arXiv:cond-mat/0103228 · Physical Review Letters

BEC Transition Temperature of a Dilute Homogeneous Imperfect Bose Gas

Peter B. Arnold, Guy D. Moore

Published Aug 28, 2001• 197 citations

Abstract

The leading-order effect of interactions on a homogeneous Bose gas is theoretically predicted to shift the critical temperature by an amount $\ensuremath{\Delta}{T}_{c}\ensuremath{\simeq}{\mathrm{ca}}_{\mathrm{sc}}{n}^{1/3}{T}_{0}$ from the ideal gas result ${T}_{0}$, where ${a}_{\mathrm{sc}}$ is the scattering length, $n$ is the density, and $c$ is a pure number. There have been several different theoretical estimates for $c$. We claim to settle the issue by measuring the numerical coefficient in a lattice simulation of O(2) ${\ensuremath{\varphi}}^{4}$ field theory in three dimensions---an effective theory which, as observed previously in the literature, can be systematically matched to the dilute Bose gas problem to reproduce nonuniversal quantities such as the critical temperature. We find $c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.32\ifmmode\pm\else\textpm\fi{}0.02$.

DOI: 10.1103/physrevlett.87.120401

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Exact 1∕N and optimized perturbative evaluation of μc for homogeneous interacting Bose gases

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Bose-Einstein condensation temperature of a homogeneous weakly interacting Bose gas: Path integral Monte Carlo study

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Transition temperature of the homogeneous imperfect Bose gas

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arXiv:cond-mat/0103228 · Physical Review Letters

BEC Transition Temperature of a Dilute Homogeneous Imperfect Bose Gas

Peter B. Arnold, Guy D. Moore

Published Aug 28, 2001• 197 citations

arXiv HTML PDF

Abstract

The leading-order effect of interactions on a homogeneous Bose gas is theoretically predicted to shift the critical temperature by an amount

\ensuremath{\Delta}{T}_{c}\ensuremath{\simeq}{\mathrm{ca}}_{\mathrm{sc}}{n}^{1/3}{T}_{0}

from the ideal gas result

{T}_{0}

, where

{a}_{\mathrm{sc}}

is the scattering length,

n

is the density, and

c

is a pure number. There have been several different theoretical estimates for

c

. We claim to settle the issue by measuring the numerical coefficient in a lattice simulation of O(2)

{\ensuremath{\varphi}}^{4}

field theory in three dimensions---an effective theory which, as observed previously in the literature, can be systematically matched to the dilute Bose gas problem to reproduce nonuniversal quantities such as the critical temperature. We find

c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.32\ifmmode\pm\else\textpm\fi{}0.02

DOI: 10.1103/physrevlett.87.120401

Related papers

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

Higher-order evaluation of the critical temperature for interacting homogeneous dilute Bose gases

