The nature of three dimensions: Non-local behavior in the three-dimensional (3D) Ising model

National Conference on Theory of Magnetism

Zhang, Z.D.^1

Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang

110016, China^1

关键词: High-temperature expansion;    Local environments;    Low-temperature expansion;    Partition functions;    Perturbative renormalization;    Theoretical approach;    Threedimensional (3-d);    Transfer matrixes;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/827/1/012001/pdf
DOI  :  10.1088/1742-6596/827/1/012001

来源: IOP

【 摘 要 】

In this work, we first reveal the nature of three dimensions: the existence of non-local behavior. Taking the three-dimensional (3D) Ising model as an example, we analyze the transfer matrices as well as the partition function to discuss in details the origin of non-local behavior in three dimensions. We then point out that fail to deal with non-local behavior is one of the main disadvantages of several theoretical approaches (such as conventional low-temperature expansions, conventional high-temperature expansions, perturbative renormalization group and Monte Carlo methods, etc), all of which take local environments into account only. This work provides a deep understanding on the natural difference between 3D bulk materials and 2D materials, and shows that how to treat the non-local behavior and the interaction between spins along the third dimension is a key point for solving exactly the problems of the 3D systems with the nearest neighboring interactions. Finally, we will briefly review recent advance in the 3D Ising model and related topics.