Topological Effects and Critical Phenomena in the Three-Dimensional (3D) Ising Model

Abstract

In this work, we analyze in detail the transfer matrices as well as the partition function of the three-dimensional (3D) Ising model, in order to understand the topological effects in three dimensions. We attempt to extract the non-local contributions to the physical properties of the 3D Ising system, by comparing the conjectured exact solution with the results of the conventional high-temperature and low-temperature expansions. This is because the conjectured solution takes into account both the local and non-local behaviors, while these conventional expansions consider only the local environments. Then, we summarize the results of our previous work, most of them being joint work with Professor N. H. March, on the critical exponents of the 3D Ising model and some related topics. This work provides a deep understanding on the topological contributions and the non-local behavior in the 3D Ising model and their effects on the critical exponents in the 3D Ising universality.