Universal relations among critical amplitude. Calculations up to orderε2for systems with continuous symmetry

Abstract

A general derivation of the two-scale-factor universality is given using the renormalized ${\phi }^4$ theory. As a consequence ten universal relations among critical amplitudes are obtained, any other universal relations being a combination of them. The situation is shown to be the same as for the scaling laws for critical exponents. The calculations up to order ${\epsilon }^2$ for systems with continuous symmetry are performed for some of these relations such as the specific-heat amplitudes ratio and $R_{\xi }^{+}$, which relates the amplitudes of the correlation length and of the specific heat, etc. Comparison with experiments and series results and, in particular, the discussion of the superfluid helium and RbMn${\mathrm{F}}_3$ cases, are repeated with our improved numerical results.