We study the ground states of rotating atomic Bose-Einstein condensates withdipolar interactions. We present the results of numerical studies on a periodicgeometry which show vortex lattice ground states of various symmetries:triangular and square vortex lattices, "stripe crystal" and "bubble crystal".We present the phase diagram (for systems with a large number of vortices) as afunction of the ratio of dipolar to contact interactions and of the chemicalpotential. We discuss the experimental requirements for observing transitionsbetween vortex lattice groundstates via dipolar interactions. We finallyinvestigate the stability of mean-field supersolid phases of a quasi-2Dnon-rotating Bose gas with dipolar interactions.
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