The Bose Gas Low Momentum Limit Revisited

TL;DR

This paper revisits the low momentum spectrum of a weakly interacting Bose gas, analyzing the standard Bogoliubov approach and proposing an alternative two-mode solution with a gapped mode affecting thermodynamic behavior.

Contribution

It highlights issues in the traditional Bogoliubov method related to gauge symmetry breaking and introduces a novel two-mode solution with a gapped spectrum.

Findings

Standard Bogoliubov spectrum derived as a Goldstone mode.

Identification of gauge symmetry breaking issues affecting chemical potential.

Proposal of a two-mode solution with a gap influencing specific heat behavior.

Abstract

We discuss the standard approach to the problem of the low momentum limit of the spectrum for a weakly interacting Bose gas. The Bogoliubov's spectrum is shown to be obtained as a Goldstone mode thanks to the introduction of a chemical potential $\mu$. This procedure has, however, difficulties since the breaking of the gauge symmetry implies that the corresponding chemical potential must be taken as zero, unless it is introduced before breaking the symmetry. But if this is done, after the symmetry breaking $\mu$ loses its meaning as a chemical potential. An alternative two-mode solution is suggested having two modes, one of them being the free-particle quadratic in momentum spectrum, the second bearing a gap. This gap leads to a $\lambda$-type behavior of the specific heat near the critical temperature.