Visibility of Critical-Exponent Renormalization

Abstract

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 $|{T}_{c}(x)\ensuremath{-}{T}_{c}^{0}|=\frac{x{T}_{c}}{f}$, where $f=0.6\ensuremath{-}0.9$, then we conclude that the effective exponents ${\ensuremath{\beta}}_{\mathrm{fit}}(x)$ and ${\ensuremath{\gamma}}_{\mathrm{fit}}(x)$ which will be observed experimentally, vary roughly as ${\ensuremath{\beta}}_{\mathrm{fit}}\ensuremath{\approx}\ensuremath{\beta}+x\ensuremath{\Delta}{\ensuremath{\beta}}_{X}$ and ${\ensuremath{\gamma}}_{\mathrm{fit}}\ensuremath{\simeq}\ensuremath{\gamma}+\frac{1}{5}x\ifmmode\times\else\texttimes\fi{}(1+2{x}^{2})\ensuremath{\Delta}{\ensuremath{\gamma}}_{X}$. Here $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ are the ideal exponents for the order parameter and total fluctuation or susceptibility of the pure system, while $\ensuremath{\Delta}{\ensuremath{\beta}}_{X}={\ensuremath{\beta}}_{X}\ensuremath{-}\ensuremath{\beta}$ and $\ensuremath{\Delta}{\ensuremath{\gamma}}_{X}={\ensuremath{\gamma}}_{X}\ensuremath{-}\ensuremath{\gamma}$, in which ${\ensuremath{\beta}}_{X}=\frac{\ensuremath{\beta}}{(1\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}})}$ and ${\ensuremath{\gamma}}_{X}=\frac{\ensuremath{\gamma}}{(1\ensuremath{-}\ensuremath{\alpha})}$ are the fully renormalized exponents, while $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{\ensuremath{'}}$ (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 $\ensuremath{\beta}(x)={\ensuremath{\beta}}_{X}$ and $\ensuremath{\gamma}(x)={\ensuremath{\gamma}}_{X}$ for all $x>0$.] The renormalized specific heats are found to be sensitive to $x$ but their true renormalized behavior is not evident until $x\ensuremath{\gtrsim}0.3$. Various techniques of data analysis such as logarithmic, semilogarithmic, Heller-Benedek, and Kouvel-Fisher plots have been tested.