Phase Transitions in Systems of Interacting Fermions

Abstract

Girardeau has shown that a particular class of variational approximations for the free energy of a Fermi-Dirac system may be expressed in terms of the exact free energy of a certain soluble "model Hamiltonian." This exact formula for the partition function associated with the model system involves the solution of a certain nonlinear integral equation, and in the present work it is shown in detail that there is a correspondence between phase transitions in the system and the existence of multiple solutions of this equation. The connection between the existence of phase transitions and the properties of the interaction is studied, and it is shown that these phase transitions may occur for a very wide class of interactions. It is established that there is an upper bound to the temperature, above which no phase transitions will occur at any density. The extension of these results to a classical and a Bose-Einstein system is indicated, and the lattice-gas model previously studied by Mermin is re-examined in the light of the present results.