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arXiv:cond-mat/0309060 · Physical Review A

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

Boris Kastening

Published Apr 21, 2004• 94 citations
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Abstract

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.

DOI
10.1103/physreva.69.043613

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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 \ensuremathΔ⟨\ensuremathϕ2⟩∕Nu\ensuremath{\Delta}⟨{\ensuremath{\phi}}^{2}⟩∕Nu\ensuremathΔ⟨\ensuremathϕ2⟩∕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 \ensuremathΔTc∕T0=(1.27\ifmmode±\else\textpm\fi0.11)an1∕3\ensuremath{\Delta}{T}_{c}∕{T}_{0}=(1.27\ifmmode\pm\else\textpm\fi{}0.11)a{n}^{1∕3}\ensuremathΔTc​∕T0​=(1.27\ifmmode±\else\textpm\fi0.11)an1∕3. We use seven-loop perturbative coefficients, extending earlier work by one loop order.
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.
Using a finite-temperature path integral Monte Carlo simulation (PIMC) method and finite-size scaling, we have investigated the interaction-induced shift of the phase-transition temperature for Bose-Einstein condensation of homogeneous weakly interacting Bose gases in three dimensions, which is given by a proposed analytical expression ${T}_{c}={T}_{c}^{0}{1+{c}_{1}a{n}^{1∕3}+[{c}_{2}^{\ensuremath{'}}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}(a{n}^{1∕3})+{c}_{2}^{\ensuremath{''}}]{a}^{2}{n}^{2∕3}+\mathrm{O}({a}^{3}n)}$, where Tc0{T}_{c}^{0}Tc0​ is the critical temperature for an ideal gas, aaa is the sss-wave scattering length, and nnn is the number density. We have used smaller number densities and more time slices than in the previous PIMC simulations [Gruter et al., Phys. Rev. Lett. 79, 3549 (1997)] in order to understand the difference in the value of the coefficient c1{c}_{1}c1​ between their results and the (apparently) other reliable results in the literature. Our results show that ${({T}_{c}\ensuremath{-}{T}_{c}^{0})∕{T}_{c}^{0}}∕(a{n}^{1∕3})$ depends strongly on the interaction strength an1∕3a{n}^{1∕3}an1∕3 while the previous PIMC results are considerably flatter and smaller than our results. We obtain ${c}_{1}=1.32\ifmmode\pm\else\textpm\fi{}0.14$, in agreement with results from recent Monte Carlo methods of three-dimensional O(2) scalar ${\ensuremath{\phi}}^{4}$ field theory and variational perturbation theory.
For an O(N)\mathrm{O}(N)O(N)-symmetric scalar field theory with Euclidean action $\ensuremath{\int}{d}^{3}\phantom{\rule{0.3em}{0ex}}x[\frac{1}{2}\ensuremath{\mid}\ensuremath{\nabla}\ensuremath{\phi}{\ensuremath{\mid}}^{2}+\frac{1}{2}r{\ensuremath{\phi}}^{2}+\frac{1}{4!}u{\ensuremath{\phi}}^{4}]$, where $\ensuremath{\phi}=({\ensuremath{\phi}}_{1},\dots{},{\ensuremath{\phi}}_{N})$ is a vector of NNN real-field components, variational perturbation theory through seven loops is employed for N=0,1,2,3,4N=0,1,2,3,4N=0,1,2,3,4 to compute the renormalized value of r∕(N+2)u2r∕(N+2){u}^{2}r∕(N+2)u2 at the phase transition. Its exact large-NNN limit is determined as well. We also extend an earlier computation of the interaction-induced shift $\ensuremath{\Delta}⟨{\ensuremath{\phi}}^{2}⟩∕Nu$ from N=1,2,4N=1,2,4N=1,2,4 to N=0,3N=0,3N=0,3. For N=2N=2N=2, the results for the two quantities are used to compute the second-order shift of the condensation temperature of a dilute Bose gas, both in the homogenous case and for the wide limit of a harmonic trap. Our results are in agreement with earlier Monte Carlo simulations for N=1,2,4N=1,2,4N=1,2,4. The Appendix contains previously unpublished numerical seven-loop data.
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.