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The Hohenberg-Martin dilemma of conserving versus gapless theories for systems with Bose-Einstein condensate is considered. This dilemma states that, generally, a theory characterizing a system with broken global gauge symmetry, which is necessary for BoseEinstein condensation, is either conserving, but has a gap in its spectrum, or is gapless, but does not obey conservation laws. In other words, such a system either displays a gapless spectrum, which is necessary for condensate existence, but is not conserving, which implies that it corresponds to an unstable system, or it respects conservation laws, describing a stable system, but the spectrum acquires a gap, which means that the condensate cannot appear. An approach is described, resolving this dilemma, and it is shown to give good quantitative agreement with experimental data. Calculations are accomplished in the Hartree-Fock-Bogolubov approximation. 1 Condensate existence and stability The necessary and sufficient condition for Bose-Einstein condensation is the spontaneous breaking of global gauge symmetry U(1) (see [1–3]). This implies that below the Bose condensation temperature Tc the statistical average of a field operator 〈ψ̂〉 becomes nonzero, since it is proportional to √ ρ0, where ρ0 ≡ N0/V is the average condensate density. According to the Bogolubov-Ginibre theorem [4–6] the condensate density ρ0 is a minimizer of thermodynamic potential. The same concerns the density of uncondensed particles ρ1 that can be expressed through ρ0 as ρ1 = ρ− ρ0, with ρ being a fixed average particle density. Keeping this in mind, it is possible to formulate the conditions of condensate existence and stability. Below, we use the system of units, where the Planck and Boltzmann constants are set to one. The general criterion of Bose-Einstein condensation is the occurrence of a nonzero thermodynamic limit lim V→∞ N0 V = ρ0 > 0 . (1)

Hartree–Fock–Bogolubov Method in the Theory of Bose-Condensed Systems