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Using modified spin wave (MSW) method, we study the $J_1-J_2$ Heisenbergmodel with first and second neighbor antiferromagnetic exchange interactions.For symmetric $S=1/2$ model, with the same couplings for all the equivalentneighbors, we find three phase in terms of frustration parameter${\bar\alpha=J_2/J_1}$: (1) a commensurate collinear ordering with staggeredmagnetization (N{\'e}el.I state) for $0\leq{\bar\alpha}\lesssim 0.207$ , (2) amagnetically gapped disordered state for $0.207\lesssim{\bar\alpha}\lesssim0.369$, preserving all the symmetries of the Hamiltonian and lattice, hence bydefinition is a quantum spin liquid (QSL) state and (3) a commensuratecollinear ordering in which two out of three nearest neighbor magnetizationsare antiparallel and the remaining pair are parallel (N{\'e}el.II state), for$0.396\lesssim{\bar\alpha}\leq 1$. We also explore the phase diagram ofdistorted $J_1-J_2$ model with $S=1/2$. Distortion is introduced as aninequality of one nearest neighbor coupling with the other two. This yields aricher phase diagram by the appearance of a new gapped QSL, a gapless QSL andalso a valence bond crystal (VBC) phase in addition to the previously threephases found for undistorted model.

Quantum phase diagram of $J_1-J_2$ Heisenberg $S=1/2$ antiferromagnet in honeycomb lattice: a modified spin wave study