New Inequalities among the Critical-Point Exponents for the Spin-Spin and Energy-Energy Correlation Functions
Luke Lokia Liu, Richard I. Joseph, Eugene Stanley
Two new inequalities, (i) ${\ensuremath{\gamma}}^{\ensuremath{'}}\ensuremath{\ge}\frac{2\ensuremath{\beta}(2\ensuremath{-}\ensuremath{\eta})}{(d\ensuremath{-}2+\ensuremath{\eta})}+2\ensuremath{\varphi}[\frac{2\ensuremath{\beta}}{(d\ensuremath{-}2+\ensuremath{\eta})}\ensuremath{-}{\ensuremath{\nu}}_{\ensuremath{\varphi}}^{\ensuremath{'}}]$ and (ii) $\frac{(\ensuremath{\delta}\ensuremath{-}1)}{\ensuremath{\delta}}\ensuremath{\ge}\frac{2(2\ensuremath{-}\ensuremath{\eta})}{\ensuremath{\delta}(d\ensuremath{-}2+\ensuremath{\eta})}+2\ensuremath{\varphi}[\frac{2}{\ensuremath{\delta}(d\ensuremath{-}2+\ensuremath{\eta})}\ensuremath{-}{\ensuremath{\mu}}_{\ensuremath{\varphi}}]$, are derived among critical-point exponents that describe the behavior of the two-spin correlation function ${C}_{2}(T, H, \stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})\ensuremath{\equiv}〈{s}_{0}^{z}{s}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{z}〉\ensuremath{-}〈{s}_{0}^{z}〉〈{s}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{z}〉$, subject to plausible assumptions (rigorous for Ising magnets). Here ${\ensuremath{\nu}}_{\ensuremath{\varphi}}^{\ensuremath{'}}$ and ${\ensuremath{\mu}}_{\ensuremath{\varphi}}$ describe the divergence as $T\ensuremath{\rightarrow}{T}_{c}^{\ensuremath{-}}$ and as $H\ensuremath{\rightarrow}{0}^{+}$, respectively, of the "generalized correlation length" ${\ensuremath{\xi}}_{\ensuremath{\varphi}}(T, H)$, defined as the $2\ensuremath{\varphi}\mathrm{th}$ root of the normalized $2\ensuremath{\varphi}\mathrm{th}$ spatial moment of ${C}_{2}(T, H, \stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$. Also derived are the corresponding inequalities among exponents that describe the behavior of the energy-energy correlation function. Inequality (i) is shown to lead to an inequality between primed and unprimed exponents. Moreover, if ${\ensuremath{\nu}}_{\ensuremath{\varphi}}^{\ensuremath{'}}$ is independent of $\ensuremath{\varphi}$, then (i) implies that ${\ensuremath{\nu}}^{\ensuremath{'}}\ensuremath{\ge}\frac{2\ensuremath{\beta}}{(d\ensuremath{-}2+\ensuremath{\eta})}$ and ${\ensuremath{\gamma}}^{\ensuremath{'}}\ensuremath{\ge}(2\ensuremath{-}\ensuremath{\eta}){\ensuremath{\nu}}^{\ensuremath{'}}$, while if ${\ensuremath{\mu}}_{\ensuremath{\varphi}}$ is independent of $\ensuremath{\varphi}$, then (ii) implies $\ensuremath{\mu}\ensuremath{\ge}\frac{2}{\ensuremath{\delta}(d\ensuremath{-}2+\ensuremath{\eta})}$ and $\frac{(\ensuremath{\delta}\ensuremath{-}1)}{\ensuremath{\delta}}\ensuremath{\ge}(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\mu}$.
Physical review. B, Solid state•Sep 1, 1972•5 citations