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)
2Cu(opba)(DMSO)
2 [DMSO = dimethyl sulfoxide; opba =
o-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 have extended the range of application of the decimation renormalization-group (RG) method, using a combination of analytic and numerical techniques in calculating thermal and magnetic properties of a wide variety of one-dimensional quantum spin-(1/2) systems. Efforts to improve the accuracy of the approach include increasing the spatial rescaling factor and investigating the effect of free versus periodic boundary conditions for each renormalization cluster. We have used the RG method to calculate the specific heat and susceptibility for chains with uniaxial exchange anisotropy (ranging from the Ising to the XY limits) in both longitudinal and transverse magnetic fields, as well as for the Heisenberg antiferromagnet with alternating couplings, and have compared our results, when possible, with either exact solutions or with numerical extrapolations from finite-chain calculations. Decimation is a high-temperature approximation, but its range of reliability is found in some cases (e.g., transverse susceptibility for the Ising chain, or susceptibility for the Heisenberg ferromagnet in a field) to extend well into the strong-coupling regime. We note that the decimation results for magnetic properties are, in general, more accurate than those for the thermal response.
The frustrated spin-1/2 Ising-Heisenberg ladder with Heisenberg intra-rung and Ising inter-rung interactions is exactly solved in a longitudinal magnetic field by taking advantage of the local conservation of the total spin on each rung and the transfer-matrix method. We have rigorously calculated the ground-state phase diagram, magnetization process, magnetocaloric effect, and basic thermodynamic quantities for the model, which can be alternatively viewed as an Ising-Heisenberg tetrahedral chain. It is demonstrated that a stepwise magnetization curve with an intermediate plateau at half of the saturation magnetization is also reflected in respective stepwise changes of the concurrence serving as a measure of bipartite entanglement. The ground-state phase diagram and zero-temperature magnetization curves of the Ising-Heisenberg tetrahedral chain are contrasted with the analogous results of the purely quantum Heisenberg tetrahedral chain, which have been obtained through density-matrix renormalization group (DMRG) calculations. While both ground-state phase diagrams fully coincide in the regime of weak inter-rung interaction, the purely quantum Heisenberg tetrahedral chain develops Luttinger spin-liquid and Haldane phases for strongly coupled rungs, which are absent in the Ising-Heisenberg counterpart model.