Phase transitions in the Ising model

H Ulman, M Athematics J Ournal

Robert H. Swendsen

The Ising model provides an extremely useful example for the investigation of phase transitions. This chapter provides both an introduction to the properties of the Ising model and an overview of the complex phenomena exhibited at phase transitions in general. The one-dimensional model is solved exactly using transfer matrices. Models in higher dimensions are treated in the mean field approximation.

Christopher J. Monahan, Axel Klomparens

The Ising model serves as a model for magnetic phase transitions and as a testing ground for the study of strongly-coupled systems. The model is exactly solvable in two dimensions and can be simulated with relatively small computing resources. We investigate the phase transition of the Ising model through a novel scaling procedure first proposed to explore the phase structure of a theory with four $SU(2)$ doublet fermions in Butt et al.~(PRL 134, 2025, 031602). We present a numerical validation of this scaling procedure by demonstrating that the procedure aligns with the known properties of the two-dimensional Ising model.

Tomoyasu Tanaka

Fa Yueh Wu

Tianshu Gu

Phase transitions are widespread physical phenomena in nature and the phase transition of ferromagnetic substances which can be described by Ising model is an important part of physics. This paper discusses the phase transition of the low-dimensional Ising model. For one-dimensional Ising model, the exact solution is given and no phase transition is found when there is no external magnetic field. For two-dimensional Ising model, Monte Carlo method along with importance sampling, careful balance conditions and the Metropolis algorithm is introduced to numerically simulate it. The trends of order parameters like magnetization, specific heat and magnetic susceptibility are analyzed and the mutation of some order parameters is found around critical temperature. The influences of external magnetic field and the scale of system are discussed and critical temperature of phase transition is explored by several methods, which is eventually found around 2.269 ( kBT/J). Lastly, compared with theoretical values, the disadvantages are pointed out and optimization suggestions are raised for improvement.