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

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.