Replica study of pinned bubble crystals

TL;DR

This study calculates the frequency-dependent conductivity of bubble and Wigner crystal states in higher Landau levels, incorporating disorder effects, and compares results with microwave experiments to understand their dynamical properties.

Contribution

It introduces a combined elastic, Hartree-Fock, and replica approach to analyze the dynamical conductivity of bubble crystals in disordered quantum Hall systems.

Findings

Computed pinning frequencies matching experimental data

Identified the evolution of conductivity with filling factor

Provided a microscopic basis for disorder effects in bubble states

Abstract

In higher Landau levels ($N>1$), the ground state of the two-dimensional electron gas in a strong perpendicular magnetic field evolves from a Wigner crystal for small filling $\nu $ of the partially filled Landau level, into a succession of bubble states with increasing number of guiding centers per bubble as $\nu $ increases, to a modulated stripe state near $\nu =0.5$. In this work, we compute the frequency-dependent longitudinal conductivity $% \sigma_{xx}(\omega) $ of the Wigner and bubble crystal states in the presence of disorder. We apply an elastic theory to the crystal states which is characterized by a shear and a bulk modulus. We obtain both moduli from the microscopic time-dependent Hartree-Fock approximation. We then use the replica and Gaussian variational methods to handle the effects of disorder. Within the semiclassical approximation we get the dynamical conductivity as well as the pinning frequency as functions of the Landau level filling factor and compare our results with recent microwave experiments.