In this thesis, we study S=1/2 Heisenberg model on honeycomb lattice with first and second neighbor anti-ferromagnetic exchange interactions ( J 1 -J 2 model), by Variational Monte Carlo(VMC), Exact Diagonalization (ED) and Bond Operator (BO) methods. First, using VMC method, the stability of Neel order, against the frustrating second neighbor interactio J 2 is investigated. We find that the ground state of model remains Neel ordered up to J 2 ? 0.2. Beyond this value, the ground state becomes and Nearest Neighbor Valence Bond (NNVB) bases, provide a body of evidence in favor of Plaquette valence bond ordering for 0.2 J 2 0.3. This state undergoes a transition to the staggered dimerized state at J 2 ?0.35. The BO calculations also support the ED results, except that this according to this method, the Neel to Plaquette transition occur at J 2 ?0.08. Keywords : Heisenberg model, Frustration, Spin liquid, staggered dimerized order, plaquette order.
