Critical exponents of ferromagnets with dipolar interactions: Second-orderεexpansion

Abstract

The zero-field critical behavior of ferromagnets above $T_c$, with both isotropic exchange coupling and dipolar interactions, is studied by renormalization-group techniques in $d=4-\epsilon$ dimensions with $n \mathrm{}(=d)\mathrm{}$ component spins. The critical exponents are calculated to order ${\epsilon }^2$, and are found to be numerically very close to their nondipolar analogs. In particular the correlation length exponent is given by $2\nu =1+\left(\frac{9}{34}\right)\epsilon +\left(\frac{7013}{58956}\right){\epsilon }^2+O\left({\epsilon }^3\right)$. The value of the specific-heat exponent ${\alpha }_s (=2-d\nu )$ seems inconsistent with experimental data (at $d=3\mathrm{}$, $\epsilon =1\mathrm{}$). The critical scattering function is shown to have the form ${\Gamma }^{\alpha \beta }\left(\overrightarrow{\mathrm{q}}\right)=C{t}^{-\gamma }\hat{D}\left({\xi }^2{q}^2\right)({\delta }_{\alpha \beta }-\frac{q^{\alpha }{q}^{\beta }}{q^2})$ to all orders in $\epsilon$. The exponents related to the leading correction to scaling, to crossovers to anisotropic exchange behavior, and to cubic dipolar behavior, are also calculated, and are found to differ significantly from their nondipolar counterparts.