Fredholm Determinant Evaluations of the Ising Model Diagonal Correlations and their λ Generalization

摘要

The diagonal spin-spin correlations of the square lattice Ising model, originally expressed as Toeplitz determinants, are given by two distinct Fredholm determinants-one with an integral operator having an Appell function kernel and another with a summation operator having a Gauss hypergeometric function kernel. Either determinant allows for a Neumann expansion possessing a natural λ-parameter generalization and we prove that both expansions are in fact equal, implying a continuous and a discrete representation of the form factors. Our proof employs an extension of the classic study by Geronimo and Case [1], applying scattering theory to orthogonal polynomial systems on the unit circle, to the bi-orthogonal situation.