We study the effect of quantum fluctuations by means of a transverse magnetic field ($Gamma$) on the antiferromagnetic $J_1-J_2$ Ising model on the checkerboard lattice, the two dimensional version of the pyrochlore lattice. The zero-temperature phase diagram of the model has been obtained by employing a plaquette operator approach (POA). The plaquette operator formalism bosonizes the model, in which a single boson is associated to each eigenstate of a plaquette and the inter-plaquette interactions define an effective Hamiltonian. The excitations of a plaquette would represent an-harmonic fluctuations of the model, which lead not only to lower the excitation energy compared with a single-spin flip but also to lift the extensive degeneracy in favor of a plaquette ordered solid (RPS) state, which breaks lattice translational symmetry, in addition to a unique collinear phase for $J_2>J_1$. The bosonic excitation gap vanishes at the critical points to the N{e}el ($J_2 < J_1$) and collinear ($J_2 > J_1$) ordered phases, which defines the critical phase boundaries. At the homogeneous coupling ($J_2=J_1$) and its close neighborhood, the (canted) RPS state, established from an-harmonic fluctuations, lasts for low fields, $Gamma/J_1lesssim 0.3$, which is followed by a transition to the quantum paramagnet (polarized) phase at high fields. The transition from RPS state to the N{e}el phase is either a deconfined quantum phase transition or a first order one, however a continuous transition occurs between RPS and collinear phases.
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