arXiv:cond-mat/9304001 · Modern Physics Letters B

TWO-DIMENSIONAL BOSE GAS AT LOW DENSITY

А. А. Овчинников

Published Jun 30, 1993• 1 citation

Abstract

We propose a new method to describe the interacting bose gas at zero temperature. For three-dimensional system, the correction to the ground state energy in density is reproduced. For the two-dimensional dilute bose gas, the ground state energy in the leading order in the parameter | ln α2ρ|−1, where α is a two-dimensional scattering length, is obtained.

DOI: 10.1142/s0217984993001028

Related papers

On the description of a two-dimensional Bose gas at low densities

Alexander A. Ovchinnikov

We propose a new method to describe the interacting Bose gas at zero temperature. For the three-dimensional system the correction to the ground-state energy in density is reproduced. For a two-dimensional dilute Bose gas the ground-state energy in the leading order in the parameter mod ln p alpha 2 mod -1 where alpha is the scattering length is obtained.

Journal of Physics Condensed MatterNov 15, 199316 citations

Dilute Bose gas in two dimensions: Density expansions and the Gross-Pitaevskii equation

Alexander Yu. Cherny, Arkady A Shanenko

arXiv:cond-mat/0103632Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJul 27, 200145 citations

The Ground State Energy of a Two-Dimensional Bose Gas

Søren Fournais, Théotime Girardot, Lukas Junge +2 more

Abstract We prove the following formula for the ground state energy density of a dilute Bose gas with density $$\rho $$ ρ in 2 dimensions in the thermodynamic limit $$\begin{aligned} e^{\text {2D}}(\rho ) = 4\pi \rho ^2 Y\Big (1 - Y \vert \log Y \vert + \Big ( 2\Gamma + \frac{1}{2} + \log (\pi ) \Big ) Y \Big ) + o(\rho ^2 Y^{2}), \end{aligned}$$ e 2D ( ρ ) = 4 π ρ 2 Y ( 1 - Y | log Y | + ( 2 Γ + 1 2 + log ( π ) ) Y ) + o ( ρ 2 Y 2 ) , as $$\rho a^2 \rightarrow 0$$ ρ a 2 → 0 . Here $$Y= |\log (\rho a^2)|^{-1}$$ Y = | log ( ρ a 2 ) | - 1 and a is the scattering length of the two-body potential. This result in 2 dimensions corresponds to the famous Lee–Huang–Yang formula in 3 dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular, it covers the crucial case of the hard core potential.

Communications in Mathematical PhysicsFeb 23, 202410 citations

A lower bound on the ground state energy of dilute Bose gas

Ji Oon Lee, Jun Yin

Consider an N-boson system interacting via a two-body repulsive short-range potential V in a three dimensional box Λ of side length L. We take the limit N,L→∞ while keeping the density ρ=N/L3 fixed and small. We prove a new lower bound for its ground state energy per particle, E(N,Λ)/N≥4πaρ[1−O(ρ1/3|log ρ|3)], as ρ→0, where a is the scattering length of V.

arXiv:0908.0109Journal of Mathematical PhysicsMay 1, 20107 citations

Theory of the weakly interacting Bose gas

Jens O. Andersen

arXiv:cond-mat/0305138Reviews of Modern PhysicsJul 1, 2004506 citations

Quantum Corrections to the Energy Density of a Homogeneous Bose Gas

Eric Braaten, Agustín Nieto

arXiv:cond-mat/9712041Dec 3, 19971 citation

arXiv:cond-mat/9304001 · Modern Physics Letters B

TWO-DIMENSIONAL BOSE GAS AT LOW DENSITY

А. А. Овчинников

Published Jun 30, 1993• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.1142/s0217984993001028

Related papers

On the description of a two-dimensional Bose gas at low densities

Alexander A. Ovchinnikov

Journal of Physics Condensed MatterNov 15, 199316 citations

Dilute Bose gas in two dimensions: Density expansions and the Gross-Pitaevskii equation

Alexander Yu. Cherny, Arkady A Shanenko

A dilute homogeneous two-dimensional (2D) Bose gas at zero temperature is studied with the method developed earlier by the authors. This method allows for considering renormalization of an arbitrary pairwise potential in a self-consistent manner, without the pseudopotential delta-function representation. Low-density expansions are derived for the chemical potential, ground-state energy, pair distribution function, kinetic and interaction energies. The expansion parameter is found to be a dimensionless in-medium scattering amplitude u obeying the equation 1/u+ln u=-ln(na(2)pi)-2gamma, where na(2) and gamma are the gas parameter and the Euler constant, respectively. It is shown that the ground-state energy is mostly kinetic in the low-density limit. This result does not depend on a specific form of the pairwise interaction potential, contrary to the 3D case. A new form of the 2D Gross-Pitaevskii equation is proposed within our scheme.A dilute homogeneous two-dimensional (2D) Bose gas at zero temperature is studied with the method developed earlier by the authors. This method allows for considering renormalization of an arbitrary pairwise potential in a self-consistent manner, without the pseudopotential delta-function representation. Low-density expansions are derived for the chemical potential, ground-state energy, pair distribution function, kinetic and interaction energies. The expansion parameter is found to be a dimensionless in-medium scattering amplitude u obeying the equation 1/u+ln u=-ln(na(2)pi)-2gamma, where na(2) and gamma are the gas parameter and the Euler constant, respectively. It is shown that the ground-state energy is mostly kinetic in the low-density limit. This result does not depend on a specific form of the pairwise interaction potential, contrary to the 3D case. A new form of the 2D Gross-Pitaevskii equation is proposed within our scheme.

