Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0212590

Transition Temperature of Dilute, Weakly Repulsive Bose Gas

Masudul Haque, Andrei E. Ruckenstein

Published Dec 25, 2002• 0 citations
arXivHTMLPDF

Abstract

Within a quasiparticle framework, we reconsider the issue of computing the Bose-Einstein condensation temperature ($T_c$) in a weakly non-ideal Bose gas. The main result of this and previous investigations is that $T_c$ increases with the scattering length $a$, with the leading dependence being either linear or log-linear in $a$. The calculation of $T_c$ reduces to that of computing the excitation spectrum near the transition. We report two approaches to regularizing the infrared divergence at the transition point. One leads to a $a\sqrt{|\ln{a}|}$-like shift in $T_c$, and the other allows numerical calculations for the shift.

DOI
10.48550/arxiv.cond-mat/0212590

Related papers

The Transition Temperature of the Dilute Interacting Bose Gas

Gordon Baym, Jean-Paul Blaizot, Markus Holzmann +2 more

We show that the critical temperature of a uniform dilute Bose gas increases linearly with the $s$-wave scattering length describing the repulsion between the particles. Because of infrared divergences, the magnitude of the shift cannot be obtained from perturbation theory, even in the weak coupling regime; rather, it is proportional to the size of the critical region in momentum space. By means of a self-consistent calculation of the quasiparticle spectrum at low momenta at the transition, we find an estimate of the effect in reasonable agreement with numerical simulations.

arXiv:cond-mat/9905430•Physical Review Letters•Aug 30, 1999•248 citations

Weakly Non-ideal Bose Gas: Comments on Critical Temperature Calculations

Masudul Haque

A number of calculations have appeared for the Bose-Einstein condensation temperature (Tc) of a weakly repulsive dilute Bose gas. After a short survey of previous work, I point out several issues related to these approaches, and outline future calculations.

arXiv:cond-mat/0302076•Feb 4, 2003•1 citation

The transition temperature of the dilute interacting Bose gas for N internal states

Gordon Baym, Jean-Paul Blaizot, Jean Zinn‐Justin

We calculate explicitly the variation $δT_c$ of the Bose-Einstein condensation temperature $T_c$ induced by weak repulsive two-body interactions to leading order in the interaction strength. As shown earlier by general arguments, $δT_c/T_c$ is linear in the dimensionless product $an^{1/3}$ to leading order, where $n$ is the density and $a$ the scattering length. This result is non-perturbative, and a direct perturbative calculation of the amplitude is impossible due to infrared divergences familiar from the study of the superfluid helium lambda transition. Therefore we introduce here another standard expansion scheme, generalizing the initial model which depends on one complex field to one depending on $N$ real fields, and calculating the temperature shift at leading order for large $N$. The result is explicit and finite. The reliability of the result depends on the relevance of the large $N$ expansion to the situation N=2, which can in principle be checked by systematic higher order calculations. The large $N$ result agrees remarkably well with recent numerical simulations.

arXiv:cond-mat/9907241•Europhysics Letters (EPL)•Jan 15, 2000•107 citations

The transition temperature of the dilute interacting Bose gas for $N$ internal degrees of freedom

Gordon Baym, C Jean-paul Blaizot, Jean Zinn‐Justin

We calculate explicitly the variation δTc of the Bose-Einstein condensation temperature Tc induced by weak repulsive two-body interactions to leading order in the interaction strength. As shown earlier by general arguments, δTc/Tc is linear in the dimensionless product an1/3 to leading order, where n is the density and a the scattering length. This result is non-perturbative, and a direct perturbative calculation of the amplitude is impossible due to infrared divergences familiar from the study of the superfluid helium lambda transition. Therefore we introduce here another standard expansion scheme, generalizing the initial model which depends on one complex field to one depending on N real fields, and calculating the temperature shift at leading order for large N. The result is explicit and finite. The reliability of the result depends on the relevance of the large N expansion to the situation N = 2, which can in principle be checked by systematic higher order calculations. The large N result agrees remarkably well with recent numerical simulations.

•Jan 1, 1999•0 citations

Shift of BEC Temperature of Homogenous Weakly Interacting Bose Gas

Boris Kastening

We report on the computation of the shift of the Bose-Einstein condensation temperature for a homogenous weakly interacting Bose gas in leading order in the diluteness parameter a n^(1/3), where `a' is the scattering length and `n' is the particle density. The perturbative series, which is afflicted by infrared divergences, is resummed by means of variational perturbation theory. Using coefficients through seven loops, we arrive at Delta T_c/T_c = 1.27 +/- 0.11 a n^(1/3), which compares favorably with recent Monte-Carlo data.

