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arXiv:1203.4774 · Physical Review A

Thermal entanglement between non-nearest-neighbor spins on fractal lattices

Yu-Liang Xu, Lu-Shun Wang, Xiang-Mu Kong

Published Jan 16, 2013• 19 citations
arXivHTMLPDF

Abstract

We investigate the thermal entanglement between two end sites in spin chains and on fractal lattices by taking the negativity as a measure and using the decimation renormalization-group method. The effects of the temperature $T$, the anisotropy parameter $\ensuremath{\Delta}$, and the size of the system, $L$, on the entanglement are examined in detail. It is found that the entanglement decreases monotonically with increasing $T$ and vanishes beyond a critical value ${T}_{c}$. Our results also show that with increasing $\ensuremath{\Delta}$ from $\ensuremath{-}\ensuremath{\infty}$ to zero the entanglement first increases to the maximum and then decreases sharply to zero. Different from the cases of spin chains and Koch curves, the entanglement on the diamond-type hierarchical (DH) lattices presents some interesting behaviors. As the sizes of the DH lattices, $L$, become large, the entanglement is rather robust and there exists sizable entanglement between long-distance end sites. This result indicates that different fractal structures can result in various entanglement properties.

DOI
10.1103/physreva.87.012312

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Adopting concurrence as entanglement measure, we study entanglement and quantum phase transition of the Ising spin systems on the triangular lattice and Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski fractal lattices by using the quantum renormalization-group method. It is found that the ground-state entanglement between two spins (or spin blocks) depends on the following factors: the size of system, the magnetic field, the exchange coupling, and the structure of lattice. As the size of the system becomes large, (a) the range of the magnetic field, in which the entanglement exists, contracts gradually and focuses on the critical point; and (b) the first derivative of entanglement shows singular behavior, and its maximum or minimum is approaching to the critical point gradually. The scaling behaviors of entanglement on the different lattices are similar but the scaling relations are diverse. For the triangular lattice, the space dimensionality determines the scaling relationship between the critical exponent of the entanglement and the critical exponent correlation length. However, for fractal lattices, it is the fractal dimensionality but not the space one to determine this relationship.
The entanglement quantum properties of a spin-1/2 Ising-Heisenberg model on a symmetrical diamond chain were analyzed. Due to the separable nature of the Ising-type exchange interactions between neighboring Heisenberg dimers, calculation of the entanglement can be performed exactly for each individual dimer. Pairwise thermal entanglement was studied in terms of the isotropic Ising-Heisenberg model and analytical expressions for the concurrence (as a measure of bipartite entanglement) were obtained. The effects of external magnetic field H and next-nearest neighbor interaction J(m) between nodal Ising sites were considered. The ground state structure and entanglement properties of the system were studied in a wide range of coupling constant values. Various regimes with different values of ground state entanglement were revealed, depending on the relation between competing interaction strengths. Finally, some novel effects, such as the two-peak behavior of concurrence versus temperature and coexistence of phases with different values of magnetic entanglement, were observed.
We investigate the thermal entanglement in a spin-1/2 Ising–Heisenberg diamond chain, in which the vertical Heisenberg spin dimers alternate with single Ising spins. Due to the fact that the Dzyaloshinskii–Moriya (DM) interaction contributes to unusual and interesting magnetic properties in actual materials, and moreover it plays a significant role in the degree of the entanglement of the Heisenberg quantum spin systems, we focus on the effects of different DM interactions, including D z and D x , on the thermal entanglement of the Heisenberg spin dimer. The concurrence, as a measure of spin dimer entanglement, is calculated for different values of exchange interactions, DM interaction, external magnetic field, and temperature. It is found that the critical temperature and the critical magnetic field corresponding to the vanishing of entanglement increase with DM interaction, and the entanglement revival region gets larger by increasing DM interaction, thus DM interaction favors the formation of the thermal entanglement. It is observed that different DM interaction parameters ( D z and D x ) have remarkably different influences on the entanglement. Different from the case D z , there is the non-monotonic variation of the concurrence with temperature in the case D x , and additionally the DM interaction D x can induce the entanglement near zero temperature in the case that the antiferromagnetic Ising-type interaction constant is larger than the antiferromagnetic Heisenberg interaction constant. It is also shown that for the same value of DM interaction the critical magnetic field of the case D x is larger than that of the case D z .
Abstract The spin-1/2 antiferromagnetic (AF) Heisenberg systems are studied on three typical diamond-type hierarchical lattices (systems A, B and C) with fractal dimensions 1.63, 2 and 2.58, respectively, and the phase diagrams, critical phenomena and quantum correlations are calculated by a combination of the equivalent transformation and real-space renormalization group methods. We find that there exist a reentrant behavior for system A and a finite temperature phase transition in the isotropic Heisenberg limit for system C (not for system B). Unlike the ferromagnetic case, the Néel temperatures of AF systems A and B are inversely proportional to ln Δ c − Δ (when Δ → Δ c ) and ln Δ (when Δ → 0), respectively. And we also find that there is a turning point of quantum correlation in the isotropic Heisenberg limit Δ = 0 where there is a ‘peak’ of the contour and no matter how large the size of system is, quantum correlation will change to zero in the Ising limit for the three systems. The quantum correlation decreases with the increase of lattice size L and it is almost zero when L ⩾ 30 for system A, and for systems B and C, they still exist when L is larger than that of system A. Moreover, we discuss the effects of quantum fluctuation and analyze the errors of results in the above three systems, which are induced by the noncommutativity.
We examine the entanglement of thermal states of nnn spins interacting through different types of XYXYXY couplings in the presence of a uniform magnetic field, by evaluating the negativities of all possible bipartite partitions of the whole system and of subsystems. We consider both the case where every qubit interacts with all others and where just nearest neighbors interact in a one-dimensional chain. Limit temperatures for nonzero negativities are also evaluated and compared with the mean field critical temperature. It is shown that limit temperatures of global negativities are strictly independent of the magnetic field in all XXZXXZXXZ models, in spite of the quantum transitions that these models may exhibit at zero temperature, while in anisotropic models they always increase for sufficiently large fields. Results also show that these temperatures are higher than those limiting pairwise entanglement.
The entanglement thermal (equilibrium) properties in spin-1/2 Ising-Heisenberg model on an ideal diamond chain is analyzed. Due to the classical character of Ising-type exchange interactions between two neighboring antiferromagnetic Heisenberg dimers, the calculation of quantum entanglement (by tracing out Ising spins) can be performed for each of the dimers separately. The concurrence, as a measure of entanglement is obtained and different regimes depending on the values of exchange interactions are revealed. The effects of the magnetic field are incorporated and critical temperatures corresponding to the vanishing or arising of entanglement are considered.