Single Mode Approximation Spectrum for a Quantum Hall Liquid Crystal

The single mode approximation (SMA) has been a useful tool for the determination of the collective excitation spectra of various systems. For example, Feynman used it successfully for the calculation of the spectrum of sound waves in superfluid 4He [1]; and Girvin, MacDonald and Platzman employed a projected SMA to determine that of the fractional quantum Hall (QH) effect [2] for Laughlin-like states [3]. In this paper we calculate the excitation spectrum for density fluctuations in a QH liquid crystal. We consider nematic, tetratic, and hexatic generalizations of Laughlin's trial wave function [3] having two-, four- and six-fold broken rotational symmetry (BRS) respectively. We perform extensive Monte Carlo (MC) simulations to compute the density-coupled spectrum of these liquid crystalline states in the lowest Landau levels. We find significant angular dependence of the spectrum with singular behavior for long wavelengths and considerable deepening of the magnetoroton as the anisotropy is increased.