Analysis of a Boson Model Displaying a Phase Transition

Abstract

The simple boson model proposed by Foldy, which displayed a discontinuity in its energy as a function of interaction strength, is studied using symmetry arguments, the method of linearized equations of motion, and perturbation theory. The particular symmetry responsible for the discontinuity, or phase transition, is pointed out. The method of linearized equations is shown to predict the transition value of the interaction strength and, like the Bogoliubov approximation, to accurately describe the energy of the first excited state. It is further shown that simple perturbation theory is quite accurate except in the transition region. The validity of that feature of the Bogoliubov method in which excited states are described by independent quasiparticles is examined. This description is found to be excellent in the normal region. All comparisons are made with a numerical solution which provided the complete energy spectrum and the associated eigenvectors. It is hoped that these arguments can be applied in considerations of the validity of the various approximations for realistic systems.