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

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 $〈\delta L\left(\overrightarrow{0}\right)\delta Q\left(\overrightarrow{\mathrm{R}}\right)〉\approx R^{\frac{\mathrm{}(d-1\mathrm{})\mathrm{}}{2}} e^{-\kappa R}(A_0+A_1{R}^{-1\mathrm{}}+\cdots )+R^{-x}{e}^{-{\kappa }^{\prime }R}(B_0+B_1{R}^{-1\mathrm{}}+\cdots )+\cdots$ as $R\to \infty$. For sufficiently small $H$ and high temperature $T$, the exponent $x$ is equal to the dimension of the lattice and ${\kappa }^{\prime }=2\mathrm{}\kappa$. The coefficients $A_n$ and $B_n$ factor as $A_n\mathrm{}\left(Q\right)\mathrm{}{A}_n\mathrm{}\left(L\right)\mathrm{}$ and $B_n\mathrm{}\left(Q\right)\mathrm{}{B}_n\mathrm{}\left(L\right)\mathrm{}$, respectively. If $\overrightarrow{\mathrm{Q}}$ is an operator involving an odd number of closely spaced spins, $A_n\mathrm{}\left(Q\right)\mathrm{}$ tends to a finite limit and $B_n\mathrm{}\left(Q\right)\mathrm{}$ tends to zero as $H$ tends to zero. In contrast, if $\overrightarrow{\mathrm{Q}}$ involves an even number of closely spaced spins, $A_n\mathrm{}\left(Q\right)\mathrm{}$ tends to zero and $B_n\mathrm{}\left(Q\right)\mathrm{}$ 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 $\overrightarrow{\mathrm{Q}}$ and $\overrightarrow{\mathrm{L}}$ are products of odd numbers of spin operators,