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arXiv:quant-ph/0102137 · Physical Review A

Thermal concurrence mixing in a one-dimensional Ising model

Daniel Gunlycke, Viv M. Kendon, Vlatko Vedral, Sougato Bose

Published Sep 6, 2001• 274 citations
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Abstract

We investigate the entanglement arising naturally in a one-dimensional Ising chain in a magnetic field in an arbitrary direction. We find that for different temperatures, different orientations of the magnetic field give maximum entanglement. In the high-temperature limit, this optimal orientation corresponds to the magnetic field being perpendicular to the Ising orientation $(z$ direction). In the low-temperature limit, we find that varying the angle of the magnetic field very slightly from the z direction leads to a rapid rise in entanglement. We also find that the orientation of the magnetic field for maximum entanglement varies with the field amplitude. Furthermore, we have derived a simple rule for the mixing of concurrences (a measure of entanglement) due to the mixing of pure states satisfying certain conditions.

DOI
10.1103/physreva.64.042302

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We investigate the entanglement arising naturally in a one-dimensional Ising chain in a magnetic field in an arbitrary direction. We find that for different temperatures, different orientations of the magnetic field give maximum entanglement. In the high-temperature limit, this optimal orientation corresponds to the magnetic field being perpendicular to the Ising orientation (z(z(z direction). In the low-temperature limit, we find that varying the angle of the magnetic field very slightly from the z direction leads to a rapid rise in entanglement. We also find that the orientation of the magnetic field for maximum entanglement varies with the field amplitude. Furthermore, we have derived a simple rule for the mixing of concurrences (a measure of entanglement) due to the mixing of pure states satisfying certain conditions.
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)].
We study the entanglement properties in a one-dimensional Ising chain with a tilted magnetic field that is capable of showing both integrable and nonintegrable behaviors. Here the pairwise entanglement is characterized by concurrence and the multipartite entanglement is characterized by the Q measure. According to the entanglement properties of the ground state in the Ising mode, which have tilt angle, we can find that the Q measure decreases with the increasing of the strength of external field. And the phase transition property of the system is changed with the increase of tilt angle for the external magnetic field. We also consider the evolution of entanglement in this model, and find that the nonintegrability can suppress the pairwise entanglement but promotes the multipartite entanglement with the integrable system.
We study the thermal entanglement by means of concurrence in a two-qubit isotropic XY model in the presence of site-dependent external magnetic fields in arbitrary directions. We find that at a given temperature and magnetic field strength, the mirror symmetry of the two fields about the x-y plane is a necessary condition for maximum entanglement. However, if there is no constraint on the field strengths, then the necessary condition for maximum entanglement reduces to the configuration that the two fields are vertical, anti-parallel and with the same strength. We also investigate the anisotropic XY model and find that the above conclusion more or less holds.
We investigate the entanglement properties of a one-dimensional chain of qubits coupled via nearest-neighbor spin-spin interactions. The entanglement measure used is the -concurrence, which is distinct from other measures on spin chains such as bipartite entanglement in that it can quantify ``global'' entanglement across the spin chain. Specifically, it computes the overlap of a quantum state with its time-reversed state. As such, this measure is well suited to study ground states of spin-chain Hamiltonians that are intrinsically time-reversal-symmetric. We study the robustness of -concurrence of ground states when the interaction is subject to a time-reversal antisymmetric magnetic field perturbation. The -concurrence in the ground state of the isotropic model is computed and it is shown that there is a critical magnetic field strength at which the entanglement experiences a jump discontinuity from the maximum value to zero. The -concurrence for thermal mixed states is derived and a threshold temperature is computed below which the system has nonzero entanglement.