Nematic—to—smectic-Atransitions with planar director fluctuations

Nematic—to—smectic-$A$ transitions in liquid crystals with director fluctuations confined to an easy plane are studied via a scaling analysis and an expansion in $\epsilon =4-d$. The lower critical dimension for these systems is $d^{*}=2.5\mathrm{}$. In three dimensions, scaling predicts anisotropically diverging correlation lengths, with exponents in the ratio 3:2. The $\epsilon$ expansion suggests a fluctuation-driven first-order phase transition for an $n$-component generalized model, provided $n<\mathrm{}110.8\mathrm{}$. For $n>\mathrm{}110.8\mathrm{}$, we find a stable, anisotropic fixed point with, however, a much weaker anisotropy than that predicted by scaling. Two sets of lengths are then necessary to describe order-parameter correlations below the transition. The constraint of planar director fluctuations can, in principle, be imposed experimentally by applying electric fields to nematics with a large transverse dielectric constant. The theory may also be applicable to transitions from smectic-$C$ liquid crystals to bismectics.