Two-scale-factor universality and the renormalization group

The hypothesis of two-scale-factor universality, originally proposed by Stauffer, Ferer, and Wortis, is shown to follow from the renormalization-group approach, for systems close to their critical point. Values of the universal ratios involving correlation length and specific-heat amplitudes are obtained from the $\epsilon$ expansion, for Ising, $X-Y$, and Heisenberg models. In the latter two cases the correlation function has a power-law behavior at large distances below $T_c$, and the (transverse) correlation length is defined in terms of the stiffness constant ${\rho }_s$. Experimental values of the correlation lengths and amplitude ratios are determined for superfluid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, which is $X-Y$-like, and for the Heisenberg antiferromagnet ${\mathrm{RbMnF}}_3$. Comparisons are made between the values of the amplitude ratios coming from $\epsilon$ expansions, series, and experiments.