The crossover behavior of the susceptibilities ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}$ of spin systems with anisotropic exchange coupling is discussed on the basis of an "extended" scaling hypothesis which gives ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}(T,g)\ensuremath{\approx}A{t}^{\ensuremath{-}\ensuremath{\gamma}}{X}_{\ensuremath{\alpha}}(\frac{Bg}{{t}^{\ensuremath{\varphi}}})$, where
g is the anisotropy parameter, $t=\frac{(T\ensuremath{-}{T}_{c0})}{{T}_{c0}}$, with
Tc0 and $\ensuremath{\gamma}$ being the iso tropic critical temperature, and susceptibility exponent, while
A and
B are model-dependent amplitudes. Analysis of high-temperature series expansions for ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}(T,g)$ as polynomials in
g for rhombic and axial anisotropy in the fcc, bcc, and simple cubic classical Heisenberg (
n=3) and classical
XY and planar-spin models (
N=2), verifies the scaling with crossover exponents $\ensuremath{\varphi}=1.25\ifmmode\pm\else\textpm\fi{}0.015$ (
n=3) and $\ensuremath{\varphi}=1.175\ifmmode\pm\else\textpm\fi{}0.015$ (
n=2). The universality of the scaling functions ${X}_{\ensuremath{\alpha}}(x)$ is demonstrated both for small
x and in the anisotropic limit $T\ensuremath{\rightarrow}{T}_{c}(g)$, where $x\ensuremath{\rightarrow}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}$; for
n=3, accurate representations are constructed in the form ${X}_{\ensuremath{\alpha}}(x)\ensuremath{\simeq}{P}_{\ensuremath{\alpha}}(\frac{x}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}}){(1\ensuremath{-}\frac{x}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}})}^{\ensuremath{-}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}}$, where $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}$ is the anisotropic susceptibility exponent, while ${P}_{\ensuremath{\alpha}}(z)$ varies smoothly from ${P}_{\ensuremath{\alpha}}(0)=1$ to ${P}_{\ensuremath{\alpha}}(1)\ensuremath{\simeq}1.10$.
The susceptibility of simple-cubic-lattice ferromagnetic Ising films of
n two-dimensional lattice layers is studied by extrapolation of high-temperature series expansions (to eleventh and twelfth orders) for
n=3,4,…,14 layers with periodic boundary conditions, and
n=3,4,…,10 layers with free-surface conditions. The corresponding surface susceptibility series for sc, bcc, and fcc lattices are analyzed in the light of universality hypotheses. On the basis of finite-size scaling theory explicit scaling functions are constructed describing the crossover from two-dimensional to three-dimensional critical behavior in terms of the ratio $\frac{n}{\ensuremath{\xi}(T)}$, where $\ensuremath{\xi}\ensuremath{\sim}\ensuremath{\Delta}{T}^{\ensuremath{-}\ensuremath{\nu}}$ is the bulk correlation length.
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 ${\mathcal{H}}_{l,\mathrm{anis}}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j}\ensuremath{\equiv}\ensuremath{-}{J}_{\mathrm{xy}}(\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}+R\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j})$ where ${s}_{i}=\ifmmode\pm\else\textpm\fi{}1$, the first summation is over all nearest-neighbor pairs in the
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
Jxy and
Jz 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 $R\ensuremath{\equiv}\frac{{J}_{z}}{{J}_{\mathrm{xy}}}=0$, both the sc and fcc lattices reduce to two-dimensional square lattices, while in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the sc reduces to noninteracting linear chains.
Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K; for the triangular lattice to v 16 , for the square lattice to v 21 and for the honeycomb lattice to v 32 , inclusive. The asymptotic behaviour of the ferromagnetic and antiferromagnetic susceptibility is studied. It is concluded that the ferromagnetic singularity is not exactly factorizible. The antiferromagnetic susceptibility of the square and honeycomb lattices has a singularity of the same type as the energy at the antiferromagnetic critical temperature.
The anisotropic Heisenberg model has been investigated by interpolating anisotropy between the Ising and isotropic Heisenberg limits via general spin high temperature series (HTS) for the zero-field susceptibility of the ferromagnets for the square lattice. This is done by locating the critical temperature ( T c (2) ) and the susceptibility exponent (γ) from the analysis of the series by Pade approximant and ratio methods. The critical temperature is found to be spin dependent. It decreases as anisotropy changes from the Ising limit towards the isotropic limit and remains finite (lower than that of Stanley and Kaplan) for all the spins except for S =1/2 at the isotropic limit. Exponent γ is almost Ising like for the anisotropic system and anomalous jump in it occurs for the isotropic case. In addition, expressions for the magnetization are derived by using spin-wave theory. The limit of complete isotropy has been found not to be a special case for the susceptibility, in contrast to magnetization.
Series for the reduced susceptibility $\overline{\ensuremath{\chi}}$, the reduced specific heat
CH, and second moment ${\ensuremath{\mu}}_{2}$ of the static correlation function for the three-dimensional
S=21 Ising model on both the simple cubic (sc) and face-centered cubic (fcc) lattices with different coupling strengths in different lattice directions have been analyzed to determine the crossover exponent $\ensuremath{\phi}$ describing the behavior of the critical temperature as a function of the anisotropy parameter
R in the Hamiltonian $\mathcal{H}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j}\ensuremath{\equiv}\ensuremath{-}{J}_{\mathrm{xy}}(\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}+R\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j})$. Here
si=+1, the first sum is over all nearest-neighbor pairs in the
xy plane, and the second sum is over all pairs coupled in the
z direction. The constant gap exponent we obtain for successive derivatives of $\overline{\ensuremath{\chi}}$ and
CH with respect to
R confirms the exponent predictions of scaling in the parameter
R for thermodynamic functions, while the results of the ${\ensuremath{\mu}}_{2}$ series confirm the exponent predictions of scaling with respect to
R for the two-spin correlation function. Our results agree with the predictions for $\ensuremath{\phi}$ of Abe and Suzuki, and also with rigorous relations satisfied by the exponents describing the derivatives of the various functions. Our results do not agree with previously published results, which are based on an analysis of only the susceptibility on only the sc lattice.