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

Abstract

The topological interaction method is applied to the evaluation of boundary–boundary correlation functions, which yield the square of the spontaneous magnetization mb at the boundary in the thermodynamic limit. One of the remarkable results is that the boundary–boundary correlation function CM is found to be nonmonotonic with respect to the system size M for T<Tc, because of boundary effects. On the other hand, it is monotonic above Tc, CM≃(A+(T)∕M)exp(−M∕ξ) with the correlation length ξ and with A+(T)∼T−Tc near the critical point Tc. At the critical point, CM∼1∕M, and below Tc, CM≃mb2+(−A−(T)M+B(T)+C(T)∕M)exp(−M∕ξ), where A−(T)∼mb3, B(T)∼mb2, C(T)∼mb∼T−Tc. Fisher’s finite-size scaling is confirmed in this respect.