Order-by-disorder degeneracy lifting of interacting bosons on the dice lattice

Motivated by recent experimental progress in the realization of synthetic gauge fields in systems of ultracold atoms, we consider interacting bosons on the dice lattice with half flux per plaquette. All bands of the noninteracting spectrum of this system were previously found to have the remarkable property of being completely dispersionless. We show that degeneracies remain when interactions are treated at the level of mean-field theory, and the ground state exhibits vortex lattice configurations already established in the simpler XY model in the same geometry. We argue that including quantum and thermal fluctuations will select a unique vortex lattice up to overall symmetries based on the order-by-disorder mechanism. We verify the stability of the selected state by analyzing the condensate depletion. The latter is shown to exhibit an unusual nonmonotonic behavior as a function of the interaction parameters which can be understood as a consequence of the dispersionless nature of the noninteracting spectrum. Finally, we comment on the role of domain walls which have interactions mediated through fluctuations.