Models for quasi-two-dimensional helium and magnets

Abstract

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 ($D=3\mathrm{}$) and planar ($D=2\mathrm{}$) models with lattice anisotropy. The Hamiltonian is $\{\mathcal{H},=-J_{\mathrm{xy}}\Sigma \stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{\overrightarrow{\mathrm{S}}}_i^{\mathrm{}\left(D\right)\mathrm{}}·{\overrightarrow{\mathrm{S}}}_j^{\mathrm{}\left(D\right)\mathrm{}}-J_z\Sigma \stackrel{z}{〈\mathrm{ij}〉}{\overrightarrow{\mathrm{S}}}_i^{\mathrm{}\left(D\right)\mathrm{}}·{\overrightarrow{\mathrm{S}}}_j^{\mathrm{}\left(D\right)\mathrm{}}\}\{\equiv -J_{\mathrm{xy}}\left(\Sigma \stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{\overrightarrow{\mathrm{S}}}_i^{\mathrm{}\left(D\right)\mathrm{}}·{\overrightarrow{\mathrm{S}}}_j^{\mathrm{}\left(D\right)\mathrm{}}+R\Sigma \stackrel{z}{〈\mathrm{ij}〉}{\overrightarrow{\mathrm{S}}}_i^{\mathrm{}\left(D\right)\mathrm{}}·{\overrightarrow{\mathrm{S}}}_j^{\mathrm{}\left(D\right)\mathrm{}}\right)\}$ where ${\overrightarrow{\mathrm{S}}}_i^{\mathrm{}\left(D\right)\mathrm{}}$ is a classical spin of $D$ dimensions, the first summation is over all nearest-neighbor pairs in the $\mathrm{xy}$ plane, and the second sum is over pairs of spins coupling adjacent planes. The two-spin correlation functions are used to obtain the susceptibility ($\chi$), specific heat ($C_H$) and spherical moments (${\mu }_t$) as double power series in $\frac{J_{\mathrm{xy}}}{k_{B}T}$ and $\frac{J_z}{k_{B}T}$ 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 $n\mathrm{th}$ derivatives with respect to $R$ are analyzed for the susceptibility and the spherical moments. By considering these functions in the limit $R=0\mathrm{}$, we obtain evidence concerning the possibility of a phase transition for the two-dimensional ($d=2\mathrm{}$) lattice. Our evidence rests upon standard methods, as well as on two new sequences (based on scaling in the parameter $R$): ${\Delta }_{n,l}\equiv {\rho }_{n,l}-{\rho }_{n-1,l}\sim \frac{\varphi T_c}{l{T}_c^{\mathrm{MF}}}$ and ${\tilde{\Delta }}_{n,l}\equiv n{\rho }_{n-1,l}-(n-1\mathrm{})\mathrm{}{\rho }_{n,l}\sim \left(\frac{T_c}{T_c^{\mathrm{MF}}}\right)[\frac{({\gamma }_0-1)}{l}+1]$. Here ${\rho }_{n,l}\equiv \frac{\left(\frac{a_{l+n,n}}{a_{l+n-1,n}}\right)}{a_{l,0\mathrm{}}}$, where $a_{l,n}$ are the coefficients in ${\overline{\chi }}^{\mathrm{}\left(n\right)\mathrm{}}\equiv \frac{{\partial }^n\overline{\chi }}{\partial R^n}\propto \Sigma {l=n}^{\infty }{a}_{l,n}{\left(\frac{J_{\mathrm{xy}}}{\mathrm{kT}}\right)}^l$. Much of the evidence for the cases considered in this work ($D=2, 3$) is strengthened by comparison with the exactly known situations $D=1\mathrm{}$ (Ising model) and $D=\infty$ (spherical model). Subject to the assumption that scaling in $R$ holds, we estimate that the susceptibility exponent for the classical planar model is ${\gamma }_0\mathrm{}(D=2\mathrm{})={2.53}_{-0.28\mathrm{}}^{+0.30\mathrm{}}$. 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 ${\gamma }_0\mathrm{}(D=3\mathrm{})\mathrm{}\cong 3.5$.