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arXiv:quant-ph/0208133 · Physical Review A

Entanglement in the one-dimensional Kondo necklace model

A. Saguia, M. S. Sarandy

Published Jan 23, 2003• 20 citations
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Abstract

We discuss the thermal and magnetic entanglement in the one-dimensional Kondo necklace model. First, we show how the entanglement naturally present at zero temperature is distributed among pairs of spins according to the strength of the two couplings of the chain, namely, the Kondo exchange interaction and the hopping energy. The effect of temperature and the presence of an external magnetic field are then investigated. We also discuss the adjustment of these variables in order to control the entanglement available in the system. In particular, it is indicated that there exists a critical magnetic field above which the entanglement undergoes a sharp variation, leading the ground state to a completely unentangled phase.

DOI
10.1103/physreva.67.012315

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We discuss the thermal and magnetic entanglement in the one-dimensional Kondo necklace model. First, we show how the entanglement naturally present at zero temperature is distributed among pairs of spins according to the strength of the two couplings of the chain, namely, the Kondo exchange interaction and the hopping energy. The effect of temperature and the presence of an external magnetic field are then investigated. We also discuss the adjustment of these variables in order to control the entanglement available in the system. In particular, it is indicated that there exists a critical magnetic field above which the entanglement undergoes a sharp variation, leading the ground state to a completely unentangled phase.
"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."
The magnetization process of one-dimensional Kondo necklace model with next-nearest-neighbor interaction has been studied by the use of exact-diagonalization method. The magnetization plateaus appear at zero ( m =0) and at half ( m =1/2) of the saturation magnetization. The ground state in the absence of a magnetic field is regarded as an aggregate of intra-atomic Kondo singlets. On the other hand, the spin configuration at the 1/2-plateau represents the intra-chain dimer (σ-dimer) phase with broken translational symmetry. The picture for spin configurations in the uniform magnetic fields other than m =0 and 1/2 are also given. We also determined the phase diagram for the ground state of the system in the magnetic field.
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.
We analyze several thermodynamic properties of the two-dimensional Kondo necklace using finite-temperature stochastic series expansion. In agreement with previous zero-temperature findings the model is shown to exhibit a quantum critical point (QCP), separating an antiferromagnetic from a paramagnetic dimerized state at a critical Kondo exchange-coupling strength ${J}_{c}\ensuremath{\approx}1.4$. We evaluate the temperature dependent uniform and staggered structure factors as well as the uniform and staggered susceptibilities and the local ``impurity'' susceptibility close to the QCP as well as in the ordered and quantum disordered phase. The crossover between the classical, renormalized classical, and quantum critical regime is analyzed as a function of temperature and Kondo coupling.
We study the low-energy sector of the Heisenberg necklace model. Using the field-theory methods, we estimate how the coupling of the electronic spins with the paramagnetic Kondo spins affects the overall spin dynamics and evaluate its dependence on a magnetic field. We are motivated by the experimental realizations of the spin-1/2 Heisenberg chains in SrCuO2{\mathrm{SrCuO}}_{2}SrCuO2​ and Sr2CuO3{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}Sr2​CuO3​ cuprates, which remain one-dimensional Luttinger liquids down to temperatures much lower than the in-chain exchange coupling JJJ. We consider the perturbation of the energy spectrum caused by the interaction $\ensuremath{\gamma}$ with nuclear spins (I=3/2I=3/2I=3/2) present on the same sites. We find that the resulting necklace model has a characteristic energy scale, $\mathrm{\ensuremath{\Lambda}}\ensuremath{\sim}{J}^{1/3}{(\ensuremath{\gamma}I)}^{2/3}$, at which the coupling between (nuclear) spins of the necklace and the spins of the Heisenberg chain becomes strong. This energy scale is insensitive to a magnetic field BBB. For ${\ensuremath{\mu}}_{B}B>\mathrm{\ensuremath{\Lambda}}$ we find two gapless bosonic modes that have different velocities, whose ratio at strong fields approaches a universal number, 2+1\sqrt{2}+12​+1.