Supersolidity and crystallization of a dipolar Bose gas in an infinite tube

We calculate the ground states of a dipolar Bose gas confined in an infinite tube potential. We use the extended Gross-Pitaevskii equation theory and present a novel numerical method to efficiently obtain solutions. A key feature of this method is an analytic result for a truncated dipole-dipole interaction potential that enables the long-ranged interactions to be accurately evaluated within a unit cell. Our focus is on the transition of the ground state to a crystal driven by dipole-dipole interactions as the short ranged interaction strength is varied. We find that the transition is continuous or discontinuous depending upon average system density. These results give deeper insight into the supersolid phase transition observed in recent experiments, and validate the utility of the reduced three-dimensional theory developed in [Phys. Rev. Res. 2, 043318 (2020)] for making qualitatively accurate predictions.