Recent experiment on spin-3/2 bilayer honeycomb lattice antiferromagnet shows a spin liquid behavior down to very low temperatures . This behavior can be ascribed to the frustration effect due to competitions between first and second nearest neighbor antiferromagnet interaction . Motivated by the experiment , we study Antiferromagnet Heisenberg model , using Mean field theory and Holestein-Primakov transformation . In mean field theory we use magnetization as a local field in a spin system and fourier transform the hamiltonian to find an interaction matrix . By diagonalizing this interaction matrix and choosing the lowest eigenvalue we can deduce the ground state of the system . This minimum eigenvalue changes by variation of J2 and J2| we found that two eigenvalues determine the phases of the system . Although both of eigenvalues have minimum at Q=0 , they show different state for spins . One of these minima shows antiferromagnetic phase and the other shows layer antiferromagnetic phase . Moreover they have coexistence line which is a line of first order transition . Furthermore degenerate phases have two region due to these two eigenvalues . They have no coexistence line in the phase diagram . Through this calculation shows that this system has highly degenerate ground states in both eigenvalues . Second nearest neighbor calculation in adjacent layer and drawing phase diagram of this system in J2-J2| shows that these neighbors also increase frustration in the system . Moreover by usingHolestein-Primakov transformation we can add the quantum effect of magnon states to the system . The manganite Bi3Mn4O12(NO3) Which was shows no spin ordering down to low temperatures has coupling constants in antiferromagnetic area of our phase diagram . Using Heisenberg hamiltonian we cannot explain the frustration effects for this manganite . The quantum effects or other interaction in this hamiltonian may explain the absence of ordering down to low temperature for the manganite . Keywords: Antiferromagnet Heisenberg Model , Spin-Liquid,Frustration , Mean field theory , Holestein-Primakov Transformation, Manganite Bi3Mn4O12(NO3)
