Correlation Functions for the Two-Dimensional Ising Model

The energy-density---energy-density and energy-density---spin correlation functions are evaluated for the two-dimensional square Ising model. Results are obtained at zero magnetic field and near the critical point, i.e., for ε=(4J/kT)|(T-TcTc)|1. The correlation functions are ${〈{E}R{E}₀〉{-}{〈E〉}²,={({2J}{{π}})}²{{ε}}²[{{K}₁}²({ε}R){-}{{K}₀}²({ε}R)],}{〈{E}R{{σ}}₀〉{-}〈E〉〈{σ}〉,=({2J}{{π}}){ε}〈{σ}〉{∫}{2{ε}R}^{{∞}}ds {s}^{{-}2}{e}^{{-}8},}$ where $R$ is the distance between the points in question and ${K}₀$ and ${K}₁$ are modified Bessel functions of the second kind. Very similar results hold if the coupling constants are different in the two directions. The above results agree in form with the structure predicted by the "scaling-law" approach.