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arXiv:2203.01841 · Journal of Mathematical Physics

A second order upper bound on the ground state energy of a Bose gas beyond the Gross–Pitaevskii regime

Giulia Basti

Published Jul 1, 2022• 4 citations
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Abstract

We consider a system of N bosons in a unitary box in the grand-canonical setting interacting through a potential with the scattering length scaling as N−1+κ, κ ∈ (0, 2/3). This regimes interpolate between the Gross–Pitaevskii regime (κ = 0) and the thermodynamic limit (κ = 2/3). In the work of Basti et al. [Forum Math., Sigma 9, E74 (2021)], as an intermediate step to prove an upper bound in agreement with the Lee–Huang–Yang formula in the thermodynamic limit, a second order upper bound on the ground state energy for κ < 5/9 was obtained. In this paper, thanks to a more careful analysis of the error terms, we extend the mentioned result to κ < 7/12.

DOI
10.1063/5.0089790

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We consider a system of N bosons in a unitary box in the grand-canonical setting interacting through a potential with the scattering length scaling as N−1+κ, κ ∈ (0, 2/3). This regimes interpolate between the Gross–Pitaevskii regime (κ = 0) and the thermodynamic limit (κ = 2/3). In the work of Basti et al. [Forum Math., Sigma 9, E74 (2021)], as an intermediate step to prove an upper bound in agreement with the Lee–Huang–Yang formula in the thermodynamic limit, a second order upper bound on the ground state energy for κ < 5/9 was obtained. In this paper, thanks to a more careful analysis of the error terms, we extend the mentioned result to κ < 7/12.
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.
We consider a homogeneous Bose gas in the Gross-Pitaevskii limit at temperatures that are comparable to the critical temperature for Bose-Einstein condensation. Recently, an upper bound for the grand canonical free energy was proved in Boccato et al. (SIAM J Math Anal 56(2):2611-2660, 2024) capturing two novel contributions. First, the free energy of the interacting condensate is given in terms of an effective theory describing the probability distribution of the number of condensed particles. Second, the free energy of the thermally excited particles equals that of a temperature-dependent Bogoliubov Hamiltonian. We extend this result to a more general class of interaction potentials, including interactions with a hard core. Our proof follows a different approach than the one in Boccato et al. (SIAM J Math Anal 56(2):2611-2660, 2024): We model microscopic correlations between the particles by a Jastrow factor and exploit a cancellation in the computation of the energy that emerges due to the different length scales in the system.
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.
Consider NNN bosons in a finite box $\ensuremath{\Lambda}={[0,L]}^{3}\ensuremath{\subset}{\mathbb{R}}^{3}$ interacting via a two-body non-negative soft potential $V=\ensuremath{\lambda}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ with $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ fixed and $\ensuremath{\lambda}>0$ small. We will take the limit $L,N\ensuremath{\rightarrow}\ensuremath{\infty}$ by keeping the density $\ensuremath{\varrho}=N∕{L}^{3}$ fixed and small. We construct a variational state, which gives an upper bound on the ground-state energy per particle $\ensuremath{\epsilon}$, $\ensuremath{\epsilon}\ensuremath{\leqslant}4\ensuremath{\pi}\ensuremath{\varrho}a[1+(128∕15\sqrt{\ensuremath{\pi}}){(\ensuremath{\varrho}{a}^{3})}^{1∕2}{S}_{\ensuremath{\lambda}}]+O({\ensuremath{\varrho}}^{2}\ensuremath{\mid}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varrho}\ensuremath{\mid})$, as $\ensuremath{\varrho}\ensuremath{\rightarrow}0$, with a constant satisfying $1\ensuremath{\leqslant}{S}_{\ensuremath{\lambda}}\ensuremath{\leqslant}1+C\ensuremath{\lambda}$. Here aaa is the scattering length of VVV and thus depends on $\ensuremath{\lambda}$. In comparison, the prediction by Lee and Yang [Phys. Rev. 105, 1119 (1957)] and Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957)] asserts that ${S}_{\ensuremath{\lambda}}=1$ independent of $\ensuremath{\lambda}$.