An upper bound on the condensate fraction in a Bose gas with weak repulsion

For a spinless Bose gas moving in three-dimensional free space with an interaction V(r) which is everywhere positive, I show that the condensate fraction f is bounded above by an expression parametrized by the quantity nV0/kTc(0)≡η, where n is the density, V0 the space integral of the interaction potential and Tc(0) the critical temperature of the corresponding noninteracting gas. The only assumption needed is that fluctuations in the interacting system have the standard extensivity properties. In particular, for T = Tc(0) the upper bound on f is proportional to η1/3, while for η1/3<<1-Tc(0)/T<<1 it is proportional to η/(1-(Tc(0)/T)3/2)2.


23.2
(equation (18) of his paper [6]), while imposing no restriction on the transition temperature T c as such, places an upper bound on the condensate fraction f as a function of T : where T (0) c is the transition temperature of the noninteracting gas of the same mass and density n. Using an improved version of the lemma, Roepstorff [7] was able to obtain a slightly tighter upper limit which replaces γ in equation (1) by 2. The bound (1) (or the improved version of [7]) is completely independent of the strength and even the sign of the interparticle interactions; however, while it is nonvacuous for any value of T , for T ∼ T (0) c the limit on f is quite weak (f 0.8). Finally, I note that Penrose [8] obtained an upper limit for the condensate fraction at zero temperature, f (0), for a hard-core Bose gas in terms of the fraction of the volume occupied by the hard cores, and Pitaevskii [9] has established a more general upper limit on f (0) in terms of the excitation spectrum (see also [10]); not surprisingly, this limit tends to 1 as the slope of the k → 0 spectrum (thus, presumably, the strength of the (effective) interaction) tends to zero.
The purpose of this paper is to establish a second upper bound on f (T ) which in certain regimes is stronger than (1), and in particular, for T T (0) c , tends to zero with the strength of the (repulsive) interaction [11] †. I consider a system of N identical spinless bosons moving freely (i.e. in zero external potential) in a volume Ω, impose the standard periodic boundary conditions and take the thermodynamic limit N, Ω → ∞, N/Ω → constant ≡ n. The wave function of the lowest single-particle state is then simply a constant, Ω −1/2 , and corresponds to zero kinetic energy; I denote the creation operator for this state by a + 0 , and writeN 0 ≡ a + 0 a 0 . Then the condensate fraction f is defined ‡ as N 0 /N , where for any operatorQ we define Q ≡ TrQρ whereρ is, here and below, the true N -body density matrix of the interacting system. An assumption which will be essential in the argument below is that in the thermodynamic equilibrium state of the interacting system the fluctuations inN 0 have the usual behaviour as regards extensivity in the thermodynamic limit; more formally, that we can bound the probability P (N 0 ) of finding a given eigenvalue ofN 0 different from the average N 0 by the inequality where ζ and K tend to constants of order 1 in the thermodynamic limit. While the inequality (2) is actually false for the noninteracting gas below T c , it is overwhelmingly plausible that for any system with finite (repulsive; cf. below) interactions it should hold. Concerning the interparticle interaction potential V (r), I assume and define (4) † The motivation for emphasizing this feature is the fact that most perturbation calculations seem to indicate that the effect of a repulsive interaction is to raise T c , with the increase tending to zero with the strength of the repulsion. See e.g. [11].
‡ For pedagogical simplicity I make, here, the usual assumption that condensation occurs in the state k = 0 (only). However, it is clear that the ensuing argument carries through for condensation into an arbitrary single-particle state or set of states.

