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arXiv:quant-ph/0009060 · Physical Review Letters

Natural Thermal and Magnetic Entanglement in the 1D Heisenberg Model

M. C. Arnesen, Sougato Bose, Vlatko Vedral

Published Jun 14, 2001• 796 citations
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Abstract

We investigate the entanglement between any two spins in a one dimensional Heisenberg chain as a function of temperature and the external magnetic field. We find that the entanglement in an antiferromagnetic chain can be increased by increasing the temperature or the external field. Increasing the field can also create entanglement between otherwise disentangled spins. This entanglement can be confirmed by testing Bell's inequalities involving any two spins in the solid.

DOI
10.1103/physrevlett.87.017901

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We investigate the entanglement between any two spins in a one dimensional Heisenberg chain as a function of temperature and the external magnetic field. We find that the entanglement in an antiferromagnetic chain can be increased by increasing the temperature or the external field. Increasing the field can also create entanglement between otherwise disentangled spins. This entanglement can be confirmed by testing Bell's inequalities involving any two spins in the solid.
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The thermal pairwise entanglement (TE) of the S=1/2S=1/2S=1/2 XYXYXY chain in a transverse magnetic field is exactly resolved by means of the Jordan-Wigner transformation in the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. It is found that the TE vanishes at a fixed point with temperature ${T}_{c}\ensuremath{\simeq}0.484\text{ }3J$, which is independent of the magnetic field. A thermal quantity is proposed to witness the entangled state. Furthermore, the TE of the S=1/2S=1/2S=1/2 antiferromagnetic-ferromagnetic (AF-F) Heisenberg chain is studied by the transfer-matrix renormalization-group method. The TEs of the spins coupled by AF and F interactions are found to behave distinctively. The vanishing temperature of the field-induced TE of the spins coupled by F interactions is observed dependent on the magnetic field. The results are further confirmed and analyzed within a mean-field framework.
The thermal entanglement in the Heisenberg XYZ\mathrm{XYZ}XYZ chain is investigated in the presence of an external magnetic field B. In the two-qubit system, the critical magnetic field Bc{B}_{c}Bc​ is increased by introducing the interaction of the z component of two neighboring spins Jz.{J}_{z}.Jz​. This interaction not only improves the critical temperature Tc,{T}_{c},Tc​, but also enhances the entanglement for a particular fixed B. We also analyze the pairwise entanglement between nearest neighbors in three qubits. The pairwise entanglement, for a fixed T, can become strong by controlling B and Jz.{J}_{z}.Jz​.
Typically two particles (spins) could be maximally entangled at zero temperature, and for a certain temperature the phenomenon of entanglement vanishes at the threshold temperature. For the Heisenberg coupled model or even the Ising model with a transverse magnetic field, one can observe some rise of entanglement even for a disentangled region at zero temperature. So we can understand this emergence of entanglement at finite temperature as being due to the mixing of some maximally entangled states with some other untangled states. Here, we present a simple one-dimensional Ising model with alternating Ising and Heisenberg spins in an arbitrarily oriented magnetic field, which can be mapped onto the classical Ising model with a magnetic field. This model does not show any evidence of entanglement at zero temperature, but surprisingly at finite temperature rise a pairwise thermal entanglement between two untangled spins at zero temperature when an arbitrarily oriented magnetic field is applied. This effect is a purely magnetic field, and the temperature dependence, as soon as the temperature increases, causes a small increase in concurrence, achieving its maximum at around 0.1. Even for long-range entanglement, a weak concurrence still survives. There are also some real materials that could serve as candidates that would exhibit this effect, such as Dy(NO3{}_{3}3​)(DMSO)2{}_{2}2​Cu(opba)(DMSO)2{}_{2}2​ [DMSO = dimethyl sulfoxide; opba = ooo-phenylenebis(oxamoto)] [J. Stre\ifmmode \check{c}\else \v{c}\fi{}ka, M. Hagiwara, Y. Han, T. Kida, Z. Honda, and M. Ikeda, Condens. Matter Phys. 15, 43002 (2012)].