Collective modes of the two-dimensional Wigner crystal in a strong magnetic field

We present a calculation of the dispersion relation of the intra-Landau-level and inter-Landau-level collective modes of the two-dimensional Wigner crystal in a strong magnetic field. Our analysis is based on the time-dependent Hartree-Fock approximation (TDHFA) for the density-density response function and allows for an arbitrary large degree of anharmonicity. We derive the density-density response function from an equation-of-motion approach that uses identities valid in the strong-field limit. The TDHFA response function is shown to depend only on the ground-state density as calculated in the Hartree-Fock approximation. The intra-Landau-level collective excitation and the excitation near the cyclotron frequency are shown to correspond, respectively, to the low- and high-energy branches of the harmonic phonon spectrum. Additional collective-excitation branches occur near larger multiples of the cyclotron frequency. In contrast with the Hartree-Fock approximation, the TDHFA excitation spectrum does not contain transitions between the Landau-level subbands created by the self-consistent Hartree-Fock potential in the electron lattice.