Scaling with Respect to a Parameter for the Gibbs Potential and Pair Correlation Function of theS=12Ising Model with Lattice Anisotropy

Abstract

Series for the reduced susceptibility $\overline{\chi }$, the reduced specific heat ${\overline{C}}_H$, and second moment ${\mu }_2$ of the static correlation function for the three-dimensional $S=\frac{1}{2}$ Ising model on both the simple cubic (sc) and face-centered cubic (fcc) lattices with different coupling strengths in different lattice directions have been analyzed to determine the crossover exponent $\varphi$ describing the behavior of the critical temperature as a function of the anisotropy parameter $R$ in the Hamiltonian $\mathcal{H}=-J_{\mathrm{xy}}\Sigma {〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_i{s}_j-J_z\Sigma {〈\mathrm{ij}〉}^z{s}_i{s}_j\equiv -J_{\mathrm{xy}}(\Sigma {〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_i{s}_j+R\Sigma {〈\mathrm{ij}〉}^z{s}_i{s}_j)$. Here $s_i=+1\mathrm{}$, the first sum is over all nearest-neighbor pairs in the $\mathrm{xy}$ plane, and the second sum is over all pairs coupled in the $z$ direction. The constant gap exponent we obtain for successive derivatives of $\overline{\chi }$ and ${\overline{C}}_H$ with respect to $R$ confirms the exponent predictions of scaling in the parameter $R$ for thermodynamic functions, while the results of the ${\mu }_2$ series confirm the exponent predictions of scaling with respect to $R$ for the two-spin correlation function. Our results agree with the predictions for $\varphi$ of Abe and Suzuki, and also with rigorous relations satisfied by the exponents describing the derivatives of the various functions. Our results do not agree with previously published results, which are based on an analysis of only the susceptibility on only the sc lattice.