New Collective Mode in the Fractional Quantum Hall Liquid

We apply the methods of continuum mechanics to the study of the collective modes of the fractional quantum Hall liquid. Our main result is that at long-wavelength, there are two distinct modes of oscillations, while previous theories predicted only one. The two modes are shown to arise from the internal dynamics of shear stresses created by the Coulomb interaction in the liquid. Our prediction is supported by recent light scattering experiments, which report the observation of two long-wavelength modes in a quantum Hall liquid.

Figure 1

Dispersion of the collective modes calculated from Eqs. (4, 5) with $\nu =1/3$, $\Delta =0.15$, ${\mu }_{\infty }/n=0.075$, ${\Lambda }_0/(\Delta {\mu }_{\infty })=0.7$, and $K/n=0.025$. $q$ is in units of ${\ell }^{-1}$, and $\omega$ is in units of $\frac{e^2}{\hslash {ϵ}_b\ell }$. Next to each dispersion, we show schematically the direction of rotation of the displacement field and the balance of the forces that sustain the mode: “L” denotes the Lorentz force, “S” the ordinary shear force, and “LS” the Lorentz shear force. The shaded region indicates the conjectured two-roton continuum, which limits the dispersion of the $+$ mode. The inset compares the calculated frequency splitting (solid line) with the experimental data of Ref. 11 (solid and open squares) for two samples of different electron density. The vertical bars indicate the experimental uncertainty.