Order-by-disorder of interacting bosons on the dice lattice under a synthetic gauge field

We consider a gas of interacting bosons in the two-dimensional dice lattice in the presence of a half-elementary magnetic flux threading each plaquette. The single particle spectrum of the system consists of three doubly-degenerate completely flat bands, which indicates a large ground state degeneracy. It is shown how this degeneracy is partially lifted in the superfluid regime at the mean-field level. Furthermore, it is shown how quantum and thermal fluctuations conclusively remove the remaining accidental degeneracy between the mean field states, thus selecting a unique state up to overall symmetries. This can be elegantly described by means of the distribution of condensate vortices in the Kagom\'{e} vortex lattice, which is dual to the dice lattice.