The boundary correlation function of fixed-to-free boundary-condition-changing operatorsin a square-lattice Ising model

Abstract

We study the boundary correlation function of the fixed-to-free boundary-condition-changing(bcc) operators in the square-lattice Ising model. First, we find a formula forrepresenting a large class of two-point boundary correlation functions using a2 × 2 block Toeplitz determinant. Using this formula the correlation function of the fixed-to-freebcc operator is represented using block Toeplitz determinants, for arbitrary, uniformlyanisotropic couplings. This block Toeplitz determinant is transformed into a scalar Toeplitzdeterminant when the size of the matrix is an even number. We use Szegö’s theorem and theFisher–Hartwig theorem to identify the asymptotic behavior of this scalar Toeplitzdeterminant.