arXiv:cond-mat/0302076

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

Masudul Haque

Published Feb 4, 2003• 1 citation

Abstract

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.

DOI: 10.48550/arxiv.cond-mat/0302076

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arXiv:cond-mat/0302076

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

Masudul Haque

Published Feb 4, 2003• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.cond-mat/0302076

Related papers

Transition Temperature of Dilute, Weakly Repulsive Bose Gas

Masudul Haque, Andrei E. Ruckenstein

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.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.

arXiv:cond-mat/0212590Dec 25, 20020 citations

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Jan 1, 20017 citations

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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.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/0310460Comptes Rendus PhysiqueJan 1, 200434 citations

Tc for homogeneous dilute Bose gases: A second-order result

Peter B. Arnold, Guy D. Moore, Boris Tomášik

The transition temperature for a dilute, homogeneous, three-dimensional Bose gas has the expansion ${T}_{\mathrm{c}}{=T}_{0}{{1+c}_{1}{\mathrm{an}}^{1/3}+[{c}_{2}^{\ensuremath{'}}\mathrm{ln}{(an}^{1/3}{)+c}_{2}^{\ensuremath{''}}]{a}^{2}{n}^{2/3}{+O(a}^{3}n)},$ where a is the scattering length, n the number density, and ${T}_{0}$ the ideal gas result. The first-order coefficient ${c}_{1}$ depends on nonperturbative physics. In this paper, we show that the coefficient ${c}_{2}^{\ensuremath{'}}$ may be computed perturbatively. We also show that the remaining second-order coefficient ${c}_{2}^{\ensuremath{''}}$ depends on non-perturbative physics but may be related, by a perturbative calculation, to quantities that have previously been measured using lattice simulations of three-dimensional $O(2)$ scalar field theory. Making use of those simulation results, we find ${T}_{\mathrm{c}}\ensuremath{\simeq}{T}_{0}{1+(1.32\ifmmode\pm\else\textpm\fi{}0.02){\mathrm{an}}^{1/3}+[19.7518\mathrm{ln}{(an}^{1/3})+(75.7\ifmmode\pm\else\textpm\fi{}0.4)]{a}^{2}{n}^{2/3}{+O(a}^{3}n)}.$The transition temperature for a dilute, homogeneous, three-dimensional Bose gas has the expansion ${T}_{\mathrm{c}}{=T}_{0}{{1+c}_{1}{\mathrm{an}}^{1/3}+[{c}_{2}^{\ensuremath{'}}\mathrm{ln}{(an}^{1/3}{)+c}_{2}^{\ensuremath{''}}]{a}^{2}{n}^{2/3}{+O(a}^{3}n)},$ where a is the scattering length, n the number density, and ${T}_{0}$ the ideal gas result. The first-order coefficient ${c}_{1}$ depends on nonperturbative physics. In this paper, we show that the coefficient ${c}_{2}^{\ensuremath{'}}$ may be computed perturbatively. We also show that the remaining second-order coefficient ${c}_{2}^{\ensuremath{''}}$ depends on non-perturbative physics but may be related, by a perturbative calculation, to quantities that have previously been measured using lattice simulations of three-dimensional $O(2)$ scalar field theory. Making use of those simulation results, we find ${T}_{\mathrm{c}}\ensuremath{\simeq}{T}_{0}{1+(1.32\ifmmode\pm\else\textpm\fi{}0.02){\mathrm{an}}^{1/3}+[19.7518\mathrm{ln}{(an}^{1/3})+(75.7\ifmmode\pm\else\textpm\fi{}0.4)]{a}^{2}{n}^{2/3}{+O(a}^{3}n)}.$

arXiv:cond-mat/0107124Physical Review ADec 13, 200165 citations

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.

(T_c-T_c^0)/T_c^0 = -0.93 aρ^{1/3}

, where T_c^0 is the transition temperature of the corresponding ideal Bose gas, a is the scattering length, and

ρ

is the particle number density.

{T}_{0}

the ideal gas result. The first-order coefficient

{c}_{1}

depends on nonperturbative physics. In this paper, we show that the coefficient ${c}_{2}^{\ensuremath{'}}$ may be computed perturbatively. We also show that the remaining second-order coefficient ${c}_{2}^{\ensuremath{''}}$ depends on non-perturbative physics but may be related, by a perturbative calculation, to quantities that have previously been measured using lattice simulations of three-dimensional

O(2)

scalar field theory. Making use of those simulation results, we find ${T}_{\mathrm{c}}\ensuremath{\simeq}{T}_{0}{1+(1.32\ifmmode\pm\else\textpm\fi{}0.02){\mathrm{an}}^{1/3}+[19.7518\mathrm{ln}{(an}^{1/3})+(75.7\ifmmode\pm\else\textpm\fi{}0.4)]{a}^{2}{n}^{2/3}{+O(a}^{3}n)}.$