Three-Dimensional Ising Model and Transfer Matrices

Abstract

The use of a transfer matrix method to solve the 3D Ising model is straightforwardly generalized from the 2D case. We follow B.Kaufman's approach. No approximation is made, however the largest eigenvalue cannot be identified. This problem comes from the fact that we follow the choice of directions of 2-dimensional rotations in the direct product space of the 2D Ising model such that all eigenvalue equations reduce miraculously to only one equation. Other choices of directions of 2-dimensional rotations for finding the largest eigenvalue may lose this fascinating feature. Comparing the series expansion of internal energy per site at the high temperature limit with the series obtained from the computer graphic method, we find these two series have very similar structures. A possible correct via a factor Φ(x) is suggested to fit the result of the graphic method.