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Derivation of Hartree theory for two-dimensional attractive Bose gases in almost Gross-Pitaevskii regime

Lukas Junge, François Louis Antoine Visconti

TL;DR

This work provides a rigorous derivation of Hartree theory and the nonlinear Schrödinger (NLS) limit for trapped two-dimensional Bose gases with attractive mean-field interactions, valid for broad interaction scalings including polynomial and near-Gross–Pitaevskii exponential forms. The authors combine momentum-space localization, finite-dimensional quantum de Finetti reductions, and careful control of high- and low-occupancy sectors to prove the convergence of the ground-state energy per particle e_N to the NLS energy e^{nls}, and to establish Bose–Einstein condensation into NLS minimizers. Importantly, stability of second kind is extended to all β>0 (polynomial) and κ∈(0,1) (exponential), enabling results in regimes previously out of reach for attractive interactions. The results apply to both defocusing and focusing cases, yield quantitative condensation descriptions, and sharpen the connection between microscopic many-body Hamiltonians and macroscopic NLS/Hartree descriptions in 2D. The techniques enhance understanding of the GP regime in lower dimensions and broaden the applicability of NLS-based effective theories for attractive Bose gases.

Abstract

We study the ground state energy of trapped two-dimensional Bose gases with mean-field type interactions that can be attractive. We prove the stability of second kind of the many-body system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger (NLS) energy functional. Notably, we can take any polynomial scaling of the interaction, and even exponential scalings arbitrarily close to the Gross--Pitaevskii regime, which is a drastic improvement on the best-known result for systems with attractive interactions. As a consequence of the stability of second kind we also obtain Bose-Einstein condensation for the many-body ground states for a much improved range of the diluteness parameter.

Derivation of Hartree theory for two-dimensional attractive Bose gases in almost Gross-Pitaevskii regime

TL;DR

This work provides a rigorous derivation of Hartree theory and the nonlinear Schrödinger (NLS) limit for trapped two-dimensional Bose gases with attractive mean-field interactions, valid for broad interaction scalings including polynomial and near-Gross–Pitaevskii exponential forms. The authors combine momentum-space localization, finite-dimensional quantum de Finetti reductions, and careful control of high- and low-occupancy sectors to prove the convergence of the ground-state energy per particle e_N to the NLS energy e^{nls}, and to establish Bose–Einstein condensation into NLS minimizers. Importantly, stability of second kind is extended to all β>0 (polynomial) and κ∈(0,1) (exponential), enabling results in regimes previously out of reach for attractive interactions. The results apply to both defocusing and focusing cases, yield quantitative condensation descriptions, and sharpen the connection between microscopic many-body Hamiltonians and macroscopic NLS/Hartree descriptions in 2D. The techniques enhance understanding of the GP regime in lower dimensions and broaden the applicability of NLS-based effective theories for attractive Bose gases.

Abstract

We study the ground state energy of trapped two-dimensional Bose gases with mean-field type interactions that can be attractive. We prove the stability of second kind of the many-body system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger (NLS) energy functional. Notably, we can take any polynomial scaling of the interaction, and even exponential scalings arbitrarily close to the Gross--Pitaevskii regime, which is a drastic improvement on the best-known result for systems with attractive interactions. As a consequence of the stability of second kind we also obtain Bose-Einstein condensation for the many-body ground states for a much improved range of the diluteness parameter.

Paper Structure

This paper contains 8 sections, 6 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Notations
  4. Statement of the main result
  5. Strategy of the proof
  6. Proof of Theorem \ref{['th:convergence_nls_theory']}
  7. Plain wave estimate and other results
  8. Proof of the main technical estimate: Lemma \ref{['lemma:bound_interaction_low_occupancy']}

Key Result

Theorem 2

Let $w_N$ and $V$ satisfy Assumption ass:potentials and the condition eq:interaction_potential_negative_assumption. Then, and for some constant $C > 0$ that depends only on $w$ and $V$. Moreover, for a sequence $\{\Psi_N\}_N$ of ground states of $H_N$, there exists a Borel probability measure $\mu$ supported on the minimisers of $\mathcal{E}^{\textmd{nls}}$ such that, along a subsequence, If $\

Theorems & Definitions (12)

  • Theorem 2: Convergence to NLS theory
  • Theorem 3
  • proof
  • Proposition 4: Plane wave estimate
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • proof : Proof of Proposition \ref{['prop:plane_wave_estimate']}
  • proof : Proof of Lemma \ref{['lemma:state_two_projections_full_trace']}
  • proof : Proof of Lemma \ref{['lemma:state_one_projection_new_state']}
  • ...and 2 more