Nonlinear elasticity of the sliding columnar phase

The sliding columnar phase is a liquid-crystalline phase of matter composed of two-dimensional smectic lattices stacked one on top of the other. This phase is characterized by strong orientational but weak positional correlations between lattices in neighboring layers and a vanishing shear modulus for sliding lattices relative to each other. A simplified elasticity theory of the phase only allows intralayer fluctuations of the columns and has three important elastic constants: the compression, rotation, and bending moduli B , Ky , and K . The rotationally invariant theory contains anharmonic terms that lead to long-wavelength renormalizations of the elastic constants similar to the Grinstein-Pelcovits renormalization of the elastic constants in smectic liquid crystals @Phys. Rev. Lett. 47, 856 ~1981!; Phys. Rev. A 26, 915 ~1982!#. We calculate these renormalizations at the critical dimension d53 and find that Ky(q);K (q);B(q);@ ln (1/q)#, where q is a wave number. The behavior of B , Ky , and K in a model that includes fluctuations perpendicular to the layers is identical to that of the simple model with rigid layers. We use dimensional regularization rather than a hard-cutoff renormalization scheme because ambiguities arise in the one-loop integrals with a finite cutoff. @S1063-651X~98!11011-5#