23.3
Then, subject to assumptions (2) and (3), I shall establish the following inequality: where F 0 (N, Ω, T ) is the free energy of the corresponding noninteracting gas. Equation (5) is an implicit bound on f (cf. below) and is the central result of this note; note that in the limit V 0 → 0 it is compatible (as an equality) with the known thermodynamics of the noninteracting Bose gas. The demonstration of the inequality (5) proceeds via the establishment of upper and lower bounds on the free energy F (N, Ω, T ) of the interacting system. LetĤ 0 denote the kinetic energy operator, and let k label the plane-wave states which are eigenstates ofĤ 0 . Then an upper limit on F follows straightforwardly from the Peierls-Bogoliubov inequality (see e.g. [12] †), namely where in the second step we used the fact that in view of equation (3) The derivation of a lower bound on F is a little more tricky; it relies on the fact that F cannot be less than the free energy of a noninteracting gas at the same T and Ω but with a total particle number N (1 − f ); otherwise we could construct a trial density matrix for the latter system which does better than the standard formρ 0 ≡ Z −1 exp(−βĤ 0 ). To see this, define an operatorŶ by its matrix elements in the basis whereN 0 is diagonal, as follows: In words,Ŷ removes all the particles in the condensate while leaving the rest untouched. Note that while the operatorŶ is obviously nonunitary (in fact does not even possess an inverse), nevertheless when operating on a manifold of given fixed total particle number N it preserves the orthonormality of the many-particle states. Because the actual N -particle density operator of the interacting system in general fails to commute withN 0 , the density matrixρ defined (witĥ Y + the Hermitian conjugate ofŶ ) bŷ in general corresponds to a superposition of states with different total particle number; however, since all physical operators such as the kinetic energy commute withN , in evaluating the expectation values of such operators it is adequate to replace the superposition by the corresponding mixture. Note that in view of equation (2), the fluctuations, in the ensemble defined byρ , in the 'final' value of N f ofN around its expectation value N (1−f ) are a negligible fraction of the latter in the thermodynamic limit; in fact, if Prob denotes the probability in the ensemble defined byρ , then the probability Prob (N f ) is given in terms of the P (N 0 ) defined in equation (2) by 23.4 while from equation (3) the potential energy (zero!) cannot have increased. Consequently we reach the conclusion that the free energy of the noninteracting system described byρ cannot exceed F (N, Ω, T ). There remains one final, rather delicate step. Consider the problem of constructing the optimum density matrix for a mixture of states of the noninteracting gas with different N f , with the probability of a given N f being given by equation (9); note that the problem is as it were posed in the context of the macrocanonical ensemble, so we do not assume a priori a grand canonical distribution. It is clear that the solution is a mixture of the N f -particle macrocanonical noninteracting gas density matrices with weights Prob (N f ), and from equations (9) and (2) it is then clear that, to within terms whose relative contribution vanishes in the thermodynamic limit, the associated free energy is just F 0 (N (1 − f ), Ω, T ) (note that this statement is true whether or not T > T (0) c ). Thus, the free energy of the state described byρ cannot lie below F 0 , and since we already saw that it cannot exceed F (N, Ω, T ), we conclude that F (N, Ω, T ) is bounded below by F 0 (N (1 − f ), Ω, T ). Combining this result with the inequality (6), we arrive at equation (5).
Before examining the implications of equation (5) for the quantity f itself, it is worth noting (with a view to possible future developments) that (5) is actually a special case of a more general inequality, which can be proved by a slight modification of the above argument in which the upper bound on F is relative to F 0 replaced by the exact relation and the actual mean value V of the potential is kept in the lower bound: where K ≡ H 0 − T S (S = entropy) and the notation ∆ϕ(N, Ω, T ) represents the value of ϕ in the true (interacting) system relative to that in the corresponding noninteracting gas.
To turn the inequality (5) I will consider here only the case of T ≥ T (0) c (N ), where N is the actual number of particles in the system of interest (the opposite case involves even messier algebra and is less interesting). Then using the standard relation (above) between T (0) c (N ) and N and using (11), we can bound the LHS of (5) by an integral over and find after some algebra the result where ≡ (N ). Substituting (12) into (5), we find the implicit bound Using standard results [13] for the solution of cubic equations we can obtain from (13) the explicit bound In particular, for T = T (0) c while for η 1/3 1 the bound (15) reduces approximately to The above results are not directly applicable to the system of most experimental interest, namely dilute atomic alkali gases in a harmonic trap, both because we cannot assume that condensation occurs in the lowest-single-particle state of the noninteracting system and because the actual interatomic potential by no means satisfies equation (3). To take the latter point first, it is clear, to the extent that the standard approximation of replacing the true potential by a pseudopotential proportional to the s-wave scattering length remains valid for this problem, that one recovers (15) for the free-space problem with a s > 0, with V 0 replaced as usual by 4π 2 a s /m; thus at T (0) c the condensate fraction would be bounded above by an expression proportional to (na 3 s ) 1/9 . The former complication seems more serious, and I have not at the time of writing succeeded in generalizing the argument to a trap geometry.
Finally, I note that if a lower limit could be found on the actual potential energy relative to the Hartree-Fock value, it might well be possible to improve the bound (15) considerably. However, even with such an improvement it looks unlikely that an agreement of the above form will yield a bound on the transition temperature T c , as distinct from the condensate fraction f (T ).

23.6
To establish the needed result we use the inequality e −x ≥ 1 − x and set ≡ int(μ −1 ) (int ≡ integral part of); then we can write which is the result needed in the text.