F. Cruz, Marcus Benghi Pinto, Rudnei O. Ramos +1 more

We use the nonperturbative linear $\ensuremath{\delta}$ expansion method to evaluate analytically the coefficients ${c}_{1}$ and ${c}_{2}^{\ensuremath{''}}$ that appear in the expansion for the transition temperature for a dilute, homogeneous, three-dimensional Bose gas given by ${T}_{c}{=T}_{0}({1+c}_{1}{\mathrm{an}}^{1/3}+[{c}_{2}^{\ensuremath{'}}\mathrm{ln}{(an}^{1/3}{)+c}_{2}^{\ensuremath{''}}]{a}^{2}{n}^{2/3}{+O(a}^{3}n)),$ where ${T}_{0}$ is the result for an ideal gas, a is the s-wave scattering length, and n is the number density. In a previous work the same method has been used to evaluate ${c}_{1}$ to order ${\ensuremath{\delta}}^{2}$ with the result ${c}_{1}=3.06.$ Here, we push the calculation to the next two orders obtaining ${c}_{1}=2.45$ at order ${\ensuremath{\delta}}^{3}$ and ${c}_{1}=1.48$ at order ${\ensuremath{\delta}}^{4}.$ Analyzing the topology of the graphs involved we discuss how our results relate to other nonperturbative analytical methods such as the self-consistent resummation and the $1/N$ approximations. At the same orders we obtain ${c}_{2}^{\ensuremath{''}}=101.4,$ ${c}_{2}^{\ensuremath{''}}=98.2,$ and ${c}_{2}^{\ensuremath{''}}=82.9.$ Our analytical results seem to support the recent Monte Carlo estimates ${c}_{1}=1.32\ifmmode\pm\else\textpm\fi{}0.02$ and ${c}_{2}^{\ensuremath{''}}=75.7\ifmmode\pm\else\textpm\fi{}0.4.$We use the nonperturbative linear $\ensuremath{\delta}$ expansion method to evaluate analytically the coefficients ${c}_{1}$ and ${c}_{2}^{\ensuremath{''}}$ that appear in the expansion for the transition temperature for a dilute, homogeneous, three-dimensional Bose gas given by ${T}_{c}{=T}_{0}({1+c}_{1}{\mathrm{an}}^{1/3}+[{c}_{2}^{\ensuremath{'}}\mathrm{ln}{(an}^{1/3}{)+c}_{2}^{\ensuremath{''}}]{a}^{2}{n}^{2/3}{+O(a}^{3}n)),$ where ${T}_{0}$ is the result for an ideal gas, a is the s-wave scattering length, and n is the number density. In a previous work the same method has been used to evaluate ${c}_{1}$ to order ${\ensuremath{\delta}}^{2}$ with the result ${c}_{1}=3.06.$ Here, we push the calculation to the next two orders obtaining ${c}_{1}=2.45$ at order ${\ensuremath{\delta}}^{3}$ and ${c}_{1}=1.48$ at order ${\ensuremath{\delta}}^{4}.$ Analyzing the topology of the graphs involved we discuss how our results relate to other nonperturbative analytical methods such as the self-consistent resummation and the $1/N$ approximations. At the same orders we obtain ${c}_{2}^{\ensuremath{''}}=101.4,$ ${c}_{2}^{\ensuremath{''}}=98.2,$ and ${c}_{2}^{\ensuremath{''}}=82.9.$ Our analytical results seem to support the recent Monte Carlo estimates ${c}_{1}=1.32\ifmmode\pm\else\textpm\fi{}0.02$ and ${c}_{2}^{\ensuremath{''}}=75.7\ifmmode\pm\else\textpm\fi{}0.4.$

arXiv:cond-mat/0112306Physical Review AMay 9, 200252 citations

Exact 1∕N and optimized perturbative evaluation of μc for homogeneous interacting Bose gases

Jean-Loı̈c Kneur, Marcus Benghi Pinto

In the framework of the $\mathrm{O}(N)$ three-dimensional effective scalar field model for homogeneous dilute weakly interacting Bose gases we use the $1∕N$ expansion, within the large- $N$ limit, to evaluate the parameter ${r}_{c}$ which is directly related to the critical chemical potential ${\ensuremath{\mu}}_{c}$. This quantity enters the order- ${a}^{2}{n}^{2∕3}$ coefficient contributing to the critical temperature shift $\ensuremath{\Delta}{T}_{c}$ where $a$ represents the $s$ -wave scattering length and $n$ represents the density. Compared to the recent precise numerical lattice simulation results, our calculation suggests that the large- $N$ approximation performs rather well even for the physical case $N=2$ . We then calculate the same quantity but using different forms of the optimized perturbative (variational) method, showing that these produce excellent results for both the finite- $N$ and large- $N$ cases.In the framework of the $\mathrm{O}(N)$ three-dimensional effective scalar field model for homogeneous dilute weakly interacting Bose gases we use the $1∕N$ expansion, within the large-$N$ limit, to evaluate the parameter ${r}_{c}$ which is directly related to the critical chemical potential ${\ensuremath{\mu}}_{c}$. This quantity enters the order-${a}^{2}{n}^{2∕3}$ coefficient contributing to the critical temperature shift $\ensuremath{\Delta}{T}_{c}$ where $a$ represents the $s$-wave scattering length and $n$ represents the density. Compared to the recent precise numerical lattice simulation results, our calculation suggests that the large-$N$ approximation performs rather well even for the physical case $N=2$. We then calculate the same quantity but using different forms of the optimized perturbative (variational) method, showing that these produce excellent results for both the finite-$N$ and large-$N$ cases.

