Moments of the total magnetization and conformal invariance in the finite two-dimensional Ising model

Abstract

We consider Ising strips with width N and periodic boundary conditions and Ising squares with edge length N and special partially periodic boundaries. Assuming invariance of the spin correlations under conformal mappings of the infinite plane onto the strip and square, we determine the second and fourth moments of the total magnetization M from the known bulk two- and four-spin correlation functions at criticality. For both geometries the predictions of conformal invariance for universal asymptotic forms involving the ratio 〈$M^4$〉〈$M^2$ ${〉}^{\mathrm{-}2}$ as N→∞ are in excellent agreement with transfer-matrix results.