arXiv:cond-mat/0103632Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJul 27, 200145 citations

The Ground State Energy of a Two-Dimensional Bose Gas

Søren Fournais, Théotime Girardot, Lukas Junge +2 more

Abstract We prove the following formula for the ground state energy density of a dilute Bose gas with density $\rho$ ρ in 2 dimensions in the thermodynamic limit $\begin{aligned} e^{\text {2D}}(\rho ) = 4\pi \rho ^2 Y\Big (1 - Y \vert \log Y \vert + \Big ( 2\Gamma + \frac{1}{2} + \log (\pi ) \Big ) Y \Big ) + o(\rho ^2 Y^{2}), \end{aligned}$ e 2D ( ρ ) = 4 π ρ 2 Y ( 1 - Y | log Y | + ( 2 Γ + 1 2 + log ( π ) ) Y ) + o ( ρ 2 Y 2 ) , as $\rho a^2 \rightarrow 0$ ρ a 2 → 0 . Here $Y= |\log (\rho a^2)|^{-1}$ Y = | log ( ρ a 2 ) | - 1 and a is the scattering length of the two-body potential. This result in 2 dimensions corresponds to the famous Lee–Huang–Yang formula in 3 dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular, it covers the crucial case of the hard core potential.Abstract We prove the following formula for the ground state energy density of a dilute Bose gas with density $$\rho $$ ρ in 2 dimensions in the thermodynamic limit $$\begin{aligned} e^{\text {2D}}(\rho ) = 4\pi \rho ^2 Y\Big (1 - Y \vert \log Y \vert + \Big ( 2\Gamma + \frac{1}{2} + \log (\pi ) \Big ) Y \Big ) + o(\rho ^2 Y^{2}), \end{aligned}$$ e 2D ( ρ ) = 4 π ρ 2 Y ( 1 - Y | log Y | + ( 2 Γ + 1 2 + log ( π ) ) Y ) + o ( ρ 2 Y 2 ) , as $$\rho a^2 \rightarrow 0$$ ρ a 2 → 0 . Here $$Y= |\log (\rho a^2)|^{-1}$$ Y = | log ( ρ a 2 ) | - 1 and a is the scattering length of the two-body potential. This result in 2 dimensions corresponds to the famous Lee–Huang–Yang formula in 3 dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular, it covers the crucial case of the hard core potential.

Communications in Mathematical PhysicsFeb 23, 202410 citations

A lower bound on the ground state energy of dilute Bose gas

Ji Oon Lee, Jun Yin

arXiv:0908.0109Journal of Mathematical PhysicsMay 1, 20107 citations

Theory of the weakly interacting Bose gas

Jens O. Andersen

This article reviews recent advances in the theory of the three-dimensional dilute homogeneous Bose gas at zero and finite temperature. Effective-field-theory methods are used to formulate a systematic perturbative framework that can be used to calculate the properties of a system at $T=0.$ The perturbative expansion of these properties is essentially an expansion in the gas parameter $\sqrt{{\mathrm{na}}^{3}},$ where $a$ is the $s$ -wave scattering length and $n$ is the number density. In particular, the leading quantum corrections to the ground-state energy density, the condensate depletion, and long-wavelength collective excitations are rederived in an efficient and economical manner. Nonuniversal effects are also discussed. These effects are higher-order corrections that depend on properties of the interatomic potential other than the scattering length, such as the effective range. The article critically examines various approaches to the dilute Bose gas in equilibrium at finite temperature. These include the Bogoliubov approximation, the Popov approximation, the Hartree-Fock-Bogoliubov approximation, the \ensuremath{\Phi}-derivable approach, optimized perturbation theory, and renormalization-group techniques. The article ends with a look at recent calculations of the critical temperature of the dilute Bose gas, which include $1/N$ techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.This article reviews recent advances in the theory of the three-dimensional dilute homogeneous Bose gas at zero and finite temperature. Effective-field-theory methods are used to formulate a systematic perturbative framework that can be used to calculate the properties of a system at $T=0.$ The perturbative expansion of these properties is essentially an expansion in the gas parameter $\sqrt{{\mathrm{na}}^{3}},$ where $a$ is the $s$-wave scattering length and $n$ is the number density. In particular, the leading quantum corrections to the ground-state energy density, the condensate depletion, and long-wavelength collective excitations are rederived in an efficient and economical manner. Nonuniversal effects are also discussed. These effects are higher-order corrections that depend on properties of the interatomic potential other than the scattering length, such as the effective range. The article critically examines various approaches to the dilute Bose gas in equilibrium at finite temperature. These include the Bogoliubov approximation, the Popov approximation, the Hartree-Fock-Bogoliubov approximation, the \ensuremath{\Phi}-derivable approach, optimized perturbation theory, and renormalization-group techniques. The article ends with a look at recent calculations of the critical temperature of the dilute Bose gas, which include $1/N$ techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.

arXiv:cond-mat/0305138Reviews of Modern PhysicsJul 1, 2004506 citations

Quantum Corrections to the Energy Density of a Homogeneous Bose Gas

Eric Braaten, Agustín Nieto

Quantum corrections to the properties of a homogeneous interacting Bose gas at zero temperature can be calculated as a low-density expansion in powers of $\sqrt{ρa^3}$ , where $ρ$ is the number density and $a$ is the S-wave scattering length. We calculate the ground state energy density to second order in $\sqrt{ρa^3}$ . The coefficient of the $ρa^3$ correction has a logarithmic term that was calculated in 1959. We present the first calculation of the constant under the logarithm. The constant depends not only on $a$ , but also on an extra parameter that describes the low energy $3\to 3$ scattering of the bosons. In the case of alkali atoms, we argue that the second order quantum correction is dominated by the logarithmic term, where the argument of the logarithm is $ρa \ell_V^2$ , and $\ell_V$ is the length scale set by the van der Waals potential.Quantum corrections to the properties of a homogeneous interacting Bose gas at zero temperature can be calculated as a low-density expansion in powers of $\sqrt{ρa^3}$, where $ρ$ is the number density and $a$ is the S-wave scattering length. We calculate the ground state energy density to second order in $\sqrt{ρa^3}$. The coefficient of the $ρa^3$ correction has a logarithmic term that was calculated in 1959. We present the first calculation of the constant under the logarithm. The constant depends not only on $a$, but also on an extra parameter that describes the low energy $3\to 3$ scattering of the bosons. In the case of alkali atoms, we argue that the second order quantum correction is dominated by the logarithmic term, where the argument of the logarithm is $ρa \ell_V^2$, and $\ell_V$ is the length scale set by the van der Waals potential.

arXiv:cond-mat/9712041Dec 3, 19971 citation

Abstract We prove the following formula for the ground state energy density of a dilute Bose gas with density

\rho

ρ in 2 dimensions in the thermodynamic limit

\begin{aligned} e^{\text {2D}}(\rho ) = 4\pi \rho ^2 Y\Big (1 - Y \vert \log Y \vert + \Big ( 2\Gamma + \frac{1}{2} + \log (\pi ) \Big ) Y \Big ) + o(\rho ^2 Y^{2}), \end{aligned}

e 2D ( ρ ) = 4 π ρ 2 Y ( 1 - Y | log Y | + ( 2 Γ + 1 2 + log ( π ) ) Y ) + o ( ρ 2 Y 2 ) , as

\rho a^2 \rightarrow 0

ρ a 2 → 0 . Here

Y= |\log (\rho a^2)|^{-1}

Y = | log ( ρ a 2 ) | - 1 and a is the scattering length of the two-body potential. This result in 2 dimensions corresponds to the famous Lee–Huang–Yang formula in 3 dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular, it covers the crucial case of the hard core potential.

T=0.

The perturbative expansion of these properties is essentially an expansion in the gas parameter

\sqrt{{\mathrm{na}}^{3}},

where

a

is the

s

-wave scattering length and

n

is the number density. In particular, the leading quantum corrections to the ground-state energy density, the condensate depletion, and long-wavelength collective excitations are rederived in an efficient and economical manner. Nonuniversal effects are also discussed. These effects are higher-order corrections that depend on properties of the interatomic potential other than the scattering length, such as the effective range. The article critically examines various approaches to the dilute Bose gas in equilibrium at finite temperature. These include the Bogoliubov approximation, the Popov approximation, the Hartree-Fock-Bogoliubov approximation, the \ensuremath{\Phi}-derivable approach, optimized perturbation theory, and renormalization-group techniques. The article ends with a look at recent calculations of the critical temperature of the dilute Bose gas, which include

1/N

techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.

Quantum corrections to the properties of a homogeneous interacting Bose gas at zero temperature can be calculated as a low-density expansion in powers of

\sqrt{ρa^3}

, where

ρ

is the number density and

a

is the S-wave scattering length. We calculate the ground state energy density to second order in

\sqrt{ρa^3}

. The coefficient of the

ρa^3

correction has a logarithmic term that was calculated in 1959. We present the first calculation of the constant under the logarithm. The constant depends not only on

a

, but also on an extra parameter that describes the low energy

3\to 3

scattering of the bosons. In the case of alkali atoms, we argue that the second order quantum correction is dominated by the logarithmic term, where the argument of the logarithm is

ρa \ell_V^2

, and

\ell_V

is the length scale set by the van der Waals potential.