Hexatic-Herringbone Coupling at the Hexatic Transition in Smectic Liquid Crystals: 4-$\epsilon$ Renormalization Group Calculations Revisited

TL;DR

This study revisits the renormalization group analysis of the smectic-A to hexatic smectic-B phase transition, confirming it is a fluctuation-driven first order transition with new unstable fixed points.

Contribution

The paper identifies two previously overlooked nontrivial fixed points in the model, but confirms the transition remains first order due to fluctuations.

Findings

Two new nontrivial fixed points found but are unstable.

Transition remains fluctuation-driven first order.

Discusses potential missing order parameter effects.

Abstract

Simple symmetry considerations would suggest that the transition from the smectic-A phase to the long-range bond orientationally ordered hexatic smectic-B phase should belong to the XY universality class. However, a number of experimental studies have constantly reported over the past twenty years "novel" critical behavior with non-XY critical exponents for this transition. Bruinsma and Aeppli argued in Physical Review Letters {\bf 48}, 1625 (1982), using a $4-\epsilon$ renormalization-group calculation, that short-range molecular herringbone correlations coupled to the hexatic ordering drive this transition first order via thermal fluctuations, and that the critical behavior observed in real systems is controlled by a `nearby' tricritical point. We have revisited the model of Bruinsma and Aeppli and present here the results of our study. We have found two nontrivial strongly-coupled herringbone-hexatic fixed points apparently missed by those authors. Yet, those two new nontrivial fixed-points are unstable, and we obtain the same final conclusion as the one reached by Bruinsma and Aeppli, namely that of a fluctuation-driven first order transition. We also discuss the effect of local two-fold distortion of the bond order as a possible missing order parameter in the Hamiltonian.