A gapped $Z_2$ spin liquid phase with a $U(1)$ mean field ansatz: a Bosonic resonating valence-bond description

Gapped $Z_2$ spin liquid as the simplest spin liquid has been proved to be the most difficult to realize in realistic models. Here we show that the frustration from a third-neighbor exchange $J_3$ on the spin-$\frac{1}{2}$ $J_1-J_2$ model on the square lattice may serve to stabilize such a long-sought state. We argue that a Bosonic RVB description is more appropriate than a Fermonic RVB description for such a gapped spin liquid phase. We show that while the mean field ansatz of the proposed state has a $U(1)$ gauge symmetry, the gauge fluctuation spectrum on it is actually gapped and the state should be understood rather as a $Z_2$ spin liquid state with topological order. The state is thus locally stable with respect to gauge fluctuation and can emerge continuously from a collinear Neel ordered phase.