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arXiv:2305.00797 · Reviews in Mathematical Physics

Lower bounds on the energy of the Bose gas

Søren Fournais, Théotime Girardot, Lukas Junge, Leo Morin, Marco Olivieri

Published Jul 20, 2023• 3 citations
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Abstract

We present an overview of the approach to establish a lower bound to the ground state energy for the dilute, interacting Bose gas in a periodic box. In this paper, the size of the box is larger than the Gross–Pitaevskii length scale. The presentation includes both the two- and three-dimensional cases, and catches the second-order correction, i.e. the Lee–Huang–Yang term. The calculation on a box of this length scale is the main step to calculate the energy in the thermodynamic limit. However, the periodic boundary condition simplifies many steps of the argument considerably compared to the localized problem coming from the thermodynamic case.

DOI
10.1142/s0129055x23600048

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We present an overview of the approach to establish a lower bound to the ground state energy for the dilute, interacting Bose gas in a periodic box. In this paper, the size of the box is larger than the Gross–Pitaevskii length scale. The presentation includes both the two- and three-dimensional cases, and catches the second-order correction, i.e. the Lee–Huang–Yang term. The calculation on a box of this length scale is the main step to calculate the energy in the thermodynamic limit. However, the periodic boundary condition simplifies many steps of the argument considerably compared to the localized problem coming from the thermodynamic case.
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Abstract We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second-order term, as predicted by the Lee–Huang–Yang formula. This result was first established in [20] by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported V∈L3(R3)V \in L^3 (\mathbb {R}^3)V∈L3(R3) , gives better rates and, in our opinion, is substantially simpler.
We consider a system of N N bosons interacting in a three-dimensional box endowed with periodic boundary condition that is strongly confined in one direction such that the normalized thickness of the box d ≪ 1 d\ll 1 . We assume particles to interact through a repulsive, radially symmetric and short-range interaction potential with scattering length scale a ≪ d a\ll d . We present a comprehensive study of such system in the Gross-Pitaevskii regime, up to the second order ground state energy, starting from proving optimal Bose-Einstein condensation results which were not previously available. The fine interplay between the parameters N N , a a and d d generates three regions. Our result in one region, on the one hand, is compatible with the classical three-dimensional Lee-Huang-Yang formula. On the other hand, it reveals a new mechanism exhibiting how the second order correction compensates and modifies the first order energy, which is consistent with Schnee and Yngvason [Comm. Math. Phys. 269 (2007), pp. 659–691], and thus explains how a three-dimensional Bose gas system smoothly transits into two-dimensional system. From the analysis of this new mechanism exclusive to the second order, we find a dimensional coupling correlation effect, deeply buried away from the expected 3D and quasi-2D renormalizations, and calculate a new second order correction to the ground state energy. As a result, there are 2nd order terms not yet seen in the 3D and 2D cases.