The Bose Gas Low Momentum Limit Revisited
  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. 
H. Perez Rojas and D. Oliva