Variational description of the nuclear free energy

By means of a variational calculation, we place an upper bound on the finite-temperature free energy for nuclear systems which can be described by pseudospin Hamiltonians. The trial states are irreducible permutation invariant Gibbs states. The best trial state is the one which minimizes the free energy operator. We compare the upper bound with the numerically computed free energy for the Meshkov-Glick-Lipkin Hamiltonian for various values of nucleon number $N$ and nuclear interaction strength $V$. For large $N$ and/or $\beta (=\frac{1}{\mathrm{kT}})$ the best trial Gibbs state becomes a good approximation to the actual density operator. Somewhat surprisingly, the variational approach reveals the presence of a second order thermodynamic phase transition much more clearly than the numerical computation does, even though the former is only an approximation to the latter.