Rise of pairwise thermal entanglement for an alternating Ising and Heisenberg spin chain in an arbitrarily oriented magnetic field

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(NO${}_3$)(DMSO)${}_2$Cu(opba)(DMSO)${}_2$ [DMSO = dimethyl sulfoxide; opba = $o$-phenylenebis(oxamoto)] [J. Strečka, M. Hagiwara, Y. Han, T. Kida, Z. Honda, and M. Ikeda, Condens. Matter Phys. 15, 43002 (2012)].

Figure 1

The arising of pairwise thermal entanglement (green region). (a) The concurrence $\mathcal{C}$ between the nearest Heisenberg spin $r=1$, as a function of the magnetic field $h/\left|J\right|$ and the polar angle $\varphi$. (b) The concurrence $\mathcal{C}$ between the next-nearest Heisenberg spin $r=2$, as a function of the magnetic field $h/\left|J\right|$ and the polar angle $\varphi$.

Figure 2

The arising of pairwise thermal entanglement (green region). (a) The concurrence $\mathcal{C}$ between the nearest Heisenberg spin $r=1$, as a function of the temperature $T/\left|J\right|$ and the polar angle $\varphi$. (b) The concurrence $\mathcal{C}$ between the next-nearest Heisenberg spin $r=2$, as a function of the temperature $T/\left|J\right|$ and the polar angle $\varphi$.

Figure 3

Concurrence as a function of temperature for a fixed value of magnetic field. (a) Nearest-neighbor concurrence $r=1$. (b) Next-nearest-neighbor concurrence $r=2$.

Figure 4

Concurrence as a function of magnetic field for a fixed temperature. (a) Nearest-neighbor concurrence $r=1$. (b) Next-nearest-neighbor concurrence $r=2$.