Generalized condensation and the Bogoliubov theory of superfluidity

In this paper we scrutinize the Bogoliubov theory of superfluidity on the level of the first Bogoliubov ansatz. This ansatz preconizes the Bogoliubov truncated Hamiltonian as a starting point of this theory. Since this ansatz is based on the hypothesis concerning the Bose-Einstein condensation in zero-mode as well as on its stability with respect to interactions, we present some rigorous results, which cast doubt on this hypothesis.


Generalized condensation
Recall that the starting point of the Bogoliubov theory is based on two hypothesis: (a) there is a Bose-Einstein (BE) condensation in one mode; (b) this condensation is stable with respect to a weak interaction between particles.
These hypothesis inspired Bogoliubov (see e.g.[1,2]) into his first ansatz : the truncation of the full Hamiltonian where all sums run over the set Λ * defined by to the Hamiltonian of the Weakly Imperfect Bose-Gas (WIBG): where Here ε k = 2 k 2 /2m is the kinetic energy, and a # k = {a * k , a k } are usual boson creation and annihilation operators in the one-particle state ψ k (x) = V − 1 2 e ikx , k ∈ Λ * , x ∈ Λ; for example, a * k ≡ a * (ψ k ) = Λ dx ψ k (x) a * (x) , where a # (x) are bosonfield operators in the Fock space F Λ over L 2 (Λ) , see [3].
In this section we scrutinize the (most innocent) hypothesis (a), which concerns essentially the Perfect Bose-Gas (PBG).As it is indicated in [4] , the conventional BE condensation is a subtle matter even for the PBG.
Let instead of the cubic box We impose periodic boundary conditions on ∂Λ , and we consider a possibility of conventional BE condensation in this prism.This example is due to Casimir [5] .Now let us rewrite the equation for the particle density in the form: For ρ > ρ P c (θ) (the critical density for the PBG) the equation (1.7) gives asymptotic behaviour of the solution µ P Λ θ, ρ > ρ P c (θ) and distribution of bosons in the mode k = 0 and in its vicinity.
Cases α 1 = 1/2 and α 1 > 1/2 are provided examples of generalized condensation in the ]irardeau sense [6].This gives a motivation for the following van den Berg-Lewis-Pulè's classification of the generalized BE condensation [4,7,8]: • the condensation is called the type I when a finite number of single-particle levels are macroscopically occupied; • it is of type II when an infinite number of the levels are macroscopically occupied; • it is called the type III, or the non-extensive condensation, when none of the levels is macroscopically occupied, whereas one has: Remark 1 Returning back to the Bogoliubov theory we have to recognize that the hypothesis (a) about the one-mode BE condensation looses its grip if we pass to the PBG in a long prism.The latter is not an exotic situation if one considers superfluidity of extremely thin films of 4 He, see [9,10].

