Boundary correlation function of fixed-to-free bcc operators in square-lattice Ising model

TL;DR

This paper computes the boundary correlation function for fixed-to-free boundary condition changing operators in the square-lattice Ising model, expressing it via Toeplitz determinants and analyzing its asymptotics at criticality.

Contribution

It introduces a method to express the boundary correlation function in multiple forms and analyzes its asymptotic behavior using advanced theorems, confirming conformal field theory predictions.

Findings

Correlation function expressed as Toeplitz determinants.

Asymptotic behavior matches conformal field theory predictions.

Power-law decay at critical temperature.

Abstract

We calculate the boundary correlation function of fixed-to-free boundary condition changing operators in the square-lattice Ising model. The correlation function is expressed in four different ways using $2\times2$ block Toeplitz determinants. We show that these can be transformed into a scalar Toeplitz determinant when the size of the matrix is even. To know the asymptotic behavior of the correlation function at large distance we calculate the asymptotic behavior of this scalar Toeplitz determinant using the Szeg\"o's theorem and the Fisher-Hartwig theorem. At the critical temperature we confirm the power-law behavior of the correlation function predicted by conformal field theory.