Partial Test of the Universality Hypothesis: The Case of Different Coupling Strengths in Different Lattice Directions

Abstract

High-temperature series expansions are used to examine the dependence of critical-point exponents upon lattice anisotropy (different interaction strengths in different directions in the lattice). The two-spin correlation function $C_2\left(\overrightarrow{\mathrm{r}}\right)$ is calculated to tenth order in ($\frac{1}{k_{B}T}$) for the Ising Hamiltonian ${\mathcal{H}}_{l \mathrm{anis}}=-J_{\mathrm{xy}}\Sigma \stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{S}_i^z{S}_j^z-J_z\Sigma \stackrel{z}{〈\mathrm{ij}〉}{S}_i^z{S}_j^z$ for a wide range of anisotropy parameters $R\equiv \frac{J_z}{J_{\mathrm{xy}}}$ and for both the sc and fcc lattices; here the first summation is over all pairs of nearest-neighbor sites whose relative displacement vector ${\overrightarrow{\mathrm{r}}}_{\mathrm{ij}}$ has no $z$ component, while the second summation is over all other pairs of nearest-neighbor sites. Hence for $R=0\mathrm{}$, both the sc and fcc lattices reduce to the two-dimensional square lattice, while in the limit $R\to \infty$, the sc becomes a one-dimensional linear chain and the fcc becomes two noninteracting three-dimensional bcc lattices. The series for $C_2\left(\overrightarrow{\mathrm{r}}\right)$ are then used to obtain series of corresponding lengths for the specific heat, susceptibility, and second moment. Analysis of these series yields results consistent with the universality hypothesis of critical-point exponents. Specifically, it is found that when lattice anisotrophy is introduced, the critical-point exponents studied (the susceptibility exponent $\gamma$ and the correlation length exponent $\nu$) do not appear to change from their values for an isotropic lattice. The problem of next-nearest-neighbor interactions is treated using similar methods in Paper II of this series (and briefly discussed in this paper).