Study on boundary-boundary correlation functions of the two-dimensional Ising model using topological interaction method

The topological interaction method is applied to the evaluation of boundary-boundary correlation functions, which yield the square of the spontaneous magnetization m _b at the boundary in the thermodynamic limit. One of the remarkable results is that the boundary-boundary correlation function C _M is found to be nonmonotonic with respect to the system size M for T< T _c, because of boundary effects. On the other hand, it is monotonic above T _c, C _M ≃(A ₊(T) √M) exp(-M/ξ) with the correlation length f and with A ₊(T) ~ √T-T _c near the critical point T _c. At the critical point, C _M~1/M, and below T _c, C _M=m _b ²+(-A_(T)√M +B(T) + C(T)√M)exp(-M/ξ), where A_(T)~m _b ³, B(T)~m _b ³, C(T)~m _b√T-T _c. Fisher's finite-size scaling is confirmed in this respect.