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

Possible phases of the spin-$1/2$ $XXZ$ model on a honeycomb lattice by boson-vortex duality