arXiv:cond-mat/0408538Physical Review AMar 9, 200518 citations

Bose-Einstein condensation temperature of a homogeneous weakly interacting Bose gas: Path integral Monte Carlo study

Kwangsik T. Nho, David Paul Landau

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.

arXiv:cond-mat/0404238Physical Review ANov 18, 200456 citations

Transition temperature of the homogeneous imperfect Bose gas

Nguyễn Văn Thụ, Pham Duy Thanh

Beyond standard approaches in existing literature, we explore the relative shift in the transition temperature of a homogeneous dilute Bose gas using the Cornwall-Jackiw-Tomboulis effective action formalism within the improved Hartree-Fock approximation. The first correction to the enhancement of the transition temperature, as compared to that of the ideal Bose gas, is expressed in a universal form: it is linear with respect to the scattering length. The slope of this linear relationship shows excellent agreement with the exact numerical calculations presented in previous studies. Additionally, we identify non-universal terms contributing to the shift.Beyond standard approaches in existing literature, we explore the relative shift in the transition temperature of a homogeneous dilute Bose gas using the Cornwall-Jackiw-Tomboulis effective action formalism within the improved Hartree-Fock approximation. The first correction to the enhancement of the transition temperature, as compared to that of the ideal Bose gas, is expressed in a universal form: it is linear with respect to the scattering length. The slope of this linear relationship shows excellent agreement with the exact numerical calculations presented in previous studies. Additionally, we identify non-universal terms contributing to the shift.

arXiv:2409.14036Sep 21, 20240 citations

Transition temperature for weakly interacting homogeneous Bose gases

F. Cruz, Marcus Benghi Pinto, Rudnei O. Ramos

We apply the nonperturbative optimized linear $\ensuremath{\delta}$ expansion method to the $O(N)$ scalar field model in three dimensions to determine the transition temperature of a dilute homogeneous Bose gas. Our results show that the shift of the transition temperature $\ensuremath{\Delta}{T}_{c}{/T}_{c}$ of the interacting model, compared with the ideal-gas transition temperature, really behaves as $\ensuremath{\gamma}{\mathrm{an}}^{1/3}$ where a is the s-wave scattering length and n is the number density. For $N=2$ our calculations yield the value $\ensuremath{\gamma}=3.059.$We apply the nonperturbative optimized linear $\ensuremath{\delta}$ expansion method to the $O(N)$ scalar field model in three dimensions to determine the transition temperature of a dilute homogeneous Bose gas. Our results show that the shift of the transition temperature $\ensuremath{\Delta}{T}_{c}{/T}_{c}$ of the interacting model, compared with the ideal-gas transition temperature, really behaves as $\ensuremath{\gamma}{\mathrm{an}}^{1/3}$ where a is the s-wave scattering length and n is the number density. For $N=2$ our calculations yield the value $\ensuremath{\gamma}=3.059.$

arXiv:cond-mat/0007151Physical review. B, Condensed matterJun 14, 200163 citations

The leading-order effect of interactions on a homogeneous Bose gas is theoretically predicted to shift the critical temperature by an amount

\ensuremath{\Delta}{T}_{c}\ensuremath{\simeq}{\mathrm{ca}}_{\mathrm{sc}}{n}^{1/3}{T}_{0}

from the ideal gas result

{T}_{0}

, where

{a}_{\mathrm{sc}}

is the scattering length,

n

is the density, and

c

is a pure number. There have been several different theoretical estimates for

c

. We claim to settle the issue by measuring the numerical coefficient in a lattice simulation of O(2)

