The pure-quantum self-consistent harmonic approximation

We present a method to study the thermodynamic behavior of quantum systems with many degrees of freedom, over the whole temperature range. It develops in the framework of the path-integral formulation of quantum statistical mechanics and leads us to an effective Hamiltonian, by means of which classical-like formulas for the thermodynamic quantities can be written. The method is the non trivial generalization of a previous variational one to the case of nonstandard systems, i.e., systems whose Hamiltonian cannot be separated in a quadratic kinetic term and a potential. This generalization is made possible by a different derivation of the final results: we retain Feynman's idea of classifying paths by the equivalence relation of having the same average phase-space point, but, instead of using a variational principle, we exploit the general procedure of the self-consistent harmonic approximation. This leads us to exactly take into account the full classical contribution to the thermodynamics of the system, as well as the quantum harmonic one, while the only nonlinear quantum part is involved in the approximation.