# Smectic-A-C transition in three dimensions

## Abstract

A theory of the smectic-A-C transition in which fluctuations in both the elastic and director degrees of freedom are treated is developed. It is argued that in zero magnetic field the asymptotic critical behavior belongs in the universality class of the threedimensional XY model, in agreement with earlier studies which did not treat fully both elastic and director fluctuations. For most experimental situations the true critical region is unobservably narrow whereupon a simple-harmonic approximation adequately describes the transition; in this approximation the static structure factor exhibits a crossover from algebraic decay characterized by a nonuniversal exponent ηA in the A phase to algebraic decay characterized by a different nonuniversal exponent ηAC at the critical temperature Tc. The splay elastic constant K1 is < and = in the A and C phases, respectively, but does not exhibit diverging pretransitional fluctuations in the A phase even arbitrarily near Tc; it jumps discontinuously at Tc. A strong magnetic field stabilizes long-range positional order in both the A and C phases. There is, nevertheless, no long-range order at Tc, where the density correlation function m(x→)m(o→) decays algebraically; schematically, m(x→)m(o→)∼x-(2+η) at Tc in the harmonic theory, while m(x→) vanishes like tβ, where t is the reduced temperature. The exponents β and η, though nonuniversal, satisfy the scaling law 4β=2+η. These finite-field results are modified at large distances (say x105 ) by the inclusion of anharmonic terms. We find m(x→)m(o→)∼x exp[-(2+η)|lnx|137] at Tc, while m(x→)∼tβ|lnt|137 as t→0. Here β and η are nonuniversal but satisfy 4β=2+η. At Tc, K1 and B (the compressional elastic constant), respectively, diverge and vanish like (lnk)237 and (lnk)-437 as the momentum k→0. The exponents 137, 237, and 437 are universal. © 1982 The American Physical Society.