arXiv:2011.05962 · Journal of Statistical Physics

Bose–Einstein Condensation for Two Dimensional Bosons in the Gross–Pitaevskii Regime

Cristina Caraci, Serena Cenatiempo, Benjamin Schlein

Published May 20, 2021• 17 citations

Abstract

We consider systems of N bosons trapped on the two-dimensional unit torus, in the Gross-Pitaevskii regime, where the scattering length of the repulsive interaction is exponentially small in the number of particles. We show that low-energy states exhibit complete Bose-Einstein condensation, with almost optimal bounds on the number of orthogonal excitations.

DOI: 10.1007/s10955-021-02766-6

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Abstract We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order $$ N^{\kappa -1}$$ N κ - 1 , for some $$\kappa >0$$ κ > 0 . For the range $$ \kappa \in [0, \frac{1}{43})$$ κ ∈ [ 0 , 1 43 ) , Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for $$ \kappa \in [0, \frac{1}{20})$$ κ ∈ [ 0 , 1 20 ) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347 ), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).

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arXiv:2011.05962 · Journal of Statistical Physics

Bose–Einstein Condensation for Two Dimensional Bosons in the Gross–Pitaevskii Regime

Cristina Caraci, Serena Cenatiempo, Benjamin Schlein

Published May 20, 2021• 17 citations

arXiv HTML PDF

Abstract

DOI: 10.1007/s10955-021-02766-6

Related papers

The Excitation Spectrum of Two-Dimensional Bose Gases in the Gross–Pitaevskii Regime

Cristina Caraci, Serena Cenatiempo, Benjamin Schlein

Abstract We consider a system of N bosons, in the two-dimensional unit torus. We assume particles to interact through a repulsive two-body potential, with a scattering length that is exponentially small in N (Gross–Pitaevskii regime). In this setting, we establish the validity of the predictions of Bogoliubov theory, determining the ground state energy of the Hamilton operator and its low-energy excitation spectrum, up to errors that vanish in the limit $N \rightarrow \infty$ N → ∞ .Abstract We consider a system of N bosons, in the two-dimensional unit torus. We assume particles to interact through a repulsive two-body potential, with a scattering length that is exponentially small in N (Gross–Pitaevskii regime). In this setting, we establish the validity of the predictions of Bogoliubov theory, determining the ground state energy of the Hamilton operator and its low-energy excitation spectrum, up to errors that vanish in the limit $$N \rightarrow \infty $$ N → ∞ .

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Abstract We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order $N^{-1+\kappa }$ N - 1 + κ , for $\kappa >0$ κ > 0 . Assuming that $\kappa \in (0;1/43)$ κ ∈ ( 0 ; 1 / 43 ) , we show that low-energy states exhibit Bose–Einstein condensation and we provide bounds on the expectation and on higher moments of the number of excitations.Abstract We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order $$N^{-1+\kappa }$$ N - 1 + κ , for $$\kappa >0$$ κ > 0 . Assuming that $$\kappa \in (0;1/43)$$ κ ∈ ( 0 ; 1 / 43 ) , we show that low-energy states exhibit Bose–Einstein condensation and we provide bounds on the expectation and on higher moments of the number of excitations.

arXiv:2002.03406Annales Henri PoincaréDec 26, 202045 citations

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Élliott H. Lieb, Robert Seiringer

The ground state of a gas of Bosons confined in an external trap potential and interacting via repulsive two-body forces has recently been shown to exhibit complete Bose-Einstein condensation in the dilute limit, yielding for the first time a rigorous proof of this phenomenon in a physically realistic setting. We give here an account of this work about the Gross-Pitaevskii limit where the particle number $N$ goes to infinity with $Na$ fixed, where $a$ is the scattering length of the interparticle interaction, measured in units of the trap size.The ground state of a gas of Bosons confined in an external trap potential and interacting via repulsive two-body forces has recently been shown to exhibit complete Bose-Einstein condensation in the dilute limit, yielding for the first time a rigorous proof of this phenomenon in a physically realistic setting. We give here an account of this work about the Gross-Pitaevskii limit where the particle number $N$ goes to infinity with $Na$ fixed, where $a$ is the scattering length of the interparticle interaction, measured in units of the trap size.

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A Short Proof of Bose–Einstein Condensation in the Gross–Pitaevskii Regime and Beyond

Christian Brennecke, Morris Brooks, Cristina Caraci +1 more

Abstract We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order $N^{\kappa -1}$ N κ - 1 , for some $\kappa >0$ κ > 0 . For the range $\kappa \in [0, \frac{1}{43})$ κ ∈ [ 0 , 1 43 ) , Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for $\kappa \in [0, \frac{1}{20})$ κ ∈ [ 0 , 1 20 ) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347 ), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).Abstract We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order $$ N^{\kappa -1}$$ N κ - 1 , for some $$\kappa >0$$ κ > 0 . For the range $$ \kappa \in [0, \frac{1}{43})$$ κ ∈ [ 0 , 1 43 ) , Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for $$ \kappa \in [0, \frac{1}{20})$$ κ ∈ [ 0 , 1 20 ) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347 ), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).

arXiv:2401.00784Annales Henri PoincaréJul 1, 20245 citations

N \rightarrow \infty

N → ∞ .

Abstract We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order

N^{-1+\kappa }

N - 1 + κ , for

\kappa >0

κ > 0 . Assuming that

\kappa \in (0;1/43)

κ ∈ ( 0 ; 1 / 43 ) , we show that low-energy states exhibit Bose–Einstein condensation and we provide bounds on the expectation and on higher moments of the number of excitations.

N

goes to infinity with

Na

fixed, where

a

is the scattering length of the interparticle interaction, measured in units of the trap size.

Abstract We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order

N^{\kappa -1}

N κ - 1 , for some

\kappa >0

κ > 0 . For the range

\kappa \in [0, \frac{1}{43})

κ ∈ [ 0 , 1 43 ) , Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for

\kappa \in [0, \frac{1}{20})

κ ∈ [ 0 , 1 20 ) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347 ), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).