The high-temperature series for the two-spin correlation functions on both the B-site spinel and diamond lattices have been calculated through tenth order for the Hamiltonian, H=−JΣ〈 ij 〉 Σα=1n(n≤ D) SiαSjα. Here Siα is the αth component of the D-dimensional classical spin and we define n as the spin space symmetry parameter. By <ij> we limit the first summation to nearest-neighbor pairs of sites i and j on the lattice. The spin space symmetry parameter n is ≤D. We consider all six models resulting from D=1–3. Our analysis of the high-temperature series indicates that while the exponent estimates for the more loosely packed diamond lattice (q=4) indicate agreement with the universality hypothesis, the B-site spinel lattice (q=6) series are not yet displaying their critical behavior. Some conjectures concerning this peculiarity of the series for the two lattices are made.
The high-temperature series for the two-spin correlation functions on both the B-site spinel and diamond lattices have been calculated through tenth order for the Hamiltonian, H=−JΣ〈 ij 〉 Σα=1n(n≤ D) SiαSjα. Here Siα is the αth component of the D-dimensional classical spin and we define n as the spin space symmetry parameter. By <ij> we limit the first summation to nearest-neighbor pairs of sites i and j on the lattice. The spin space symmetry parameter n is ≤D. We consider all six models resulting from D=1–3. Our analysis of the high-temperature series indicates that while the exponent estimates for the more loosely packed diamond lattice (q=4) indicate agreement with the universality hypothesis, the B-site spinel lattice (q=6) series are not yet displaying their critical behavior. Some conjectures concerning this peculiarity of the series for the two lattices are made.
Using the vertex renormalized form of the technique developed by Englert,1 high-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heisenberg model are calculated for various lattices and anisotropies through order T−8 (close-packed lattices) and T−9 (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices γ (zero-field susceptibility), ν (correlation range), and α (specific heat) as functions of anisotropy. The results are consistent with the hypothesis that the critical indices only change when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended. Series for the limiting cases (spin-infinity Ising, XY and isotropic Heisenberg models) as well as for the spin-½ Ising model2 have been derived for the B-site spinel lattice. Analysis of these series shows that the anomalies in the critical indices, reported on the basis of six-term expansions,3 disappear with the use of longer series.
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 ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ is calculated to tenth order in (kBT1) for the Ising Hamiltonian ${\mathcal{H}}_{l \mathrm{anis}}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}\stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}\stackrel{z}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}$ for a wide range of anisotropy parameters $R\ensuremath{\equiv}\frac{{J}_{z}}{{J}_{\mathrm{xy}}}$ and for both the sc and fcc lattices; here the first summation is over all pairs of nearest-neighbor sites whose relative displacement vector ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{\mathrm{ij}}$ has no z component, while the second summation is over all other pairs of nearest-neighbor sites. Hence for R=0, both the sc and fcc lattices reduce to the two-dimensional square lattice, while in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the sc becomes a one-dimensional linear chain and the fcc becomes two noninteracting three-dimensional bcc lattices. The series for ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ 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 $\ensuremath{\gamma}$ and the correlation length exponent $\ensuremath{\nu}$) 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).
The authors have calculated the first 16 terms of the series expansion for the high-temperature susceptibility of the Ising model for a lattice made up of the B sites of the crystobalite or spinel structures (the 'tetrahedron' lattice). By means of a transformation technique which allows the enumeration of graphs to be carried out on the simpler diamond lattice six new terms have been added to previously published results. They estimate that the critical temperature is given by nu c =0.23300+or-0.00001 and that the exponent gamma is gamma =1.250+or-0.001, a result which gives further confirmation of the universality hypothesis.
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 ${\mathcal{H}}_{\mathrm{nnn}}=\ensuremath{-}{J}_{1}\ensuremath{\Sigma}\stackrel{\mathrm{nn}}{〈\mathrm{ij}〉}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}^{(D)}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}^{(D)}\ensuremath{-}{J}_{2}\ensuremath{\Sigma}\stackrel{\mathrm{nnn}}{〈\mathrm{ij}〉}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}^{(D)}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}^{(D)},$ where the first and second sums are over pairs of nearest-neighbor (nn) and next-nearest-neighbor (nnn) sites, and where the spins ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}^{(D)}$ are D-dimensional unit vectors. The two-spin correlation function, ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, is calculated to tenth, ninth, and eighth order in kBT1 for the Ising (D=1), classical-planar (D=2), and classical-Heisenberg (D=3) models, respectively, for various values of the parameter ${R}^{\ensuremath{'}}\ensuremath{\equiv}\frac{{J}_{2}}{{J}_{1}}$ 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 ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ we obtain series for the specific heat, susceptibility, and second moment. Analysis of these series and detailed comparisons with the exactly soluble spherical model ($D=\ensuremath{\infty}$) lead us to conclude that the exponents $\ensuremath{\gamma}$ (susceptibility) and $\ensuremath{\nu}$ (correlation length) may be independent of ${R}^{\ensuremath{'}}$; this suggestion is consistent with the universality hypothesis.
We present the high-temperature series for a nearest-neighbor model with O(2) symmetry on a simple cubic lattice with the most general single-site potential. In particular, the magnetic susceptibility and the second-moment correlation length are computed to 22nd order. The series specialized to some particular improved Hamiltonians have been already analyzed in the paper M. Campostrini, M. Hasenbusch, A. Pelissetto, and E. Vicari, Phys. Rev. B 74, 144506 (2006) [cond-mat/0605083], to determine the critical exponents and other universal quantities of the three-dimensional XY universality class.
High-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heinsenberg model are calculated for various lattices and anisotropies through order ${T}^{\ensuremath{-}8}$ (close-packed lattices) and ${T}^{\ensuremath{-}9}$ (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices $\ensuremath{\gamma}$ (susceptibility), $\ensuremath{\nu}$ (correlation range), and $\ensuremath{\alpha}$ (specific heat) as functions of anisotropy. Our results are consistent with the hypothesis that the critical indices change only when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended.