A generalized thermal random phase approximation with the Lipkin model

An accuracy and a range of validity of the recently proposed generalized thermal random phase approximation (GTRPA) are studied by comparing with the exact grand canonical calculations for the two-level Lipkin model. A system of nonlinear equations that couple a parameter of the Hartree-Fock (HF) transformation, energies of HF quasi particles, energy of collective phonon excitation, amplitudes of a phonon wave function and a number of thermal quasi particles in a thermal vacuum state (or a ground thermal state of the heated Lipkin system) is evaluated. The dependence of the intrinsic energy < H > on the temperature T is calculated. At all temperatures and all values of a coupling strength GTRPA gives better results than other approximations (e.g. the thermal HF method or the thermal RPA or the thermal renormalized RPA). Advantages of GTRPA are most obvious at the values of T near the phase transition point. In GTRPA the phase transition takes place at lower T than in other approximations. Moreover, a statistical behaviour of the Lipkin model is described within GTRPA with an appropriate accuracy even in the case when the HF transformation parameter is fixed at a zero value, thus corresponding to the 'spherical' phase of a mean field. The reason is that within GTRPA HF and collective variables of the system are coupled and change their values consistently