Theory of Toeplitz Determinants and the Spin Correlations of the Two-Dimensional Ising Model. II

We study in detail the asymptotic behavior, for large separations, of the correlation between two spins in the same row of the two-dimensional Ising model with the particular value of a purely imaginary magnetic field H=1 / 2iπkT. We derive the partition function and the magnetization per spin first proposed by Lee and Yang and find that the correlation function approaches its limiting value exponentially. In contrast to the case of no magnetic field, the approach is shown to be oscillatory rather than monotonic. The first three terms of the asymptotic behavior are explicitly given. The correlation function is obtained by expressing it as a block Toeplitz determinant whose behavior for large separation is studied by solving a pair of coupled Wiener-Hopf sum equations.