where f$_{k}$ is the Bose distribution function of quasi-particles, f$_{k}$ = 1 / [exp( βE$_{k}$ ) − 1]. This self-consistency equation can be written explicitly as where the term − 1 / 2 ϵ$_{k}$ in the integrand on the right-hand side is a counter term due to vacuum renormalization. The chemical potential µ and the order parameter Δ can be solved from Eq. (4) and the following equation for the density n , There is a gap in the atomic excitation energy given by If the molecular-BEC state is stable, one necessary condition is that the gap E$_{0}$ is real, or − µ ′ ≥ Δ. When − µ ′ = Δ, the atomic excitation becomes gapless, E$_{0}$ = 0, which marks the lower boundary of the molecular-BEC phase. The upper phase boundary is located at the superfluid transition temperature T$_{p}$ where Δ = 0 and E$_{0}$ = − µ $^{′}$. In addition to atomic excitations, there are collective excitations of Feshbach molecules. Although the molecular excitation energy cannot be obtained in the mean-field approxima- tion, it can be extracted from poles of two-particle correlation functions given by where b$_{1}$ = ψ $^{2}$, b$_{2}$ = b † $_{1}$, b$_{3}$ = 2 ψ $^{†}$ψ , 0 ≤ τ ≤ β ¯ h , and T is the time-ordering operator. As in the zero-temperature case [12], the correlation function can be calculated in the random- phase approximation (RPA), where χ $^{(0)}$( k , ω ) is the correlation function calculated in the mean-field approximation. In RPA, the dispersion of Feshbach molecules satisfies the following equation, 4 det | I − gχ $^{(0)}$( k , ω ) | = 0 , (9) χ ( k , ω ) = [1 − gχ $^{(0)}$( k , ω )] − $^{1}$χ$^{(0)}$( k , ω ) , (8) χ$_{αβ}$ ( r − r $^{′}$, τ − τ $^{′}$) ≡ − 1 ¯ h 〈T [ b$_{α}$ ( r , τ ) b † $_{β}$( r $^{′}$, τ $^{′}$)] 〉 , (7) E$_{0}$ = √ µ ′ 2 − Δ $^{2}$. (6) n = 1 V ∑ k [ v 2 k + ( u 2 k + v 2 $_{k}$) f$_{k}$ ] = ∫ d $^{3}$k (2 π ) 3 [ ϵ$_{k}$ − µ ′ 2 E$_{k}$ coth ( βE$_{k}$ 2 ) − 1 2 ] . (5) B. Excitations in the long-wavelength limit − m 4 π ¯ h $^{2}$a$_{s}$ = ∫ d $^{3}$k (2 π ) 3 [ 1 2 E$_{k}$ coth ( βE$_{k}$ 2 ) − 1 2 ϵ$_{k}$ ] , (4)