Divergence of the Correlation Length along the Critical Isotherm

Abstract

Three plausible postulates (rigorous for Ising systems) are shown to lead to three new inequalities: (i) $(2-\eta ){\mu }_{\phi }\ge \frac{(\delta -1)}{\delta }$, (ii)${\mu }_{\phi }\ge \frac{2}{\delta (d-2+\eta )}$, and (iii)$d{\mu }_{\phi }\ge \frac{(\delta +1)}{\delta }$, concerning ${\mu }_{\phi }$, the critical-point exponent characterizing the divergence along the critical isotherm of the correlation length ${\xi }_{\phi }\mathrm{}(T,H)\mathrm{}\equiv {\left[\frac{\Sigma {\overrightarrow{\mathrm{r}}}^{}{\left|\overrightarrow{\mathrm{r}}\right|}^{2\phi }{C}_2(T,H,\overrightarrow{\mathrm{r}})}{\Sigma {\overrightarrow{\mathrm{r}}}^{}{C}_2(T,H,\overrightarrow{\mathrm{r}})}\right]}^{\frac{1}{2\phi }}$. Result (iii) for ${\mu }_{\phi }$ is an analog, for the critical isotherm, of the Josephson inequalities. If we make the plausible but unproved assumption that ${\mu }_{\phi }$ is independent of $\phi$, inequality (i) becomes an equality!