Universal Ratios Among Correction-to-Scaling Amplitudes and Effective Critical Exponents

Abstract

Writing thermodynamic quantities near a critical point as ${f}_{i}\ensuremath{\sim}{A}_{i}{|t|}^{\ensuremath{-}{\ensuremath{\lambda}}_{i}}(1+{a}_{i}{|t|}^{\ensuremath{\Delta}})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$, we show that to leading order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ one has $\frac{{a}_{i}}{{a}_{j}}=\frac{({\ensuremath{\lambda}}_{i}\ensuremath{-}{{\ensuremath{\lambda}}_{i}}^{0})}{({\ensuremath{\lambda}}_{j}\ensuremath{-}{{\ensuremath{\lambda}}_{j}}^{0})}$, where ${{\ensuremath{\lambda}}_{i}}^{0}$ is the mean-field value of ${\ensuremath{\lambda}}_{i}$. This ratio is also equal to $\frac{({\ensuremath{\lambda}}_{i, \mathrm{eff}}\ensuremath{-}{\ensuremath{\lambda}}_{i})}{({\ensuremath{\lambda}}_{j, \mathrm{eff}}\ensuremath{-}{\ensuremath{\lambda}}_{j})}$, where ${\ensuremath{\lambda}}_{i, \mathrm{eff}}$ is derived from a fit of data with ${f}_{i}\ensuremath{\sim}{|t|}^{\ensuremath{-}{\ensuremath{\lambda}}_{i, \mathrm{eff}}}$. Various experiments are analyzed and compared to these predictions.