Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons

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Abstract

We consider a system of N trapped bosons with repulsive interactions in a combined semiclassical mean-field limit at positive temperature. We show that the free energy is well approximated by the minimum of the Hartree free energy functional – a natural extension of the Hartree energy functional to positive temperatures. The Hartree free energy functional converges in the same limit to a semiclassical free energy functional, and we show that the system displays Bose–Einstein condensation if and only if it occurs in the semiclassical free energy functional. This allows us to show that for weak coupling the critical temperature decreases due to the repulsive interactions.

Section snippets

Background and summary

The rigorous mathematical analysis of quantum many-particle systems has a long history, dating back to the early days of quantum mechanics. In case of (dilute) Bose gases, there has been a period of renewed interest since the first experimental observation of Bose–Einstein condensation (BEC) in trapped alkali gases in 1995 [5], [16] and the breakthrough work of Lieb and Yngvason in 1998 [58], who proved a lower bound for the ground state energy of the dilute Bose gas in the thermodynamic limit.

The Hartree free energy functional

At several points in the paper it will be convenient to use the second quantized formalism and we start by introducing the relevant notation. Afterwards, we define a canonical version of the Hartree free energy functional, which plays an important role in the proof of an upper bound for the free energy Fc in Section 5.2. We prove several statements for the two versions of the Hartree free energy functional that are used during the proof of the main results, e.g. the existence of a unique

The semiclassical free energy functional and its critical temperature

This third section is devoted to the study of the semiclassical free energy functional Fsc in (1.25). In the first part we prove the existence of a unique minimizer and establish the corresponding Euler–Lagrange equation. In the second part we use these preparations to prove Proposition 1.1.

Semiclassical mean-field limit of the Hartree free energy functional

In this section we give the proof of Theorem 1.2, which will be carried out in three steps. In the first two steps we prove upper and lower bounds on the Hartree free energy that, when combined, imply (1.35). In the third step we prove bounds for the Husimi function and the condensate fraction of the Hartree minimizer γH that imply (1.36a), (1.36b).

Bounds on the free energy

In this section we prove upper and lower bounds on the free energy (1.7) of the full quantum model that imply the claimed free energy asymptotics in Theorem 1.1. An important ingredient for our bounds is the canonical version of the Hartree free energy functional introduced in Section 2.2, and the bound on the difference of the canonical and the grand-canonical Hartree free energies in Lemma 2.5. Our bounds apply both in the canonical and the grand-canonical setting.

Bounds on the 1-pdm and the Husimi function of approximate Gibbs states

In the first part of this section we prove (1.34), which will conclude the proof of Theorem 1.1. Our approach is based on a lower bound for the bosonic relative entropy that quantifies its coercivity and was proved in [17, Lemma 4.1]. Afterwards we show how this result can be combined with Theorem 1.2 to prove Corollary 1.1, Corollary 1.2.

Proof of Theorem 1.3

In this section we show how the analysis in Sections 5 and 6 needs to be adjusted in order to prove Theorem 1.3. One main difference between the proof of Theorem 1.1 and the one of Theorem 1.3 concerns the proof of the spectral gap estimate for the Hartree operator. In case of the semiclassical MF scaling such a bound follows from the assumed bound on the Hessian of the interaction potential in (1.4), see Lemma 4.4. Here the interaction is of much shorter range and we do not have such a bound

Acknowledgements

Funding from the European Union's Horizon 2020 research and innovation programme under the ERC grant agreement No 694227 (R.S.) and under the Marie Sklodowska-Curie grant agreement No 836146 (A.D.) is gratefully acknowledged. A.D. acknowledges support of the Swiss National Science Foundation through the Ambizione grant PZ00P2 185851.

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