arXiv:cond-mat/0404354•Apr 15, 2004•16 citations

Bose–Einstein transition temperature in a dilute repulsive gas

Markus Holzmann, Jean-Noël Fuchs, Gordon Baym +2 more

We discuss certain specific features of the calculation of the critical temperature of a dilute repulsive Bose gas. Interactions modify the critical temperature in two different ways. First, for gases in traps, temperature shifts are introduced by a change of the density profile, arising itself from a modification of the equation of state of the gas (reduced compressibility); these shifts can be calculated simply within mean field theory. Second, even in the absence of a trapping potential (homogeneous gas in a box), temperature shifts are introduced by the interactions; they arise from the correlations introduced in the gas, and thus lie inherently beyond mean field theory – in fact, their evaluation requires more elaborate, non-perturbative, calculations. One illustration of this non-perturbative character is provided by the solution of self-consistent equations, which relate together non-linearly the various energy shifts of the single particle levels k . These equations predict that repulsive interactions shift the critical temperature (at constant density) by an amount which is positive, and simply proportional to the scattering length a ; nevertheless, the numerical coefficient is difficult to compute. Physically, the increase of the temperature can be interpreted in terms of the reduced density fluctuations introduced by the repulsive interactions, which facilitate the propagation of large exchange cycles across the sample.

arXiv:cond-mat/0310460•Comptes Rendus Physique•Jan 1, 2004•34 citations
Within a quasiparticle framework, we reconsider the issue of computing the Bose-Einstein condensation temperature (TcT_cTc​) in a weakly non-ideal Bose gas. The main result of this and previous investigations is that TcT_cTc​ increases with the scattering length aaa, with the leading dependence being either linear or log-linear in aaa. The calculation of TcT_cTc​ reduces to that of computing the excitation spectrum near the transition. We report two approaches to regularizing the infrared divergence at the transition point. One leads to a a∣ln⁡a∣a\sqrt{|\ln{a}|}a∣lna∣​-like shift in TcT_cTc​, and the other allows numerical calculations for the shift.
We show that the critical temperature of a uniform dilute Bose gas increases linearly with the sss-wave scattering length describing the repulsion between the particles. Because of infrared divergences, the magnitude of the shift cannot be obtained from perturbation theory, even in the weak coupling regime; rather, it is proportional to the size of the critical region in momentum space. By means of a self-consistent calculation of the quasiparticle spectrum at low momenta at the transition, we find an estimate of the effect in reasonable agreement with numerical simulations.
We calculate explicitly the variation δTcδT_cδTc​ of the Bose-Einstein condensation temperature TcT_cTc​ induced by weak repulsive two-body interactions to leading order in the interaction strength. As shown earlier by general arguments, δTc/TcδT_c/T_cδTc​/Tc​ is linear in the dimensionless product an1/3an^{1/3}an1/3 to leading order, where nnn is the density and aaa the scattering length. This result is non-perturbative, and a direct perturbative calculation of the amplitude is impossible due to infrared divergences familiar from the study of the superfluid helium lambda transition. Therefore we introduce here another standard expansion scheme, generalizing the initial model which depends on one complex field to one depending on NNN real fields, and calculating the temperature shift at leading order for large NNN. The result is explicit and finite. The reliability of the result depends on the relevance of the large NNN expansion to the situation N=2, which can in principle be checked by systematic higher order calculations. The large NNN result agrees remarkably well with recent numerical simulations.
The transition temperature of the dilute interacting Bose gas for NNN internal degrees of freedom
We calculate explicitly the variation δTc of the Bose-Einstein condensation temperature Tc induced by weak repulsive two-body interactions to leading order in the interaction strength. As shown earlier by general arguments, δTc/Tc is linear in the dimensionless product an1/3 to leading order, where n is the density and a the scattering length. This result is non-perturbative, and a direct perturbative calculation of the amplitude is impossible due to infrared divergences familiar from the study of the superfluid helium lambda transition. Therefore we introduce here another standard expansion scheme, generalizing the initial model which depends on one complex field to one depending on N real fields, and calculating the temperature shift at leading order for large N. The result is explicit and finite. The reliability of the result depends on the relevance of the large N expansion to the situation N = 2, which can in principle be checked by systematic higher order calculations. The large N result agrees remarkably well with recent numerical simulations.
We report on the computation of the shift of the Bose-Einstein condensation temperature for a homogenous weakly interacting Bose gas in leading order in the diluteness parameter a n^(1/3), where `a' is the scattering length and `n' is the particle density. The perturbative series, which is afflicted by infrared divergences, is resummed by means of variational perturbation theory. Using coefficients through seven loops, we arrive at Delta T_c/T_c = 1.27 +/- 0.11 a n^(1/3), which compares favorably with recent Monte-Carlo data.
We discuss certain specific features of the calculation of the critical temperature of a dilute repulsive Bose gas. Interactions modify the critical temperature in two different ways. First, for gases in traps, temperature shifts are introduced by a change of the density profile, arising itself from a modification of the equation of state of the gas (reduced compressibility); these shifts can be calculated simply within mean field theory. Second, even in the absence of a trapping potential (homogeneous gas in a box), temperature shifts are introduced by the interactions; they arise from the correlations introduced in the gas, and thus lie inherently beyond mean field theory – in fact, their evaluation requires more elaborate, non-perturbative, calculations. One illustration of this non-perturbative character is provided by the solution of self-consistent equations, which relate together non-linearly the various energy shifts of the single particle levels k . These equations predict that repulsive interactions shift the critical temperature (at constant density) by an amount which is positive, and simply proportional to the scattering length a ; nevertheless, the numerical coefficient is difficult to compute. Physically, the increase of the temperature can be interpreted in terms of the reduced density fluctuations introduced by the repulsive interactions, which facilitate the propagation of large exchange cycles across the sample.