Coexistence of the "bogolons" and the one-particle spectrum of excitations with a gap in the degenerated Bose gas

Properties of the weakly non-ideal Bose gas are considered without suggestion on C-number representation of the creation and annihilation operators with zero momentum. The “density-density” correlation function and the one-particle Green function of the degenerated Bose gas are calculated on the basis of the self-consistent Hartree-Fock approximation. It is shown that the spectrum of the one-particle excitations possesses a gap whose value is connected with the density of particles in the “condensate”. At the same time, the pole in the “density-density” Green function determines the phonon-roton spectrum of excitations which exactly coincides with one discovered by Bogolyubov for the collective excitations (the “bogolons”).

💡 Research Summary

The paper revisits the weakly interacting Bose gas at temperatures below the Bose‑Einstein condensation temperature without invoking the customary Bogoliubov substitution of the zero‑momentum creation and annihilation operators by c‑numbers. Instead, the authors employ a self‑consistent Hartree‑Fock (HF) approximation to derive closed equations for the one‑particle distribution function f(p)=⟨a†ₚaₚ⟩ and the effective single‑particle energy E(p). For a contact interaction U(r)=u₀δ(r) the HF equations reduce to a simple form: above the critical temperature T₀ the system behaves like an ideal Bose gas with a shifted chemical potential μ* = μ−2nu₀, while below T₀ the chemical potential adjusts to μ = (2n−n₀)u₀, where n₀ is the condensate density. In this low‑temperature regime the effective single‑particle spectrum becomes E*(p)=ε(p)+n₀u₀, which exhibits a finite gap Δ = n₀u₀ at p→0. This gap is directly proportional to the condensate density and therefore disappears only when the condensate vanishes.

Having established the one‑particle sector, the authors turn to collective excitations by calculating the density‑density response function χ_R(q,ω) within the dielectric formalism. The central object is the polarization operator Π(q,ω), which is split into a condensate contribution Π₀(q,ω) and a thermal contribution Π_T(q,ω). In the simplest random‑phase approximation (RPA) Π_T is evaluated with the ideal‑gas distribution, but the authors modify this to a “modified RPA” (MRPA) that incorporates the gapped HF distribution f_T(p) and the corresponding quasiparticle energy E*(p). At temperatures much lower than the gap (T≪Δ) the thermal part can be neglected, leaving χ_R(q,ω)≈2n₀ε(q)/(ℏ²ω²−ℏ²ω̃²(q)), where the collective mode frequency ω̃(q)=√