Gauge Transformations and the Nematic to Smectic-ATransition

Abstract

A continuum of $\theta$-parametrized gauge transformations is used to relate the order-parameter correlation function, $G_{\theta }(\overrightarrow{\mathrm{x}}, 0)$, in gauge $\theta$ to that in the transverse gauge ($\theta =\frac{\pi }{2}$). The exponent ${\eta }_{\theta }$ depends continuously on $\theta$ at an isotropic critical point. In the physical gauge ($\theta =0\mathrm{}$) in three dimensions ($d=3\mathrm{}$), Caillé's results for reduced temperature $t<\mathrm{}0\mathrm{}$, and a finite susceptibility at $t=0\mathrm{}$ at an isotropic critical point are found. For $t=0\mathrm{}$ and $d=3\mathrm{}$, the correlation lengths obtained from $G_0$ differ from those obtained from $G_{\frac{\pi }{2}}$ and satisfy ${{\xi }_{\parallel }}^{\mathrm{x}}\sim \frac{{\left({{\xi }_{\perp }}^{\mathrm{x}}\right)}^2}{\ln {{\xi }_{\perp }}^{\mathrm{x}}}$.