arXiv:1706.01822 · Europhysics Letters (EPL)

Calculation of the critical temperature of a dilute Bose gas in the Bogoliubov approximation

Marcin Napiórkowski, Robin Reuvers, J. P. Solovej

Published Jan 1, 2018• 5 citations

Abstract

Following an earlier calculation in 3D, we calculate the 2D critical temperature of a dilute, translation-invariant Bose gas using a variational formulation of the Bogoliubov approximation introduced by Critchley and Solomon in 1976. This provides the first analytical calculation of the Kosterlitz-Thouless transition temperature that includes the constant in the logarithm.

DOI: 10.1209/0295-5075/121/10007

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arXiv:1706.01822 · Europhysics Letters (EPL)

Calculation of the critical temperature of a dilute Bose gas in the Bogoliubov approximation

Marcin Napiórkowski, Robin Reuvers, J. P. Solovej

Published Jan 1, 2018• 5 citations

arXiv HTML PDF

Abstract

DOI: 10.1209/0295-5075/121/10007

Related papers

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.

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arXiv:cond-mat/0106075Physical Review LettersDec 11, 2001262 citations

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.

We consider a two-dimensional Bose gas in the dilute regime where

ρa^2

is small. For temperatures below the Berezinskii-Kosterlitz-Thouless critical temperature, we derive an explicit upper bound for the free energy density using Bogoliubov theory. Our result captures the contribution of quasiparticle modes with dispersion relation

\sqrt{p^4 + 8πρ\, δ\, p^2}

and where

δ= 2 / (|\log(ρa^2)| + \log |\log(ρa^2)|)

We study the two-dimensional dilute Bose gas using the ladder or T-matrix approximation, which is valid if ln(1/

{\mathit{na}}^{2}

)\ensuremath{\gg}1, and show that this approximation can describe also the critical region of the Kosterlitz-Thouless transition accurately. In particular, we consider the superfluid properties of the gas and find that the superfluid density shows the square-root cusp together with the universal jump at the critical temperature. The theory is applied to spin-polarized atomic hydrogen adsorbed on a superfluid helium film, since this appears to be one of the most promising experimental setups to achieve the Kosterlitz-Thouless transition in a weakly interacting Bose gas.

T

is the temperature,

m

is the mass, and

U

is the effective interaction. The dimensionless constant $\ensuremath{\xi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}380\ifmmode\pm\else\textpm\fi{}3$ is very large and thus any quantitative analysis of the experimental data crucially depends on its value. For ${\ensuremath{\xi}}_{\ensuremath{\mu}}$ our result is ${\ensuremath{\xi}}_{\ensuremath{\mu}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}13.2\ifmmode\pm\else\textpm\fi{}0.4$. We also report the study of the quasicondensate correlations at the critical point.