Models for quasi-two-dimensional helium and magnets

Recently considerable interest has focused upon materials which change spatial dimensionality as the anisotropy parameter R is varied. Here we calculate the high-temperature series of the two-spin correlation function for the classical Heisenberg (D=3) and planar (D=2) models with lattice anisotropy. The Hamiltonian is H, =-{Jₗₘ〈ij〉{S}₈^ (D) {S}₉^ (D) -Jₙ〈ij〉{S}₈^ (D) {S}₉^ (D) }-{Jₗₘ (〈ij〉{S}₈^ (D) {S}₉^ (D) +R〈ij〉{S}₈^ (D) {S}₉^ (D) ) } where {S}₈^ (D) is a classical spin of D dimensions, the first summation is over all nearest-neighbor pairs in the xy plane, and the second sum is over pairs of spins coupling adjacent planes. The two-spin correlation functions are used to obtain the susceptibility (), specific heat (C₇) and spherical moments (ₓ) as double power series in Jₗₘ{k₁T} and Jₙ{k₁T} on both the simple-cubic and face-centered-cubic lattices. All series are to tenth order except for the Heisenberg model on the simple-cubic lattice which is to ninth order. The family of nth derivatives with respect to R are analyzed for the susceptibility and the spherical moments. By considering these functions in the limit R=0, we obtain evidence concerning the possibility of a phase transition for the two-dimensional (d=2) lattice. Our evidence rests upon standard methods, as well as on two new sequences (based on scaling in the parameter R): ₍, ₋₍, ₋-₍-₁, ₋{T₂}l{T₂^MF} and { \~{}}₍, ₋₍-₁, ₋- (n-1) ₍, ₋ (T₂{T₂^MF}) ({₀-1) }l+1. Here ₍, ₋ ({a₋+₍, ₍{a₋+₍-₁, ₍}) }{a₋, ₀}, where a₋, ₍ are the coefficients in {}^ (n) {^n}{R^n}l=n^a₋, ₍ ({JₗₘkT) }^l. Much of the evidence for the cases considered in this work (D=2, 3) is strengthened by comparison with the exactly known situations D=1 (Ising model) and D= (spherical model). Subject to the assumption that scaling in R holds, we estimate that the susceptibility exponent for the classical planar model is ₀ (D=2) =2. 53-₀. ₂₈^+0. 30. The evidence for the Heisenberg model is not as convincing, but if a phase transition does exist, then our methods suggest a susceptibility exponent of ₀ (D=3) 3. 5.