{\ensuremath{\varphi}}^{4}

c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.32\ifmmode\pm\else\textpm\fi{}0.02

{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 ${T}_{\mathrm{c}}\ensuremath{\simeq}{T}_{0}{1+(1.32\ifmmode\pm\else\textpm\fi{}0.02){\mathrm{an}}^{1/3}+[19.7518\mathrm{ln}{(an}^{1/3})+(75.7\ifmmode\pm\else\textpm\fi{}0.4)]{a}^{2}{n}^{2/3}{+O(a}^{3}n)}.$

We use the nonperturbative linear $\ensuremath{\delta}$ expansion method to evaluate analytically the coefficients

{c}_{1}

and ${c}_{2}^{\ensuremath{''}}$ that appear in the expansion for the transition temperature for a dilute, homogeneous, three-dimensional Bose gas given by ${T}_{c}{=T}_{0}({1+c}_{1}{\mathrm{an}}^{1/3}+[{c}_{2}^{\ensuremath{'}}\mathrm{ln}{(an}^{1/3}{)+c}_{2}^{\ensuremath{''}}]{a}^{2}{n}^{2/3}{+O(a}^{3}n)),$ where

{T}_{0}

is the result for an ideal gas, a is the s-wave scattering length, and n is the number density. In a previous work the same method has been used to evaluate

{c}_{1}

to order ${\ensuremath{\delta}}^{2}$ with the result

{c}_{1}=3.06.

Here, we push the calculation to the next two orders obtaining

{c}_{1}=2.45

at order ${\ensuremath{\delta}}^{3}$ and

{c}_{1}=1.48

at order ${\ensuremath{\delta}}^{4}.$ Analyzing the topology of the graphs involved we discuss how our results relate to other nonperturbative analytical methods such as the self-consistent resummation and the

1/N

approximations. At the same orders we obtain ${c}_{2}^{\ensuremath{''}}=101.4,$ ${c}_{2}^{\ensuremath{''}}=98.2,$ and ${c}_{2}^{\ensuremath{''}}=82.9.$ Our analytical results seem to support the recent Monte Carlo estimates ${c}_{1}=1.32\ifmmode\pm\else\textpm\fi{}0.02$ and ${c}_{2}^{\ensuremath{''}}=75.7\ifmmode\pm\else\textpm\fi{}0.4.$

In the framework of the

\mathrm{O}(N)

three-dimensional effective scalar field model for homogeneous dilute weakly interacting Bose gases we use the

1∕N

expansion, within the large-

N

limit, to evaluate the parameter

{r}_{c}

which is directly related to the critical chemical potential ${\ensuremath{\mu}}_{c}$. This quantity enters the order-

{a}^{2}{n}^{2∕3}

coefficient contributing to the critical temperature shift $\ensuremath{\Delta}{T}_{c}$ where

a

represents the

s

-wave scattering length and

n

represents the density. Compared to the recent precise numerical lattice simulation results, our calculation suggests that the large-

N

approximation performs rather well even for the physical case

N=2

. We then calculate the same quantity but using different forms of the optimized perturbative (variational) method, showing that these produce excellent results for both the finite-

N

and large-

N

cases.

{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.

We apply the nonperturbative optimized linear $\ensuremath{\delta}$ expansion method to the

O(N)

scalar field model in three dimensions to determine the transition temperature of a dilute homogeneous Bose gas. Our results show that the shift of the transition temperature $\ensuremath{\Delta}{T}_{c}{/T}_{c}$ of the interacting model, compared with the ideal-gas transition temperature, really behaves as $\ensuremath{\gamma}{\mathrm{an}}^{1/3}$ where a is the s-wave scattering length and n is the number density. For

N=2

our calculations yield the value $\ensuremath{\gamma}=3.059.$