Finite-temperature many-body theory with the Lipkin model

We have compared exact numerical results for the Lipkin model at finite temperature with Hartree-Fock (HF) theory supplemented by the particle-particle hole-hole ring diagrams and with the static path approximation (SPA). In the simplest version of the Lipkin model the HF approach shows a ''phase transition'' which is absent in the exact results and in the SPA. For more realistic cases, HF is quite accurate at low temperature, but less so at high temperature. In contrast the SPA appears to be highly satisfactory for high temperatures, but less accurate at low temperature. The effect of the ring diagrams is small at low temperature, but it is more significant at high temperature where a moderate improvement to HF brings the results into closer agreement with the exact calculations. ((orig.))