General-RHigh-Temperature Series for the Susceptibility, Second Moment, and Specific Heat of sc and fcc Ising Models with Lattice Anisotropy

Of particular current interest is the critical behavior of functions on crossing over from one lattice dimensionality to another. To this end, we report high-temperature series for an Ising model with lattice anisotropy-i.e., with different exchange constants for different lattice directions. The Hamiltonian is ${\mathcal{H}}_{l,\mathrm{anis}}=-J_{\mathrm{xy}}\Sigma {〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_i{s}_j-J_z\Sigma {〈\mathrm{ij}〉}^z{s}_i{s}_j\equiv -J_{\mathrm{xy}}(\Sigma {〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_i{s}_j+R\Sigma {〈\mathrm{ij}〉}^z{s}_i{s}_j)$ where $s_i=\pm 1$, the first summation is over all nearest-neighbor pairs in the $\mathrm{xy}$ plane, and the second sum is over pairs coupled in the $z$ direction. The susceptibility, second moment, and specific-heat series are explicitly presented for arbitrary $J_{\mathrm{xy}}$ and $J_z$ for the simple cubic (sc) and face-centered cubic (fcc) lattices to tenth order in inverse temperature. The general-$R$ series are essential if one wishes to study the Riedel-Wegner crossover exponent appropriate to changing lattice dimensionality, since for $R\equiv \frac{J_z}{J_{\mathrm{xy}}}=0\mathrm{}$, both the sc and fcc lattices reduce to two-dimensional square lattices, while in the limit $R\to \infty$, the sc reduces to noninteracting linear chains.