Lipkin-Meshkov-Glick model at finite temperature

The Lipkin-Meshkov-Glick model is used to examine the validity of some approximate methods in the many-body theory at finite temperatures. The thermal random phase approximation and the thermal renormalized random phase approximation are discussed. The average energy of the system, the average quasispin projection and the particle number variance are calculated within these approximations and compared with the results of exact calculation with the grand canonical ensemble. It is shown that on the whole the results of thermal renormalized random phase approximation are found to be in better agreement with the exact ones. The validity of both approximations becomes better with increasing temperature and/or particle number