Visibility of Critical-Exponent Renormalization

The extent to which the renormalization of critical-point behavior should be visible experimentally is investigated on the basis of detailed numerical calculations for a three-dimensional soluble model (a mobile-electron Ising ferromagnet). If a dilution parameter x is defined such that the change in critical temperature from the "pure" or unrenormalized system is |Tc(x)-Tc⁰|=xTcf, where f=0.6-0.9, then we conclude that the effective exponents βfit(x) and γfit(x) which will be observed experimentally, vary roughly as βfit≈β+xΔβX and γfitγ+1/5x×(1+2x²)ΔγX. Here β and γ are the ideal exponents for the order parameter and total fluctuation or susceptibility of the pure system, while ΔβX=βX-β and ΔγX=γX-γ, in which βX=β(1-α^') and γX=γ(1-α) are the fully renormalized exponents, while α and α^' (assumed positive) describe the divergence of the specific heats of the pure system. [Theoretically the true limiting asymptotic behavior at the transition is described by β(x)=βX and γ(x)=γX for all $x>0$.] The renormalized specific heats are found to be sensitive to x but their true renormalized behavior is not evident until x0.3. Various techniques of data analysis such as logarithmic, semilogarithmic, Heller-Benedek, and Kouvel-Fisher plots have been tested.