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The partition function of the zero-field “Eight-Vertex” model on a square M by N lattice is calculated exactly in the limit of M, N large. This model includes the dimer, ice and zero-field Ising, F and KDP models as special cases. In general the free energy has a branch point singularity at a phase transition, with an irrational exponent.

The principles on which the two-particle convolution hyper-netted chain approximation of classical statistical mechanics is based, namely summation over a specific class of Yvon-Mayer diagrams and use of a variational principle, are used to derive a three-particle approximation. This approximation gives a symmetric three-particle correlation function and is applicable to a system with three-, as well as two-particle interactions.

We establish an equivalence between the zero-field eight-vertex model and an Ising model (with four-spin interaction) in which each spin has L possible values, labeled 1, …, L, and two adjacent spins must differ by one (to modulus L). Such an Ising model can also be thought of as a generalized ice-type model and we will later show that the eigenvectors of the transfer matrix can be obtained by a Bethe-type ansatz.

We obtain the eigenvectors of the transfer matrix of the zero-field eight vertex model. These are also the eigenvectors of the Hamiltonian of the corresponding one-dimensional anistropic Heisenberg chain.

We obtain some simple eigenvectors of the transfer matrix of the zero-field eight-vertex model. These are also eigenvectors of the Hamiltonian of the one-dimensional anisotropic Heisenberg chain. We also obtain new equations for the matrix Q ( v ) introduced in earlier papers.