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Revista Mexicana de Física

Kondo necklace model with planar and local anisotropies: entanglement approach

Juan José Mendoza-Arenas, R. Franco, Jereson Silva-Valencia

Published Jan 1, 2012• 0 citations
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Abstract

"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."

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Block entropy and quantum phase transition in the anisotropic Kondo necklace model

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We study the von Neumann block entropy in the Kondo necklace model for different anisotropies $\ensuremath{\eta}$ in the $\mathit{XY}$ 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.

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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 $\eta = 0$ , the system remains in a spin liquid state for the different Δ considered, but when $\eta \ne 0$ , 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.

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"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 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 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.
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​.
We have studied the quantum phase transition between the antiferromagnetic and spin liquid phases for the two-dimensional anisotropic Kondo necklace model. The bond operator formalism has been implemented to transform the spin Hamiltonian to a bosonic one. We have used the Green's function approach including a hard core repulsion to find the low energy excitation spectrum of the model. The bosonic excitations become gapless at the quantum critical point where the phase transition from the Kondo singlet state to long range antiferromagnetic order takes place. We have studied the effect of both intersite $(\ensuremath{\delta})$ and local $(\ensuremath{\Delta})$ anisotropies on the critical point and on the critical exponent of the excitation gap in the paramagnetic phase. We have also compared our results with previous bond operator mean field calculations.