Abstract
We investigate the microscopic structure of the long-wavelength acoustic excitations of two-dimensional dipolar supersolids with triangular and honeycomb crystal order. Using Bogoliubov–de Gennes calculations, we determine the density, phase, current, and lattice-displacement fluctuations associated with the three acoustic modes. We develop a procedure to extract the lattice displacement directly from the microscopic density perturbation, allowing the modes to be identified as one transverse and two longitudinal branches. The displacement field further separates the density fluctuations into contributions from crystal strain and particle transport relative to the lattice. This reveals that the lower longitudinal mode can have large strain and defect-density fluctuations that substantially cancel in the total density response. We show that the long-wavelength fluctuation amplitudes agree with hydrodynamic theory, which is specified entirely by the elastic coefficients of the supersolid. Our analysis provides a microscopic connection between Bogoliubov excitations and the crystalline and superfluid degrees of freedom of two-dimensional supersolids, and identifies their signatures in density- and current-sensitive probes.