Anisotropic correlation length in the eight-vertex model

We investigate the directional dependence (or anisotropy) of the correlation length of the eight-vertex model by a method which introduces the shift operator into the usual transfer matrix argument. For a given x (0 < x < 1) there are two cases with respect to a parameter q. In the case 0 < q < x3, the anisotropic correlation length (ACL) is independent of q. Noting that the eight-vertex model factors into two Ising lattices in the q → x4 limit, we can directly relate the ACL to that of the square lattice Ising model. When x3 < q < x2, the ACL depends on q. Using an algebraic curve, we show that the ACL in this case is connected with that of the Ising model on a 4–8 (or Union Jack) lattice. We suggest that a wide class of models have essentially the same ACL as the eight-vertex model has.