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Physical review. B, Solid state

Crossover scaling functions for exchange anisotropy

Pierre M. Pfeuty, David Jasnow, Michael Ellis Fisher

Published Sep 1, 1974• 256 citations
Publisher

Abstract

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 ${T}_{c0}$ 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 ${P}_{\ensuremath{\alpha}}(1)\ensuremath{\simeq}1.10$.

DOI
10.1103/physrevb.10.2088

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