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 ${H}ₙₙₙ={-}{J}₁{Σ}{nn}{〈ij〉}{{{→}}{S}}ᵢ(D)·{}{{{→}}{S}}ⱼ(D){-}{J}₂{Σ}{nnn}{〈ij〉}{{{→}}{S}}ᵢ(D)·{}{{{→}}{S}}ⱼ(D),$ where the first and second sums are over pairs of nearest-neighbor (nn) and next-nearest-neighbor (nnn) sites, and where the spins ${{{→}}{S}}(D)$ are $D$-dimensional unit vectors. The two-spin correlation function, ${C}₂({{→}}{r})$, is calculated to tenth, ninth, and eighth order in ${1}{{k}BT}$ for the Ising ($D=1$), classical-planar ($D=2$), and classical-Heisenberg ($D=3$) models, respectively, for various values of the parameter ${R}^{{'}}{≡}{{J}₂}{{J}₁}$ 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 ${C}₂({{→}}{r})$ we obtain series for the specific heat, susceptibility, and second moment. Analysis of these series and detailed comparisons with the exactly soluble spherical model ($D={∞}$) lead us to conclude that the exponents ${γ}$ (susceptibility) and ${ν}$ (correlation length) may be independent of ${R}^{{'}}$; this suggestion is consistent with the universality hypothesis.