Critical Behavior of the Anisotropicn-Vector Model

Abstract

The critical behavior of a system of $n$-component classical "spins"with anisotropic pair interactions is discussed using renormalization-group techniques for dimension $d=4-\epsilon$. To first order in $\epsilon$ the correlation and susceptibility exponents in the isotropic limit are $2\nu =\gamma =1+\left[\frac{\mathrm{}(n+2\mathrm{})\mathrm{}}{2(n+8\mathrm{})\mathrm{}}\right]\epsilon$, while the anisotropy or crossover exponent is $\phi =1+\left[\frac{n}{2(n+8\mathrm{})\mathrm{}}\right]\epsilon$. When $n\to \infty$ these expansions agree with exact spherical-model results. For $n=3\mathrm{}$ and $d=3\mathrm{}$ series expansions indicate $\phi \simeq 1.2$ (compared with $\gamma \simeq 1.38$).