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arXiv:2201.13440 · Journal of Mathematical Physics

Ground state energy of the low density Bose gas with three-body interactions

Phan Thành Nam, Julien Ricaud, Arnaud Triay

Published Jul 1, 2022• 13 citations
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Abstract

We consider the low density Bose gas in the thermodynamic limit with a three-body interaction potential. We prove that the leading order of the ground state energy of the system is determined completely in terms of the scattering energy of the interaction potential. The corresponding result for two-body interactions was proved in seminal papers of Dyson [Phys. Rev. 106, 20–26 (1957)] and of Lieb and Yngvason [Phys. Rev. Lett. 80, 2504–2507 (1998)].

DOI
10.1063/5.0087026

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We consider the low density Bose gas in the thermodynamic limit with a three-body interaction potential. We prove that the leading order of the ground state energy of the system is determined completely in terms of the scattering energy of the interaction potential. The corresponding result for two-body interactions was proved in seminal papers of Dyson [Phys. Rev. 106, 20–26 (1957)] and of Lieb and Yngvason [Phys. Rev. Lett. 80, 2504–2507 (1998)].
In the present paper we study the low density Bose gas in the thermodynamic limit interacting via two-body and three-body interaction potentials. We prove that the leading order of the ground state energy is entirely characterised by both the scattering length of the two-body potential and the scattering energy of the three-body potential. The corresponding result for two-body interactions was proven in seminal papers of Dyson (1957) and of Lieb-Yngvason (1998), and the result for three-body interactions was proven by Nam-Ricaud-Triay (2022). The present result resolves a conjecture of Nam-Ricaud-Triay (2022).
Abstract We study dilute Bose gases in the thermodynamic limit interacting via two-body and three-body interaction potentials. We prove that the leading order of the thermodynamic ground state energy is entirely characterized by both the scattering length of the two-body potential and the scattering energy of the three-body potential. The corresponding result for two-body interactions was proven in seminal papers of Dyson (1957) [1] and Lieb–Yngvason (1998) [2], and the result for three-body interactions was proven much more recently by Nam–Ricaud–Triay (2022) [3]. The present result resolves a conjecture of Nam–Ricaud–Triay (2022) [4].
We consider a gas of bosons interacting through a three-body hard-core potential in the thermodynamic limit. We derive an upper bound on the ground state energy of the system at the leading order using a Jastrow factor. Our result matches the lower bound proven by Nam-Ricaud-Triay (J Math Phys 63:071903, 2022) and therefore resolves the leading order. Moreover, a straightforward adaptation of our proof can be used for systems interacting via combined two-body and three-body interactions to generalise Theorem 1.2 from (Ann. Henri Poincaré, 2026) to hard-core potentials.
Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, ρρρ, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is 2πℏ2aρ/m2π\hbar^2 a ρ/m2πℏ2aρ/m where aaa is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange N7/5N^{7/5}N7/5 law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.
The standard calculations of the ground-state energy of a homogeneous Bose gas rely on approximations which are physically reasonable but difficult to control. Lieb and Yngvason ( Phys. Rev. Lett. 80 (1998) 2504) have proved rigorously that the commonly accepted leading-order term of the ground-state energy is correct in the zero-density limit. Here, strong indications are given that also the next-to-leading term is correct. It is shown that the first terms obtained in a perturbative treatment provide contributions which are lost in the Bogoliubov approach.
We consider a Bose gas in spatial dimension n>3n>3n>3 with a repulsive, radially symmetric two-body potential VVV. In the limit of low density ρρρ, the ground state energy per particle in the thermodynamic limit is shown to be (n−2)∣Sn−1∣an−2ρ(n-2)|\mathbb S^{n-1}|a^{n-2}ρ(n−2)∣Sn−1∣an−2ρ, where ∣Sn−1∣|\mathbb S^{n-1}|∣Sn−1∣ denotes the surface measure of the unit sphere in Rn\mathbb{R}^nRn and aaa is the scattering length of VVV. Furthermore, for smooth and compactly supported two-body potentials, we derive upper bounds to the ground state energy with a correction term (1+Cγ)8π4a6ρ2∣ln⁡(a4ρ)∣(1+Cγ)8π^4a^6ρ^2|\ln(a^4ρ)|(1+Cγ)8π4a6ρ2∣ln(a4ρ)∣ in dimension n=4n=4n=4, where γ:=∫V(x)∣x∣−2 dxγ:=\int V(x)|x|^{-2}\, dxγ:=∫V(x)∣x∣−2dx, and a correction term which is O(ρ2)\mathcal{O}(ρ^2)O(ρ2) in higher dimensions.