence, a bipartite entanglement measure, to study the criticality of the model. Absence of singularities in the derivative of concurrence in two and three dimensions in the Kondo-necklace model shows that this model features multipartite entanglement. We also discuss crossover from the one-dimensional to the two-dimensional model via the ladder structure.
Abstract Using the density matrix renormalization group method, we study the one‐dimensional Kondo necklace model including two anisotropies: η in the XY interaction of the conduction chain, and Δ in the local Kondo coupling between localized and conduction spins. For different values of the anisotropies, we calculate the energy gap and estimate the quantum critical points in which a phase transition from antiferromagnetic to spin liquid states takes place, assuming a Kosterlitz–Thouless‐like behavior. For the case η=0\eta = 0η=0 , the system remains in a spin liquid state for the different Δ considered, but when η≠0\eta \ne 0η=0 , Δ markedly affects the location of the critical points. We also calculate correlation functions for a specific case, in order to support our gap results.
We have studied the effect of anisotropies on the quantum phase transition of the Kondo necklace model in dimensions D=1D=1D=1, 2, and 3. Both the anisotropy $\ensuremath{\delta}$ of the intersite interaction term and anisotropy $\ensuremath{\Delta}$ of the on-site Kondo interaction have been included. We use a bond operator method with constraints implemented in mean-field approximation. Starting from the paramagnetic phase we determine the critical ratio (t∕J)c{(t∕J)}_{c}(t∕J)c​ of the quantum critical point and associated scaling exponents of the Kondo-singlet gap. We show that in the case of easy-axis type anisotropy $\ensuremath{\delta}>1$ a qualitatively different behavior in comparison to the conventional Kondo necklace model with $(\ensuremath{\delta},\ensuremath{\Delta})=(0,1)$ appears. We have also obtained the antiferromagnetic order parameter in the long range ordered phase for t>tct>{t}_{c}t>tc​.