Theory of layered Ising models. II. Spin correlation functions parallel to the layering

We study the spin correlation $〈{\sigma }_{00}{\sigma }_{0N}〉$, which is expressable as a block Toeplitz determinant, of a nth-order layered Ising model, in the direction parallel to the layering. The generating function of this block Toeplitz determinant can be written as a scalar function times a 2×2 matrix whose elements are nth-order trigonometric polynomials. We show that it is possible to calculate asymptotically for large $N$ the block Toeplitz determinant generated by such a matrix function. As an example we compute, for large $N$, $〈{\sigma }_{00}{\sigma }_{0N}〉$ for the second-order layered Ising model whose horizontal bonds are all equal and whose vertical bonds between the jth row and (j + 1)th row is $E_2$, if $j$ even and $E_2^{\prime }$, if $j$ odd.