Ground-state stability and excitation spectrum of a one-dimensional dipolar supersolid

We study the behavior of the excitation spectrum across the quantum phase transition from a superfluid to a supersolid phase of a dipolar Bose gas confined to a one-dimensional geometry. Including the leading beyond-mean-field effects within an effective Hamiltonian, the analysis is based on Bogoliubov theory with several order parameters accounting for the superfluid as well as solid structure. We find fast convergence of the ground-state energy in the supersolid with the number of order parameters and demonstrate a stable excitation spectrum with two Goldstone modes and an amplitude mode in the low-energy regime. Our results suggest that there exists an experimentally achievable parameter regime for dysprosium atoms, where the supersolid phase exhibits a stable excitation spectrum in the thermodynamic limit and the transition into the supersolid phase is of second order driven by the roton instability.