The model
In [11,12] it was shown that the type III condensation can be caused in the PBG by a weak external field or by a specific choice of the boundary conditions and geometry [7,13].But recently other examples of this non-extensive BE condensation were discovered for bosons in an isotropic box Λ with repulsive interaction which spreads out the conventional BE condensation of the type I into the type III, [14][15][16].
To avoid the influence of the shape on the BE condensation, let us take the cubic box Λ = L × L × L. Imposing the periodic boundary conditions on ∂Λ, we consider a simple interacting boson system with the Hamiltonian: Interaction in (2.1) results from a "severe" truncation of the full interaction H Λ .
Hereinafter we shall demonstrate that even this "small" part of the total interaction can drastically change the distribution of condensed particles spreading out the BE condensation in the one mode k = 0 into non-extensive BE condensation of the type III.First we show that the interaction in (2.1) is so gentle that it even does not modify the pressure of the perfect system.
Theorem 1 For the model (2.1) the thermodynamic limit of the pressure coincides with the pressure of the PBG: Therefore, the grand partition function for the model (2.1) takes the form : which converges for all µ ∈ R 1 .Then for the pressure in the finite volume we have the estimates: denotes the integer part of x) one gets: For µ < 0 the left-hand side of (2.5) converges (uniformly in k ∈ Λ * ) to 1 − e −β(ε k −µ) −1 when V → ∞.Hence, in the thermodynamic limit, the lower estimate in (2.4) coincides with the upper estimate: which proves the equality (2.2).Since lim ] is bounded, one extends p P (β, µ) to µ = 0 by continuity.As usual, the total density of particles for the model (2.1) is defined in the grand-canonical ensemble for a finite volume V by : Corollary 2 Due to v (0) > 0 the pressure p Λ H I Λ is defined for all chemical potentials µ ∈ R 1 , see (2.3).Moreover it is obviously a convex function of this parameter.The limit function p P (β, µ) is defined only for µ 0 and is also a convex function, but only for µ < 0. Since (2.2) is the statement regarding the convergence of convex functions of µ ∈ (−∞, 0), by the Griffiths lemma we get convergence of the corresponding derivatives (2.6)): on the very same interval µ ∈ (−∞, 0).So, in the thermodynamic limit the total particle density ρ I (β, µ) of the model (2.1) coincides with ρ P (β, µ).Since lim µ→0 − ρ P (β, µ) is bounded for d > 2, one defines ρ I (β, µ = 0) = ρ P (β, µ = 0) by continuity.
From this corollary one deduces that, as in the case of the PBG, ρ I (β, µ) attains its maximum ρ I c (θ) = lim µ→0 − ρ I (β, µ) = ρ P c (θ) at µ = 0, i.e. for densities ρ > ρ P c (θ) one must have a BE condensation.On the other hand, for the expectation of the ground-state occupation number for µ = 0 we get from (2.1): (2.7) Since the numerator and the denominator in the right-hand side of (2.7) are nothing but Darboux sums for the corresponding integrals in the limit V → ∞, we obtain that lim This means that there is no macroscopic occupation of the mode k = 0 because Theorem 3 For any parameters (β, µ 0) one has i.e. there is no macroscopic occupation of any mode k ∈ Λ * .
Proof.By the Bogoliubov convexity inequality for the pressure (see e.g.[17,18]), we get Notice that by the Cauchy-Schwarz inequality for the Gibbs state we get The estimates (2.10) and (2.11) give the assertion (2.9).
Resuming corollary 2 and theorem 3 we must admit that the model (2.1) gives an example of non-extensive (i.e. the type III) BE condensation ρ 0 and at the same time

Conclusion
The perturbation of the model (2.1) is a function of only occupation-number operators, i.e. this interaction commutes with the kinetic-energy operator.This class of interactions is called diagonal.In the series of papers [19][20][21][22][23], van den Berg, Dorlas, Lewis and Pulè gave an exhausting analysis of a hole class of diagonal models (including the exact solution of the Huang-Yang-Luttinger model ), showing how subtle and multiform is the BE condensation in presence of this type of interaction.Since the interaction is diagonal, their analysis is based on a refined Large Deviation Principle of the classical probability.For another approach to the Huang-Yang-Luttinger model see [24].
If one accepts the first Bogoliubov ansatz, one is faced with a nondiagonal interaction U ND Λ in (1.3), which does not commute with T Λ .Therefore, the existence of the k = 0 mode BE condensation in the Bogoliubov WIBG H B Λ is screaming for a rigorous analysis, see [25,26] Returning back to the theory of superfluidity we have to avow that the discussion above is rather far away from real systems.In the recent paper by Vakarchuk [27] one can find a discussion of the difficulties and of the methods going beyond the Bogoliubov theory of superfluidity.Developing some ideas by Bogoliubov, Zubarev and Yukhnovskii regarding the method of collective variables, he obtained new convincing results (see references quoted in [27]) to advance in this challenging unsolved problem of theoretical and mathematical physics.
13))Remark 2 Returning back to the Bogoliubov theory we again recognize that there is a problem with the hypothesis (a) and (b) regarding the one-mode (i.e.type I) BE condensation.Even if it exists in PBG, a very gentle interaction (e.g.(2.1)), which does not change the pressure of the PBG, may drastically change the asymptotics of the occupation number expectations, cf.(2.8).In the model (2.1) the interaction spreads out the type I condensation of the PBG into the type III BE condensation, which means that the Bogoliubov hypothesis (b) is à priori doubtful.