Decay of Order in Classical Many-Body Systems. III. Ising Model at Low Temperature

Abstract

In this work the decay of correlation in the $d$-dimensional Ising model is studied at low temperatures as a function of dimensionality of the lattice and magnetic field $h$. Except for the special case of the two-dimensional zero-field nearest-neighbor lattices, the decay of correlation verifies the Ornstein-Zernike prediction ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})\ensuremath{\approx}{D}_{\mathrm{AB}}(d,h){R}^{\ensuremath{-}\frac{(d\ensuremath{-}1)}{2}}{e}^{\ensuremath{-}\ensuremath{\kappa}R}$. For the two-dimensional zero-field case, the Ornstein-Zernike form is replaced by the "anomalous" form ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})\ensuremath{\approx}{D}_{\mathrm{AB}}{R}^{\ensuremath{-}2}{e}^{\ensuremath{-}\ensuremath{\kappa}R}$. This "anomalous" result is shown to arise from the peculiarities of the spectrum of the transfer matrix in this case and is replaced by the Ornstein-Zernike result when further-neighbor forces are present. The results presented herein agree with the previously obtained exact results for the zero-field two-dimensional Ising model.