Crystal Statistics of a Two-Dimensional Ising Lattice

The partition function $Z$ of a two-dimensional Ising lattice is evaluated by expressing it in the form $Z=\Sigma i^{}\lambda _i^{}{}_{}{}^{\mathrm{nm}}$ where ${\lambda }_i$ are the eigenvalues of a $2^n$-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 $2^n$-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 $2^n$-dimensional representative.