Vibration modes of a two-dimensional Wigner lattice coupled to ripplons on a liquid-helium surface

We present a theory of the vibration modes of a two-dimensional Wigner lattice coupled to ripplons on a liquid-helium surface based on the use of thermodynamic Green's functions. Starting from the phonon-ripplon Hamiltonian proposed by Fisher, Halperin, and Platzman, the effects of the electron-ripplon interaction (and hence the effects of the temperature and pressing electric field) on the frequencies of the coupled phonon-ripplon modes are obtained from the poles of the Green's function for the phonons of the Wigner lattice. The nature of these poles is determined by the phonon self-energy, which clearly displays the resonant coupling between the phonons and the ripplons. Our theory gives a first-principles derivation of the weights of the ripplon-induced resonances. We present approximate analytical results for the frequencies of the coupled modes. Our results are in qualitative agreement with the experiments of Grimes and Adams and the theory of Fisher et al. However, we do not find justification for the quantitative agreement with experiment that has been reported by Fisher et al. This discrepancy has to do with the fact that we show that the aforementioned weights are not given in terms of an effective Debye-Waller factor for the 2D Wigner lattice, but rather in terms of an exponential whose argument originates from the difference in electron displacement correlation functions given by $〈\overrightarrow{\mathrm{q}}·\overrightarrow{\mathrm{u}}(l_1;t_1)\overrightarrow{\mathrm{q}}·\overrightarrow{\mathrm{u}}(l_2;t_2)〉-〈{\left(\overrightarrow{\mathrm{q}}·\overrightarrow{\mathrm{u}}\right)}^2〉$. This being the case, the normal modes of the phonon-ripplon Hamiltonian have frequencies whose values are somewhat smaller than the frequencies of the resonances measured by Grimes and Adams.