We prove exponential decay of the off-diagonal correlation function in the two-dimensional homogeneous Bose gas when a^2 \rho is small and the temperature T satisfies T > 4 \pi \rho / \ln |\ln(a^2\rho). Here, a is the scattering length of the repulsive interaction potential and \rho is the density. To leading order in a^2 \rho, this bound agrees with the expected critical temperature for superfluidity. In the three-dimensional Bose gas, exponential decay is proved when \Delta T_c / T_c^0 > 5 \sqrt{a \rho^{1/3}}, where T_c^0 is the critical temperature of the ideal gas. While this condition is not expected to be sharp, it gives a rigorous upper bound on the critical temperature for Bose-Einstein condensation.
Deep Dive into Rigorous Upper Bound on the Critical Temperature of Dilute Bose Gases.
We prove exponential decay of the off-diagonal correlation function in the two-dimensional homogeneous Bose gas when a^2 \rho is small and the temperature T satisfies T > 4 \pi \rho / \ln |\ln(a^2\rho). Here, a is the scattering length of the repulsive interaction potential and \rho is the density. To leading order in a^2 \rho, this bound agrees with the expected critical temperature for superfluidity. In the three-dimensional Bose gas, exponential decay is proved when \Delta T_c / T_c^0 > 5 \sqrt{a \rho^{1/3}}, where T_c^0 is the critical temperature of the ideal gas. While this condition is not expected to be sharp, it gives a rigorous upper bound on the critical temperature for Bose-Einstein condensation.
Quantum many-body effects due to particle interactions and quantum statistics make the Bose gas a fascinating system and a challenge to theoretical physics. It is increasingly relevant to experimental physics, especially after the first realization of Bose-Einstein condensation in cold atomic gases. 1,2 It displays a stunning physical phenomenon: superfluidity. Several mechanisms that are present in the Bose gas also play a rôle in interacting electronic systems and in quantum optics.
Both the two-dimensional and the three-dimensional gas have physical relevance, and they behave rather differently. We consider them separately here. Throughout the paper, we shall assume that units are chosen in such a way that = 2m = k B = 1, where m is the particle mass.
There is no Bose-Einstein condensation in the twodimensional Bose gas at positive temperature, as was proved by Hohenberg more than forty years ago. 3 In contrast to higher dimensions, the ideal Bose gas offers no intriguing features in two dimensions. But the interacting gas is expected to display a Kosterlitz-Thouless type transition from a normal fluid to a superfluid, where the decay of off-diagonal correlations goes from exponential to power law. The critical temperature T c depends on the scattering length a of the interaction potential, which we consider to be repulsive. For dilute gases, i.e. when a 2 ρ 1, Popov 4 performed diagrammatic expansions in a functional integral approach, finding that
This formula was confirmed by Fisher and Hohenberg 5 using Bogoliubov’s theory, and by Pilati et al. 6 using Monte-Carlo simulations. No rigorous proof is available to this date, however.
In this article we prove in a mathematically rigorous fashion that there is exponential decay of the off-diagonal correlation function when the temperature satisfies
for small a 2 ρ. Thus we prove that T c cannot be bigger than the conjectured value (1), to leading order in a 2 ρ. The main novel ingredient in our proof is a rigorous bound on the grand-canonical density of the interacting Bose gas. This is explained in the next section.
A three-dimensional Bose gas is interesting even in the absence of particle interactions. Bose-Einstein condensation takes place at the critical temperature T No information on the constant c is provided, not even its sign.
1960 Glassgold, Kaufman, and Watson 9 find that the critical temperature increases as ∆T c /T
1964 Huang 10 gives an argument suggesting that ∆T c /T
1971 A Hartree-Fock computation shows that ∆T c < 0 (Fetter and Walecka 11 ).
1982 A loop expansion of the quantum field representation gives ∆T c /T (0) c ≈ -3.5(aρ 1/3 ) 1/2 (Toyoda 12 ).
1992 By studying the evolution of the interacting Bose gas, Stoof 13 finds that the change of critical temperature is linear in the scattering length with c = 16π/3ζ(3/2) 4/3 = 4.66.
1996 A diagrammatic expansion in the renormalization group yields ∆T c > 0 (Bijlsma and Stoof 14 ).
1997 A path integral Monte-Carlo simulation yields c = 0.34 ± 0.06 (Grüter, Ceperley, and Laloë 15 ).
1999 A virial expansion leads to c = 0.7 (Holzmann, Grüter, and Laloë 16 ). Another virial expansion leads Huang 17 to conclude that ∆T c /T (0) c ≈ 3.5(aρ 1/3 ) 1/2 . Interchanging the limit a → 0 with the thermodynamic limit, and using Monte-Carlo simulations, Holzmann and Krauth 18 find c = 2.3 ± 0.25. The dilute Bose gas can be mapped onto a classical field lattice model (Baym et. al. 19 ); a self-consistent approach then yields c = 2.9. 2000 An experimental realization by Reppy et. al. 20 yields c = 5.1 ± 0.9. It was later pointed out that the estimation of the scattering length between particles was not correct, however. The last articles essentially agree with one another, and also with more recent articles. 6 The case for a linear correction with constant c ≈ 1.3 is made rather convincingly; it is not beyond reasonable doubt, though. Notice that the constant c is universal in the sense that it does not depend on such special features as the mass of the particles or the details of the interactions. (The mass enters the scattering length a, however.)
The question of the critical temperature for interacting Bose gases is reviewed in Baym et. al. 26 and in Blaizot. 27 A comprehensive survey on many aspects of bosonic systems has been written by Bloch, Dalibard, and Zwerger. 28 This question is also mentioned in additional articles dealing with certain perturbation methods. The value of c is assumed to be known and its calculation serves to test the method. Some of these references can be found in Blaizot. 27 In this article we give a partial rigorous justification of the results in the literature by proving that off-diagonal correlations decay exponentially when
In particular, there is no Bose-Einstein condensation when (3) is satisfied. This rigorous result is not sharp enough to disprove any of the previous claims that have been just reviewed, although it gets close to Huang’s 1999 result. A
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