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Physical Review Letters

Critical Temperature Shift in Weakly Interacting Bose Gas

V. А. Kashurnikov, Nikolay V. Prokof'ev, Boris V. Svistunov

Published Aug 28, 2001• 223 citations
Publisher

Abstract

With a high-performance Monte Carlo algorithm we study the interaction-induced shift of the critical point in weakly interacting three-dimensional $|\ensuremath{\psi}{|}^{4}$ theory (which includes quantum Bose gas). In terms of critical density, ${n}_{c}$, mass, $m$, interaction, $U$, and temperature, $T$, this shift is universal: $\ensuremath{\Delta}{n}_{c}(T){\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}Cm}^{3}{T}^{2}U$, the constant $C$ found to be equal to $0.0140\ifmmode\pm\else\textpm\fi{}0.0005$. For quantum Bose gas with the scattering length $a$ this implies $\ensuremath{\Delta}{T}_{c}{/T}_{c}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}C}_{0}{\mathrm{an}}^{1/3}$, with ${C}_{0}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.29\ifmmode\pm\else\textpm\fi{}0.05$.

DOI
10.1103/physrevlett.87.120402

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With a high-performance Monte Carlo algorithm we study the interaction-induced shift of the critical point in weakly interacting three-dimensional ∣ψ∣4|ψ|^4∣ψ∣4-theory (which includes quantum Bose gas). In terms of critical density, ncn_cnc​, mass, mmm, interaction, UUU, and temperature, TTT, this shift is universal: Δnc(T)=−Cm3T2UΔn_c(T) = - C m^3 T^2 UΔnc​(T)=−Cm3T2U, the constant CCC found to be equal to 0.0140±0.00050.0140 \pm 0.00050.0140±0.0005. For quantum Bose gas with the scattering length aaa this implies ΔTc/Tc=C0an1/3ΔT_c/T_c = C_0 a n^{1/3}ΔTc​/Tc​=C0​an1/3, with C0=1.29±0.05C_0=1.29 \pm 0.05C0​=1.29±0.05.
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.
We study the Berezinskii-Kosterlitz-Thouless transition in a weakly interacting 2D quantum Bose gas using the concept of universality and numerical simulations of the classical $|\ensuremath{\psi}{|}^{4}$ model on a lattice. The critical density and chemical potential are given by relations ${n}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(mT/2\ensuremath{\pi}{\ensuremath{\Elzxh}}^{2})\mathrm{ln}(\ensuremath{\xi}{\ensuremath{\Elzxh}}^{2}/mU)$ and ${\ensuremath{\mu}}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(mTU/\ensuremath{\pi}{\ensuremath{\Elzxh}}^{2})\mathrm{ln}({\ensuremath{\xi}}_{\ensuremath{\mu}}{\ensuremath{\Elzxh}}^{2}/mU)$, where TTT is the temperature, mmm is the mass, and UUU 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.
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.
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.