arXiv:cond-mat/0210162 · Modern Physics Letters B

Five-Loop Critical Temperature Shift in Weakly Interacting Homogeneous Bose–Einstein Condensate

Hagen M. Kleinert

Published Aug 20, 2003• 51 citations

Abstract

Using variational perturbation theory, we calculate the shift in the critical temperature Tcup to five loops to lowest order in the scattering length a and find [Formula: see text], where n is the particle density. Our result is slightly lower than the latest Monte Carlo result (1.32±0.02) an1/3.

DOI: 10.1142/s0217984903006074

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arXiv:cond-mat/0210162 · Modern Physics Letters B

Five-Loop Critical Temperature Shift in Weakly Interacting Homogeneous Bose–Einstein Condensate

Hagen M. Kleinert

Published Aug 20, 2003• 51 citations

arXiv HTML PDF

Abstract

DOI: 10.1142/s0217984903006074

Related papers

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Boris Kastening

arXiv:cond-mat/0303486Physical Review ADec 4, 200345 citations

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Boris Kastening

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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 ${T}_{c}^{0}$ is the critical temperature for an ideal gas, $a$ is the $s$ -wave scattering length, and $n$ 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 ${c}_{1}$ 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 $a{n}^{1∕3}$ 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.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 ${T}_{c}^{0}$ is the critical temperature for an ideal gas, $a$ is the $s$-wave scattering length, and $n$ 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 ${c}_{1}$ 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 $a{n}^{1∕3}$ 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.

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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

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.

{T}_{c}^{0}

is the critical temperature for an ideal gas,

a

is the

s

-wave scattering length, and

n

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

{c}_{1}

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

a{n}^{1∕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.