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Modern Physics Letters B

PAIRWISE ENTANGLEMENT IN THE TETRAHEDRAL LATTICE WITH HEISENBERG INTERACTIONS

Ming‐Liang Hu

Published Dec 20, 2008• 3 citations
Publisher

Abstract

By using the concept of negativity, we determine entanglement features of the tetrahedral lattice with Heisenberg interactions. It is found that except [Formula: see text] for the case of (1/2-1/2-1-1) mixed spins, there is no pairwise entanglement between any two spins at any temperature when B = 0. When B increases after B = 0, the ground-state negativities exhibit square wave or stepwise structures, and experience sharply descents to lower values at some energy-level crossing points. For thermal entanglement, the negativities exhibit ridge-like or staircase structures as B increases, and there are regions in the B–T plane where an increase of temperature may help to enhance the entanglement.

DOI
10.1142/s0217984908017588

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By using the concept of negativity, we determine entanglement features of the tetrahedral lattice with Heisenberg interactions. It is found that except [Formula: see text] for the case of (1/2-1/2-1-1) mixed spins, there is no pairwise entanglement between any two spins at any temperature when B = 0. When B increases after B = 0, the ground-state negativities exhibit square wave or stepwise structures, and experience sharply descents to lower values at some energy-level crossing points. For thermal entanglement, the negativities exhibit ridge-like or staircase structures as B increases, and there are regions in the B–T plane where an increase of temperature may help to enhance the entanglement.
The bipartite entanglement of the two- and three-spin Heisenberg model was investigated by using the concept of negativity. It is found that for the ground-state entanglement of the two-spin model, the negativity always decreases as B increases if Δ γ-1, while for that of the three-spin model, the negativity exhibits square wave structures if γ=0 or Δ=0. For thermal states, there are two areas showing entanglement, namely, the main region and the sub-region. The main region exists only when Δ > Δc (Δc = γ-1 and (γ2-1)/2 for the 2- and 3-spin model respectively) and extends in terms of B and T as Δ increases, while the sub-region survives only when γ ≠ 0 and shrinks in terms of B and T as Δ increases.
The pairwise entanglement is exactly examined in the spin-1/2 Ising-Heisenberg double-tetrahedral chain with different Landé g-factors of the Ising and Heisenberg spins at zero and finite temperatures.It is shown that the phenomenon present in quantum non-chiral ground states is twice as strong as in quantum chiral ground states and that it gradually diminishes with increasing temperature until it completely vanishes at a certain threshold temperature.It is also demonstrated that the strong magnetic field maintains a weak thermal entanglement quite far from the saturation field, although the ground state is non-entangled.
The entanglement properties of a three-spin XXZ Heisenberg chain with three-spin interaction are studied by means of concurrence of pairwise entanglement. We show that ground-state pairwise entanglement, pairwise thermal entanglement, or quantum phase transition is not present in antiferromagnetic spin chain. For the ferromagnetic case, quantum phase transition takes place at Δ = 1 for anisotropic interaction and at some values of three-spin coupling strength, and pairwise thermal entanglement increases when the value of J / T increases and with anisotropic interaction and three-spin interaction decrease. In addition, we find that increasing the anisotropic interaction and the three-spin interaction will decrease critical temperature.
The entanglement characteristics of the three-qutrit spin-one Heisenberg chain in the presence of one impurity site is investigated by using the concept of negativity. The results show that the existence of impurity can lead to an enhancement of both bipartite and pairwise entanglement for certain ranges of the impurity parameter, and the amount of entanglement can be controlled and tuned by varying the impurity parameter J1. The results also show that the thermal fluctuations always suppress the entanglement and as a result there exists a threshold temperature Tth after which the entanglement vanishes. Tth increases as J1 increases except the case for that of the pairwise negativity [Formula: see text] which measures entanglement between the two norm spins in the chain.
The zero-temperature pairwise entanglement is investigated in the antiferromagnetic Heisenberg chain with one impure spin s. The impurity location is varied from the open end to a mediate site of the spin chain. It is found that the pairwise entanglement is proportional to 1/s. This means that the entanglement will disappear in the classical systems as s → ∞. Compared to the isotropic Heisenberg chain without impurity, the efficient control of the spin coupling between the impurity spin and nearest-neighbor spins can enhance the pairwise entanglement.
We have studied the symmetric and non-symmetric pairwise ground state and thermal entanglement in three-qubits system. We have considered the anisotropic Heisenberg (XXZ) model in the presence of Dzyaloshinskii–Moriya (DM) interaction in addition to the Ising model in a magnetic field with DM interaction. We have found that the increment of DM interaction and magnetic field can enhance and reduce the entanglement of the system. We have shown that the non-symmetric pairwise state has higher value concurrence and critical temperature (above which the entanglement vanishes) than the symmetric pairwise one. For the negative anisotropy, the non-symmetric entanglement is a monotonic function of DM interaction while for positive anisotropy, it has a maximum versus DM parameter and vanishes for larger values of DM interaction. The conditions for the existence of thermal entanglement are discussed in detail. The most remarkable result appears at zero temperature where the three-qubits ground state entanglement of the system (in spite of two-qubits counterpart) shows the fingerprint of the quantum phase transition for a system of infinite number of qubits.