It is shown that the Zamolodchikov model on the cubic lattice of
N sites is equivalent to a spin model on a lattice of
2N sites with just two-spin and three-spin interactions. Together with the known transfer-matrix commutativity, this implies that the partition function
Z (for
N large) factors into a product of single-variable functions. These functions, and hence
Z, are calculated by using symmetry relations. The model appears to be a critical free-fermion model.
The theory of a massless two-dimensional scalar field with a periodic boundary interaction is considered. At a critical value of the period this system defines a conformal field theory and can be reexpressed in terms of free fermions, which provide a simple realization of a hidden SU(2) symmetry of the original theory. The partition function and the boundary S matrix can be computed exactly as a function of the strength of the boundary interaction. We first consider open strings with one interacting and one Dirichlet boundary, and then with two interacting boundaries. The latter corresponds to motion in a periodic tachyon background, and the spectrum exhibits an interesting band structure which interpolates between free propagation and tight binding as the interaction strength is varied.