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 → Δc) 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.

中文翻译：

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金刚石型分层晶格上反铁磁海森堡模型的相图、量子相关性和临界现象

摘要 在分形维数分别为 1.63、2 和 2.58 的三个典型金刚石型分级晶格（系统 A、B 和 C）上研究了自旋 1/2 反铁磁 (AF) Heisenberg 系统，以及相图、临界现象和量子相关性是通过等效变换和实空间重整化群方法的组合来计算的。我们发现系统 A 存在重入行为，系统 C 的各向同性 Heisenberg 极限中存在有限温度相变（不适用于系统 B）。与铁磁外壳不同，并且我们还发现在各向同性海森堡极限Δ=0存在一个量子相关性的转折点，在此等值线有一个“峰值”，无论系统规模有多大，量子相关性在三个系统的伊辛极限。量子相关性随着晶格尺寸 L 的增加而减小，当系统 A 的 L  30 时，量子相关性几乎为零，而对于系统 B 和 C，当 L 大于系统 A 时，它们仍然存在。此外，我们讨论了影响并分析了上述三个系统中由非对易性引起的结果误差。