  Motivated by recent numerical work, we use the boson-vortex duality to study
the possible phases of the frustrated spin-$\frac{1}{2}$ $J_1-J_2$ XXZ models
on the honeycomb lattice. By condensing the vortices, we obtain various gapped
phases that either break certain lattice symmetry or preserve all the
symmetries. The gapped phases breaking lattice symmetries occur when the vortex
band structure has two minima. Condensing one of the two vortex flavors leads
to an Ising ordered phase, while condensing both vortex flavors gives rise to a
valence-bond-solid state. Both of those phases have been observed in the
numerical studies of the $J_1-J_2$ XXZ honeycomb model. Furthermore, by tuning
the band structure of vortex and condensing it at the $\Gamma$ point, we obtain
a featureless paramagnet. But the precise nature of this featureless state is
still unclear and needs future study.
