In this paper we rederive, by simpler methods, the Onsager-Kaufman formulas for the correlations and the Onsager formula for the spontaneous magnetization of the rectangular two-dimensional Ising lattice. The Pfaffian approach is used to derive the correlations in terms of Pfaffians, and for the correlations in a row a single Toeplitz determinant is obtained which is proved equivalent to the Onsager-Kaufman result. The spontaneous magnetization is obtained as the limiting value of an infinite Toeplitz determinant and its evaluation is facilitated by use of the generalization of a result first published by Szego.
The partition function for the Ising problem for a general class of plane lattices is evaluated exactly using the Pfaffian method. A classification of plane lattices in terms of basic lattices with vertices decorated by sublattices is given, and the most general class which can be solved by the Pfaffian method described. The solutions given in this paper are for the special case when neighbouring lattice points of a basic rectangular lattice are connected by single bonds only. All of the exact solutions known so far are obtained as special cases. The critical points are defined as singularities of an analytic function and equations to determine their location are found. The derivatives of the partition function are evaluated in terms of elliptic integrals.
The partition function
Z of a two-dimensional Ising lattice is evaluated by expressing it in the form $Z=\ensuremath{\Sigma}{i}^{}\ensuremath{\lambda}_{i}^{}{}_{}{}^{\mathrm{nm}}$ where ${\ensuremath{\lambda}}_{i}$ are the eigenvalues of a
2n-dimensional matrix M,
n is the number of rows and
m the number of columns of the lattice. M is a generalization of the V-shaped matrix studied by Kramers and Wannier in connection with the same problem. It is shown that M can be expressed in terms of
2n-dimensional representations of
2n-dimensional orthogonal matrices. The eigenvalues of M are determined for
n large by making use of the known relations between the eigenvalues of a
2n-dimensional orthogonal matrix and the eigenvalues of its
2n-dimensional representative.The lattice considered differs only slightly from that treated by Onsager and Kaufman. In the lattice considered here, the boundary effects are eliminated by introducing an interaction between the last lattice point of each row and the first point of the next row. The matrix V considered by Onsager and Kaufman is, except for the boundary effects, equivalent to the matrix
Mn.
Z is shown to differ from the partition function calculated by Onsager and Kaufman only in order ${n}^{\ensuremath{-}2}$.
It is shown that the transfer matrices for various Ising lattices in two dimensions commute with certain linear operators. The problem of finding an explicit form for the largest eigenvector is considerably simplified. The expansion coefficients appearing in the eigenvectors found as the solution of a set of nonlinear difference equations are Pfaffians. The connection between this type of solution and other solutions is clarified. This form for the eigenvector also simplifies the calculation of correlation functions. Some geometrical aspects of the Ising model are discussed.