Self-consistent random phase approximation in the Schutte-Da-Providencia fermion-boson model

Self-consistent random phase approximation (SCRPA) is applied to the exactly solvable model with fermion boson coupling proposed by Sch\"utte and Da-Providencia. Very encouraging results in comparison with the exact solution of the model for various observables are obtained. The transition from the normal phase to the phase with a spontaneously broken symmetry is carefully investigated. The strong reduction of the variance in SCRPA vs HF is pointed out.