Decay of Order in Classical Many-Body Systems. II. Ising Model at High Temperatures

Abstract

We employ the transfer-matrix formalism expounded in Paper I to study the decay of pair correlation functions at high temperatures in the $d$-dimensional Ising model in an arbitrary magnetic field $H$. A general correlation function decays according to $〈\ensuremath{\delta}L(\stackrel{\ensuremath{\rightarrow}}{0})\ensuremath{\delta}Q(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})〉\ensuremath{\approx}{R}^{\frac{(d\ensuremath{-}1)}{2}} {e}^{\ensuremath{-}\ensuremath{\kappa}R}({A}_{0}+{A}_{1}{R}^{\ensuremath{-}1}+\ensuremath{\cdots})+{R}^{\ensuremath{-}x}{e}^{\ensuremath{-}{\ensuremath{\kappa}}^{\ensuremath{'}}R}({B}_{0}+{B}_{1}{R}^{\ensuremath{-}1}+\ensuremath{\cdots})+\ensuremath{\cdots}$ as $R\ensuremath{\rightarrow}\ensuremath{\infty}$. For sufficiently small $H$ and high temperature $T$, the exponent $x$ is equal to the dimension of the lattice and ${\ensuremath{\kappa}}^{\ensuremath{'}}=2\ensuremath{\kappa}$. The coefficients ${A}_{n}$ and ${B}_{n}$ factor as ${A}_{n}(Q){A}_{n}(L)$ and ${B}_{n}(Q){B}_{n}(L)$, respectively. If $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ is an operator involving an odd number of closely spaced spins, ${A}_{n}(Q)$ tends to a finite limit and ${B}_{n}(Q)$ tends to zero as $H$ tends to zero. In contrast, if $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ involves an even number of closely spaced spins, ${A}_{n}(Q)$ tends to zero and ${B}_{n}(Q)$ to a finite limit as $H$ tends to zero. Thus, for finite $H$ an arbitrary pair correlation function verifies the Ornstein-Zernike (OZ) prediction; whereas in the zero field, (i) if both $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}$ are products of odd numbers of spin operators, ${G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ will verify the OZ prediction in zero field; (ii) if both $\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ involve even numbers of spins, ${G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ fails to verify the OZ prediction in zero field; and (iii) if $\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}$ involves an odd number of spins while $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ involves an even number of spins, ${G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ is identically zero in zero field for all $\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}$. The failure of the zero-field Ising model to completely verify the OZ prediction at high temperatures is due to the symmetry of this model about the $H=0$ line.