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Modern Physics Letters B

IMPURITY ENTANGLEMENT IN THE OPEN-ENDED HEISENBERG CHAINS

Ming‐Liang Hu

Published Nov 20, 2008• 6 citations
Publisher

Abstract

By using the concept of concurrence, we study pairwise entanglement between the two end spins in the open-ended Heisenberg XXX and XY chains up to ten spins. The results show that by introducing two boundary impurities, one can obtain maximum entanglement at the limit of the impurity parameter |J1| ≪ J for the even-number qubits. When |J1/J| > 0, the entanglement always decreases with the increase in the absolute value of J1/J, and for the Heisenberg XXX chain, C disappears when J1/J exceeds a certain critical point Jic, and attains an asymptotic value C0 when |J1| ≫ J(J1 < 0), while for the Heisenberg XY chain, C always disappears when |J1/J| exceeds a certain critical point Jic. Both C0 and Jic decrease with the increase of the length of the chain.

DOI
10.1142/s0217984908017357

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By using the concept of concurrence, we study pairwise entanglement between the two end spins in the open-ended Heisenberg XXX and XY chains up to ten spins. The results show that by introducing two boundary impurities, one can obtain maximum entanglement at the limit of the impurity parameter |J1| ≪ J for the even-number qubits. When |J1/J| > 0, the entanglement always decreases with the increase in the absolute value of J1/J, and for the Heisenberg XXX chain, C disappears when J1/J exceeds a certain critical point Jic, and attains an asymptotic value C0 when |J1| ≫ J(J1 < 0), while for the Heisenberg XY chain, C always disappears when |J1/J| exceeds a certain critical point Jic. Both C0 and Jic decrease with the increase of the length of the chain.
We investigate the entanglement of the three-qubit Heisenberg XXX chain in the presence of impurity and obtain the analytical expressions of the concurrence C.It is found that for impurity entanglement,C appears only when J1 J for J 0,and J1 0 for J 0,and in these two regions C increases with the increase of J1,so is the critical temperature Tc.When J1 |J|,C reaches its maximum value 0.5 and Tc reaches the asymptotic value Tc = 3.41448J1.For entanglement between the normal lattices,C appears only when J 0 and 2J J1 J,and initially increases with the increase of J1 and arrives at the maximum value Cmax =(e4JIT-3)/(e4JIT+3)before it decays to zero gradually,so is the critical temperature Tc with,however,the maximum value Tcmax = 4J/ln3.
We derive the analytic expression of the concurrence in the quantum Heisenberg XYZ model and discuss the influence of parameters J , Δ and Γ on the concurrence. By choosing different values of Γ and Δ, we obtain the XX , XY , XXX and XXZ chains. The concurrence decreases with increasing temperature. When T →0, the concurrence reaches its maximum value 1, i.e. the entangled state, |Ψ⟩ = (((2) 1/2 )/2)(|01⟩-|10⟩), is maximum entanglement. For the XXZ chain, when Γ→∞, the concurrence will meet its maximum value 𝒞 max = (sinh (1/ T ))/(cosh (1/ T )).
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 study entanglement of a pair of qubits and the bipartite entanglement between the pair and the rest within open-ended Heisenberg XXX and XY models. The open boundary condition leads to strong oscillations of entanglements with a two-site period, and the two kinds of entanglements are 180 degrees out of phase with each other. The mean pairwise entanglement and ground-state energy per site in the XXX model are found to be proportional to each other. We study the effects of a single bulk impurity on entanglement, and find that there exist threshold values of the relative coupling strength between the impurity and its nearest neighbors, after which the impurity becomes pairwise entangled with its nearest neighbors.
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.