Lipkin-Meshkov-Glick model at finite temperatures

The accuracy of some approximate methods in finite-temperature many-body theory is investigated by using the example of the Lipkin-Meshkov-Glick model. For this purpose, the thermal random-phase approximation and the thermal renormalized random-phase approximation are considered. The mean energy of the system, the mean value of the quasispin projection, and the variance of the number of particles are computed in this approximation and are compared with the results of the exact calculation of the partition function for the grand canonical ensemble. On the whole, the results obtained in the thermal renormalized random-phase approximation are closer to the exact results. With increasing temperature, the accuracy of all approximation becomes higher. The same trends are observed when either the number of particles in the system, or temperature, or both increase