Nonlinear Elasticity of the Sliding Columnar Phase

TL;DR

This paper develops a nonlinear elasticity theory for the sliding columnar phase, revealing how elastic constants renormalize at long wavelengths due to anharmonic effects, with implications for liquid-crystalline materials.

Contribution

It introduces a simplified, rotationally invariant elasticity model for the sliding columnar phase and calculates the long-wavelength renormalizations of elastic constants at the critical dimension.

Findings

Elastic constants exhibit logarithmic renormalization at d=3.

Renormalizations follow specific power-law relations with wavevector.

Model including perpendicular fluctuations behaves similarly to the simplified model.

Abstract

The sliding columnar phase is a new 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, $B$, $K_y$, 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. We calculate these renormalizations at the critical dimension $d=3$ and find that $K_y(q) \sim K^{1/2}(q) \sim B^{-1/3}(q) \sim (\ln(1/q))^{1/4}$, where $q$ is a wavenumber. The behavior of $B$, $K_y$, 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.