On the Toeplitz Determinant for Spin-Pair Correlation Function of 2-D Ising Model

Abstract

In the present paper, we consider the correlation function of two spins i and f on a two-dimensional Ising model with non-crossing interactions. We assume that the two spins are separated by n lattice spacings. By extending a theory due to Vdovi­ chenko,1) giving the spontaneous magnetization of the Ising model on the square lattice, it was shown generally that the spin-pair correlation function is expressed by a 2n X 2n determinant. 2 ) If the lattice consists of a multiple number of sub lattices and t:he two spins i and f are on the same sublattice along an axis, the expression of the correlation function takes the form of an nlv X nlv block Toeplitz determinant with 2v X 2 v blocks, where v is the number of different sub lattices on the axis. If the lattice is monatomic on the axis along which the correlation is calculated, we have an n X n block Toeplitz determinant with 2 X 2 blocks. 3 ) It is known for simple lattices in which the lattice is monatomic on the axis and the interactions have a reflection symmetry with respect to the axis or some inversion symmetry, the 2 X 2 blocks are diagonal and the spin-pair correlation function is expressed by an n X n ordinary Toeplitz determinant. 4 )-7) In the present paper, we assume that the two spins i and f are on the same sublattice along an axis and that the interactions have either a reflection symmetry with respect to the axis or such an inversion symmetry that the midpoint between every pair of lattice sites on the axis is a centre of symmetry. We then show that the correlation function is expressed by an n X n determinant, instead of a 2n X 2n determinant. If there are v different sublattices on the axis, it takes the form of an nlv X nlv block Toeplitz determinant with v X v blocks. If the lattice is monatomic on the axis along which the correlation is calculated, we have an n X n ordinary Toeplitz determinant. In the diagram approach to the two-dimensional Ising model, the partition func­ tion and the pair correlation function are expressed in terms of diagrams of loops and chains, which consist of bonds, each connecting a pair of nearest-neighbour lattice sites. 8HO ) We put directions to the loops and chains, and then a bond with direction is called a step. We express the set of all the pairs of nearest-neighbour lattice sites with a direction by B, and then a step is an element in B_ The interaction between spins at both ends of a step labeled by /3 is denoted by fp. The number of elements