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We exploit the Quantum Cluster Variational Method (QCVM) to study the$J_1$-$J_2$ model for quantum Ising spins. We first describe the QCVM anddiscuss how it is related to other Mean Field approximations. The phase diagramof the model is studied at the level of the Kikuchi approximation in squarelattices as a function of the ratio between $g = J_2/J_1$ , the temperature andthe longitudinal and transverse external fields. Our results show that quantumfluctuations may change the order of the transition and induce a gap betweenthe ferromagnetic and the stripe phases. Moreover, when both longitudinal andtransverse fields are present, thermal fluctuations and quantum effectscontribute to the appearance of a nematic phase.

The Quantum Cluster Variational Method and the Phase Diagram of the quantum ferromagnetic $J_1$-$J_2$ model