Partial Test of the Universality Hypothesis: The Case of Next-Nearest-Neighbor Interactions

High-temperature series expansions are used to examine the dependence of critical-point exponents upon the presence of second-neighbor interactions. We consider the Hamiltonian ${\mathcal{H}}_{\mathrm{nnn}}=-J_1\Sigma \stackrel{\mathrm{nn}}{〈\mathrm{ij}〉}{\overrightarrow{\mathrm{S}}}_i^{\mathrm{}\left(D\right)\mathrm{}}·{\overrightarrow{\mathrm{S}}}_j^{\mathrm{}\left(D\right)\mathrm{}}-J_2\Sigma \stackrel{\mathrm{nnn}}{〈\mathrm{ij}〉}{\overrightarrow{\mathrm{S}}}_i^{\mathrm{}\left(D\right)\mathrm{}}·{\overrightarrow{\mathrm{S}}}_j^{\mathrm{}\left(D\right)\mathrm{}},$ where the first and second sums are over pairs of nearest-neighbor (nn) and next-nearest-neighbor (nnn) sites, and where the spins ${\overrightarrow{\mathrm{S}}}^{\mathrm{}\left(D\right)\mathrm{}}$ are $D$-dimensional unit vectors. The two-spin correlation function, $C_2\left(\overrightarrow{\mathrm{r}}\right)$, is calculated to tenth, ninth, and eighth order in $\frac{1}{k_{B}T}$ for the Ising ($D=1\mathrm{}$), classical-planar ($D=2\mathrm{}$), and classical-Heisenberg ($D=3\mathrm{}$) models, respectively, for various values of the parameter $R^{\prime }\equiv \frac{J_2}{J_1}$ and for various cubic lattices (fcc, bcc, and sample cubic). These represent the first series expansions of the spin correlation function for nnn interactions. From $C_2\left(\overrightarrow{\mathrm{r}}\right)$ we obtain series for the specific heat, susceptibility, and second moment. Analysis of these series and detailed comparisons with the exactly soluble spherical model ($D=\infty$) lead us to conclude that the exponents $\gamma$ (susceptibility) and $\nu$ (correlation length) may be independent of $R^{\prime }$; this suggestion is consistent with the universality hypothesis.