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arXiv:2401.15540 · Communications of the American Mathematical Society

The second order 2D behaviors of a 3D Bose gas in the Gross-Pitaevskii regime

Xuwen Chen, Jiahao Wu, Zhifei Zhang

Published Aug 27, 2026• 0 citations

Abstract

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.

DOI: 10.1090/cams/79

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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.

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 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.

Abstract We study the dynamics of a system of N interacting bosons in a disc-shaped trap, which is realised by an external potential that confines the bosons in one spatial dimension to an interval of length of order

\varepsilon

ε . The interaction is non-negative and scaled in such a way that its scattering length is of order

\varepsilon /N

ε / N , while its range is proportional to

(\varepsilon /N)^{\beta }

( ε / N ) β with scaling parameter

\beta \in (0,1]

β ∈ ( 0 , 1 ] . We consider the simultaneous limit

\varepsilon /N

( N , ε ) → ( ∞ , 0 ) and assume that the system initially exhibits Bose–Einstein condensation. We prove that condensation is preserved by the N-body dynamics, where the time-evolved condensate wave function is the solution of a two-dimensional non-linear equation. The strength of the non-linearity depends on the scaling parameter

\beta

β . For

\beta \in (0,1)

β ∈ ( 0 , 1 ) , we obtain a cubic defocusing non-linear Schrödinger equation, while the choice

\beta =1

β = 1 yields a Gross–Pitaevskii equation featuring the scattering length of the interaction. In both cases, the coupling parameter depends on the confining potential.

For a system of

N

Fermions of spin

1/2

, with its interaction potential of scattering length

a

, the classical Huang-Yang formula states that the energy density

e(ρ)

is of the form \begin{equation*} e(ρ)=\frac{3}{5}(3π^2)^{\frac{2}{3}}ρ^{\frac{5}{3}}+2πaρ^2 +\frac{12}{35}(11-2\ln2)3^{\frac{1}{3}}π^{\frac{2}{3}}a^2ρ^{\frac{7}{3}} +o(ρ^{\frac{7}{3}}). \end{equation*} We consider a general system of

N

Fermions of spin

1/\mathbf{q}

with the scattering length of the interaction potential at the scale

a\propto N^{-1}

, that is, in the Gross-Pitaevskii regime. We prove the 2nd order ground state energy approximation corresponding to the Huang-Yang formula. The thermodynamic limit case shares a similar logic, and could be dealt with in a separate paper.

We consider many-body effects on particle scattering in one-, two-, and three-dimensional (3D) Bose gases. We show that at

T=0

these effects can be modeled by the simpler two-body T matrix evaluated off the energy shell. This is important in 1D and 2D because the two-body T matrix vanishes at zero energy and so mean-field effects on particle energies must be taken into account to obtain a self-consistent treatment of low-energy collisions. Using the off-shell two-body T matrix we obtain the energy and density dependence of the effective interaction in 1D and 2D and the appropriate Gross-Pitaevskii equations for these dimensions. Our results provide an alternative derivation of those of Kolomeisky and co-workers. We present numerical solutions of the Gross-Pitaevskii equation for a 2D condensate of hard-sphere bosons in a trap. We find that the interaction strength is much greater in 2D than for a 3D gas with the same hard-sphere radius. The Thomas-Fermi regime is, therefore, approached at lower condensate populations and the energy required to create vortices is lowered compared to the 3D case.