Exact Solution for Three-Dimensional Ising Model

TL;DR

This paper presents an exact analytical solution for the three-dimensional Ising model with specific boundary conditions, providing explicit formulas for the partition function, critical temperatures, internal energy, and specific heat.

Contribution

It introduces a novel exact solution for the 3D Ising model using operator algebras, extending Onsager's approach from 2D to 3D.

Findings

Partition function calculated rigorously for specific boundary conditions

Critical temperatures for phase transitions determined along three axes

Analytical expressions for internal energy and specific heat provided

Abstract

Three-dimensional Ising model in zero external field is exactly solved by operator algebras, similar to the Onsager's approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along two directions and the screw boundary condition along the third direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. The critical temperatures, at which order-disorder transitions in the infinite crystal occur along three axis directions, are determined. The analytical expressions for the internal energy and the specific heat are also presented.