[论文解读] A brief account of the Ising and Ising-like models: Mean-field, effective-field and exact results

Jozef Strečka, M. Jaščur|arXiv (Cornell University)|Nov 10, 2015

Complex Network Analysis Techniques参考文献 4被引用 44

一句话总结

本教程综述系统比较了在各种晶格体系中求解伊辛模型及其类伊辛模型的平均场法、有效场法和精确解法。结果表明，平均场理论可正确捕捉三临界点和量子相变；有效场理论通过簇近似显著提升了平均场法的精度；而基于转移矩阵和对偶变换的精确解法在1D和2D晶格（包括阿基米德型和修饰型结构）上可精确获得临界行为。

ABSTRACT

The article provides a tutorial review on how to treat Ising models within mean-field (MF), effective-field (EF) and exact methods. MF solutions of the spin-1 Blume-Capel (BC) model and the mixed-spin Ising model demonstrate a change of continuous phase transitions to discontinuous ones at a tricritical point. A quantum phase transition of the spin-S Ising model driven by a transverse field is explored within MF method. EF theory is elaborated within a single- and two-spin cluster approach to demonstrate an efficiency of this approximate method. The long-standing problem of this method concerned with a self-consistent determination of the free energy is addressed. EF theory is adapted for the spin-1/2 Ising model, the spin-S BC model and the transverse Ising model. The particular attention is paid to continuous and discontinuous transitions. Exact results for the spin-1/2 Ising chain, spin-1 BC chain and mixed-spin Ising chain are obtained using the transfer-matrix method, the crucial steps of which are reviewed for a spin-1/2 Ising square lattice. Critical points of the spin-1/2 Ising model on several lattices are rigorously obtained with the help of dual, star-triangle and decoration-iteration transformations. Mapping transformations are adapted to obtain exact results for the mixed-spin Ising model on planar lattices. An increase in the coordination number of the mixed-spin Ising model on decorated planar lattices gives rise to reentrant transitions, while the critical temperature of the mixed-spin Ising model on a regular honeycomb lattice is always greater than that of two semi-regular archimedean lattices. The effect of selective site dilution of the mixed-spin Ising model on a honeycomb lattice upon phase diagrams is examined. The review affords a brief account of the Ising models solved within MF, EF and exact methods along with a few comments on their future applicability.

研究动机与目标

• 提供对伊辛模型及其类伊辛模型的平均场法、有效场法和精确解法的教育性比较。
• 阐明平均场理论的局限性及其在预测三临界点和相变方面的改进之处，特别是其在相变顺序预测方面的不足。
• 解决有效场理论中自洽自由能确定的长期难题。
• 利用转移矩阵法和变换技术，推导出在各种晶格上自旋-1/2、自旋-1和混合自旋伊辛模型的精确临界点和自发磁化强度。
• 研究配位数和位点稀释对相图的影响，特别是对再entrant相变的影响。

提出的方法

• 将平均场理论应用于四种模型：纵向磁场中的自旋-1/2伊辛模型、纵向磁场中的自旋-1 Blume-Capel模型、纵向磁场中的混合自旋伊辛模型，以及横向磁场中的自旋-S伊辛模型。
• 在有效场理论中采用单自旋和双自旋簇近似以改进平均场结果，重点在于自洽自由能的估计。
• 利用Callen-Suzuki恒等式和van der Waerden恒等式推导近似热力学量。
• 采用转移矩阵法，获得一维伊辛链（自旋-1/2、自旋-1和混合自旋）的精确解。
• 应用对偶变换、星-三角形变换和修饰-迭代变换，严格确定二维伊辛模型在正方、蜂窝、三角、Kagomé及阿基米德晶格上的临界点。
• 利用映射变换，推导出在修饰晶格和三配位晶格上混合自旋伊辛模型的精确临界温度和自发磁化强度。

实验结果

研究问题

• RQ1 在三临界点附近，平均场理论与有效场理论在预测类伊辛模型相变行为方面有何异同？
• RQ2 在修饰晶格上的混合自旋伊辛模型中，配位数在诱导再entrant相变中起何种作用？
• RQ3 与近似方法相比，平面晶格上自旋-1/2伊辛模型的精确解在临界温度和磁化强度方面表现如何？
• RQ4 有效场理论能否系统性地扩展以包含自洽自由能计算？其相较于平均场理论的改进程度如何？
• RQ5 选择性位点稀释对蜂窝晶格上混合自旋伊辛模型相图有何影响？

主要发现

• 平均场理论正确预测了自旋-1 Blume-Capel模型和混合自旋伊辛模型中的三临界点，即连续相变转变为不连续相变。
• 采用双自旋簇的有效场理论比平均场理论结果更精确，尤其在正确捕捉临界行为和相变顺序方面表现更优。
• 通过转移矩阵法获得的精确解证实了自旋-1/2、自旋-1和混合自旋伊辛链在纵向磁场下的自发磁化强度和临界温度。
• 对偶变换和星-三角形变换为正方、蜂窝、三角和Kagomé晶格上的自旋-1/2伊辛模型提供了精确临界温度，其结果与已知精确值完全一致。
• 在相同配位数下，规则蜂窝晶格上混合自旋伊辛模型的临界温度高于半规则阿基米德晶格。
• 在修饰平面晶格中，随着配位数增加，混合自旋伊辛模型出现再entrant相变，该结果通过精确映射技术得到验证。