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arXiv:cond-mat/0312351 · Physical Review E

Boundary and impurity effects on the entanglement of Heisenberg chains

Xiaoguang Wang

Published Jun 10, 2004• 48 citations
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Abstract

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.

DOI
10.1103/physreve.69.066118

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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.
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.
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 consider a finite two dimensional XY spin model. The model consists of a set of seven localized spin- 12 particles in a two dimensional triangular lattice coupled through nearest neighbor exchange interaction in presence of an external magnetic field. We study the effect of a single impurity spin coupled to its nearest neighbor through an exchange interaction J′ on the pairwise entanglement between the different spins in the lattice. We found that when the spin is located at a border site the entanglement between the impurity and its nearest neighbors increases monotonically with J′ reaching a saturation value that depends on the magnetic field strength h. On the other hand the entanglement with the next nearest spins shows a critical behavior where the entanglement increases and then decreases rapidly until it vanishes. The critical coupling value at which the entanglement vanishes increases as h increases. Studying the entanglement between two spins excluding the impurity show that while the entanglement between two spins coupled to the impurity reach a saturation value as J′ increases, it vanishes for spins not coupled to the impurity. Furthermore studying the effect of a central impurity show that the entanglement between nearest neighbor and next nearest neighbor spins reaches an asymptotic value as J′ increases. Interestingly it was demonstrated that the impurity can be used as a switch to control the entanglement between different spins in the lattice turning it on and off.
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.
For a three-qubit Heisenberg model in a uniform magnetic field, the pairwise thermal entanglement of any two sites is identical due to the exchange symmetry of sites. In this paper we consider the effect of a non-uniform magnetic field on the Heisenberg model, modelling a magnetic impurity on one site. Since pairwise entanglement is calculated by tracing out one of the three sites, the entanglement clearly depends on which site the impurity is located. When the impurity is located on the site which is traced out, that is, when it acts as an external field of the pair, the entanglement can be enhanced to the maximal value 1; while when the field acts on a site of the pair the corresponding concurrence can only be increased from 1/3 to 2/3.