New Inequalities among the Critical-Point Exponents for the Spin-Spin and Energy-Energy Correlation Functions

Abstract

Two new inequalities, (i) ${\gamma }^{\prime }\ge \frac{2\beta (2-\eta )}{(d-2+\eta )}+2\mathrm{}\phi [\frac{2\beta }{(d-2+\eta )}-{\nu }_{\phi }^{\prime }]$ and (ii) $\frac{(\delta -1)}{\delta }\ge \frac{2(2-\eta )}{\delta (d-2+\eta )}+2\mathrm{}\phi [\frac{2}{\delta (d-2+\eta )}-{\mu }_{\phi }]$, are derived among critical-point exponents that describe the behavior of the two-spin correlation function $C_2(T, H, \overrightarrow{\mathrm{r}})\equiv 〈s_0^z{s}_{\overrightarrow{\mathrm{r}}}^z〉-〈s_0^z〉〈s_{\overrightarrow{\mathrm{r}}}^z〉$, subject to plausible assumptions (rigorous for Ising magnets). Here ${\nu }_{\phi }^{\prime }$ and ${\mu }_{\phi }$ describe the divergence as $T\to T_c^{-}$ and as $H\to 0^{+}$, respectively, of the "generalized correlation length" ${\xi }_{\phi }(T, H)$, defined as the $2\phi \mathrm{th}$ root of the normalized $2\phi \mathrm{th}$ spatial moment of $C_2(T, H, \overrightarrow{\mathrm{r}})$. 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 ${\nu }_{\phi }^{\prime }$ is independent of $\phi$, then (i) implies that ${\nu }^{\prime }\ge \frac{2\beta }{(d-2+\eta )}$ and ${\gamma }^{\prime }\ge (2-\eta ){\nu }^{\prime }$, while if ${\mu }_{\phi }$ is independent of $\phi$, then (ii) implies $\mu \ge \frac{2}{\delta (d-2+\eta )}$ and $\frac{(\delta -1)}{\delta }\ge (2-\eta )\mu$.