Smectic-A−Ctransition in three dimensions

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 $\mathrm{XY}$ 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 ${\eta }_A$ in the $A$ phase to algebraic decay characterized by a different nonuniversal exponent ${\eta }_{\mathrm{AC}}$ at the critical temperature $T_c$. The splay elastic constant $K_1$ is $<\infty$ and $=\infty$ in the $A$ and $C$ phases, respectively, but does not exhibit diverging pretransitional fluctuations in the $A$ phase even arbitrarily near $T_c$; it jumps discontinuously at $T_c$. A strong magnetic field stabilizes long-range positional order in both the $A$ and $C$ phases. There is, nevertheless, no long-range order at $T_c$, where the density correlation function $〈m\left(\overrightarrow{\mathrm{x}}\right)m\left(\overrightarrow{\mathrm{o}}\right)〉$ decays algebraically; schematically, $〈m\left(\overrightarrow{\mathrm{x}}\right)m\left(\overrightarrow{\mathrm{o}}\right)〉\sim x^{-(2+\eta )}$ at $T_c$ in the harmonic theory, while $m\left(\overrightarrow{\mathrm{x}}\right)$ vanishes like $t^{\beta }$, where $t$ is the reduced temperature. The exponents $\beta$ and $\eta$, though nonuniversal, satisfy the scaling law $4\beta =2+\eta$. These finite-field results are modified at large distances (say $x\gtrsim {10}^5$ Å) by the inclusion of anharmonic terms. We find