Crossover scaling functions for exchange anisotropy

Abstract

The crossover behavior of the susceptibilities ${\chi }^{\alpha \alpha }$ of spin systems with anisotropic exchange coupling is discussed on the basis of an "extended" scaling hypothesis which gives ${\chi }^{\alpha \alpha }(T,g)\approx A{t}^{-\gamma }{X}_{\alpha }\left(\frac{Bg}{t^{\phi }}\right)$, where $g$ is the anisotropy parameter, $t=\frac{(T-T_{c0})}{T_{c0}}$, with $T_{c0}$ and $\gamma$ being the iso tropic critical temperature, and susceptibility exponent, while $A$ and $B$ are model-dependent amplitudes. Analysis of high-temperature series expansions for ${\chi }^{\alpha \alpha }(T,g)$ as polynomials in $g$ for rhombic and axial anisotropy in the fcc, bcc, and simple cubic classical Heisenberg ($n=3\mathrm{}$) and classical $XY$ and planar-spin models ($N=2\mathrm{}$), verifies the scaling with crossover exponents $\phi =1.25\mathrm{}\pm 0.015$ ($n=3\mathrm{}$) and $\phi =1.175\mathrm{}\pm 0.015$ ($n=2\mathrm{}$). The universality of the scaling functions $X_{\alpha }\mathrm{}\left(x\right)\mathrm{}$ is demonstrated both for small $x$ and in the anisotropic limit $T\to T_c\mathrm{}\left(g\right)\mathrm{}$, where $x\to \dot{x}$; for $n=3\mathrm{}$, accurate representations are constructed in the form $X_{\alpha }\mathrm{}\left(x\right)\mathrm{}\simeq P_{\alpha }\left(\frac{x}{\dot{x}}\right){(1-\frac{x}{\dot{x}})}^{-\dot{\gamma }}$, where $\dot{\gamma }$ is the anisotropic susceptibility exponent, while $P_{\alpha }\mathrm{}\left(z\right)\mathrm{}$ varies smoothly from $P_{\alpha }\mathrm{}\left(0\right)=1\mathrm{}$ to $P_{\alpha }\mathrm{}\left(1\right)\mathrm{}\simeq 1.10$.