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

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

Published Jan 1, 1999• 0 citations
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Abstract

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.

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

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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.

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Bose-Einstein condensation temperature of a homogenous weakly interacting Bose gas in variational perturbation theory through seven loops

Boris Kastening

The shift of the Bose-Einstein condensation temperature for a homogenous weakly interacting Bose gas in leading order in the scattering length $a$ is computed for given particle density $n$. Variational perturbation theory is used to resum the corresponding perturbative series for $\ensuremath{\Delta}⟨{\ensuremath{\phi}}^{2}⟩∕Nu$ in a classical three-dimensional scalar field theory with coupling $u$ and where the physical case of $N=2$ field components is generalized to arbitrary $N$. Our results for $N=1,2,4$ are in agreement with recent Monte Carlo simulations; for $N=2$, we obtain $\ensuremath{\Delta}{T}_{c}∕{T}_{0}=(1.27\ifmmode\pm\else\textpm\fi{}0.11)a{n}^{1∕3}$. We use seven-loop perturbative coefficients, extending earlier work by one loop order.

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Convergence of the linear δ expansion for the shift in Tc for Bose-Einstein condensation

Eric Braaten, Eugeniu Radescu

The leading correction from interactions to the transition temperature ${T}_{c}$ for Bose-Einstein condensation can be obtained from a nonperturbative calculation in the critical $\mathrm{O}(N)$ scalar field theory in three dimensions with $N=2.$ We show that the linear $\ensuremath{\delta}$ expansion can be applied to this problem in such a way that in the large-N limit it converges to the exact analytic result. If the principle of minimal sensitivity is used to optimize the convergence rate, the errors seem to decrease exponentially with the order in the $\ensuremath{\delta}$ expansion. For $N=2,$ we calculate the shift in ${T}_{c}$ to fourth order in $\ensuremath{\delta}.$ The results are consistent with slow convergence to the results of recent lattice Monte Carlo calculations.

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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 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.
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.
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.
The leading correction from interactions to the transition temperature Tc for Bose-Einstein condensation can be obtained from a nonperturbative calculation in the critical O( N) scalar field theory in 3 dimensions with N = 2. We show that the linear δ expansion can be applied to this problem in such a way that in the large- N limit it converges to the exact analytic result. If the principal of minimal sensitivity is used to optimize the convergence rate, the errors seem to decrease exponentially with the order in the δ expansion. For N = 2, we calculate the shift in Tc to fifth order in δ. The results are consistent with slow convergence to the results of recent lattice Monte Carlo calculations. The convergence can be accelerated by using a resummation method proposed by Kleinert et al. called variational perturbation theory.
The shift of the Bose-Einstein condensation temperature for a homogenous weakly interacting Bose gas in leading order in the scattering length aaa is computed for given particle density nnn. Variational perturbation theory is used to resum the corresponding perturbative series for $\ensuremath{\Delta}⟨{\ensuremath{\phi}}^{2}⟩∕Nu$ in a classical three-dimensional scalar field theory with coupling uuu and where the physical case of N=2N=2N=2 field components is generalized to arbitrary NNN. Our results for N=1,2,4N=1,2,4N=1,2,4 are in agreement with recent Monte Carlo simulations; for N=2N=2N=2, we obtain $\ensuremath{\Delta}{T}_{c}∕{T}_{0}=(1.27\ifmmode\pm\else\textpm\fi{}0.11)a{n}^{1∕3}$. We use seven-loop perturbative coefficients, extending earlier work by one loop order.
The leading correction from interactions to the transition temperature Tc{T}_{c}Tc​ for Bose-Einstein condensation can be obtained from a nonperturbative calculation in the critical O(N)\mathrm{O}(N)O(N) scalar field theory in three dimensions with N=2.N=2.N=2. We show that the linear $\ensuremath{\delta}$ expansion can be applied to this problem in such a way that in the large-N limit it converges to the exact analytic result. If the principle of minimal sensitivity is used to optimize the convergence rate, the errors seem to decrease exponentially with the order in the $\ensuremath{\delta}$ expansion. For N=2,N=2,N=2, we calculate the shift in Tc{T}_{c}Tc​ to fourth order in $\ensuremath{\delta}.$ The results are consistent with slow convergence to the results of recent lattice Monte Carlo calculations.