Four-spin correlation function along the diagonal

Abstract

We compute the four-spin correlation function $〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{N}{\ensuremath{\sigma}}_{N+M}{\ensuremath{\sigma}}_{N+M+L}〉$ along the diagonal of an Ising square lattice. We find that it can be written in the scaling form $〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{N}{\ensuremath{\sigma}}_{N+M}{\ensuremath{\sigma}}_{N+M+L}〉\ensuremath{\sim}〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{N}〉〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{L}〉+{〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{N}〉}^{c}{〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{L}〉}^{c}{f}_{4}(tN,tL,|t|(M+\frac{1}{2N}+\frac{1}{2L}))$, where ${|t|}^{\ensuremath{-}1}$ is the correlation length. The scaling function ${f}_{4}$ is obtained as a asymptotic series with the variables $\mathrm{tN}$ and $\mathrm{tL}$ small. The remarkably simple form of this function ${f}_{4}$ suggests the reduction formula ${\ensuremath{\sigma}}_{R\ensuremath{-}\frac{r}{2}}{\ensuremath{\sigma}}_{R+\frac{r}{2}}\ensuremath{-}〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{r}〉={〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{r}〉}^{c}[{c}_{1}(r){\ensuremath{\epsilon}}_{R}+{c}_{2}(r){t}_{R}+{c}_{3}(r){u}_{R}+\ensuremath{\cdots}]$, in which ${\ensuremath{\epsilon}}_{R}={\ensuremath{\sigma}}_{R}{\ensuremath{\sigma}}_{R+1}\ensuremath{-}〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{1}〉$ is the energy-density operator, while ${t}_{R}$ and ${u}_{R}$ denote some less singular operators. The coefficients of this expansion are independent of temperature.