Anharmonic Effects in Bulk Smectic Liquid Crystals and Other "One-Dimensional Solids"

Anharmonic corrections to the familiar harmonic description of "one-dimensional solids" (i.e., liquids which develop a one-dimensional mass-density wave) are calculated by analytic renormalization-group methods in three dimensions. The elastic constants describing the compressional and undulational modes respectively vanish and diverge logarithmically at small wave vectors. Density correlations decay as ${\mathcal{r}}^{-x}$ with $x=\eta \mathrm{}\left(T\right)\mathrm{}{\left(\ln \mathcal{r}\right)}^{\tilde{\eta }}$, where the exponent $\tilde{\eta }$ assumes the universal value $\frac{1}{5}$.