Symmetry Properties of the Ashkin-Teller Model and the Eight-Vertex Model

A new approach to the statistical model of Ashkin and Teller (AT) is developed with a view to compare it with the recently solved eight-vertex (8-v) model. The four-component nearest-neighbor model of atoms is transcribed into a system of arrow graphs on a square lattice of unit cells. A bidirectional transfer matrix is constructed in terms of spin operators to represent the change in arrow directions at each unit cell. Symmetry properties of the system are considered by means of algebraic analysis on the spin operators. The existence of a dual transform permits determination of the transition temperature in the most general case of the problem. The mathematical condition on the weight factors for locating this transition temperature is found to be identical to the corresponding one for the 8-v model. Furthermore, when expressed as a function of a set of modified weights, the partition function of the AT model also obeys the same symmetry. It is conjectured that the AT model, although possessed with a similar feature of continuously varying critical indices, is topologically distinct from the 8-v model.