Correlations along a Line in the Two-Dimensional Ising Model

Abstract

The $2n$-spin correlation function for the two-dimensional Ising model at $T=T_c$ is evaluated for the special case in which all the spins lie along a straight line, separated by many lattice constants. The resulting $2n$-spin function is simply a quotient of products of two-spin correlations. A hypothesis of reducibility of fluctuations in the critical state is introduced. This hypothesis asserts that the product of any two local fluctuating quantities in the same neighborhood of space may be effectively replaced by a finite sum of local fluctuating quantities in this neighborhood. As a result, the previously found form for the $2n$-spin function may be used to evaluate the correlation function of $n$ energy densities when all $n$ points lie on a line. The $n$-energy correlation function is simply a sum of products of two-energy correlations. The quotient form for the spin correlation plus scaling is shown to immediately imply the logarithmic specific heat.