arXiv:cond-mat/0103149

Critical Temperature Shift in Weakly Interacting Bose Gas

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

Published Mar 6, 2001• 1 citation

Abstract

With a high-performance Monte Carlo algorithm we study the interaction-induced shift of the critical point in weakly interacting three-dimensional $|ψ|^4$-theory (which includes quantum Bose gas). In terms of critical density, $n_c$, mass, $m$, interaction, $U$, and temperature, $T$, this shift is universal: $Δn_c(T) = - C m^3 T^2 U$, the constant $C$ found to be equal to $0.0140 \pm 0.0005$. For quantum Bose gas with the scattering length $a$ this implies $ΔT_c/T_c = C_0 a n^{1/3}$, with $C_0=1.29 \pm 0.05$.

DOI: 10.48550/arxiv.cond-mat/0103149

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arXiv:cond-mat/0103149

Critical Temperature Shift in Weakly Interacting Bose Gas

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

Published Mar 6, 2001• 1 citation

arXiv HTML PDF

Abstract

With a high-performance Monte Carlo algorithm we study the interaction-induced shift of the critical point in weakly interacting three-dimensional

|ψ|^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:

Δn_c(T) = - C m^3 T^2 U

, the constant

C

found to be equal to

0.0140 \pm 0.0005

. For quantum Bose gas with the scattering length

a

this implies

ΔT_c/T_c = C_0 a n^{1/3}

, with

C_0=1.29 \pm 0.05

DOI: 10.48550/arxiv.cond-mat/0103149

Related papers

Critical Temperature Shift in Weakly Interacting Bose Gas

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

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 ${C}_{0}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.29\ifmmode\pm\else\textpm\fi{}0.05$.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 ${C}_{0}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.29\ifmmode\pm\else\textpm\fi{}0.05$.

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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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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.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.

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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.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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Critical Temperature of Interacting Bose Gases in Two and Three Dimensions

Sebastiano Pilati, S. Giorgini, Nikolay V. Prokof'ev

We calculate the superfluid transition temperature of homogeneous interacting Bose gases in three and two spatial dimensions using large-scale path integral Monte Carlo simulations (with up to N=10;{5} particles). In 3D we investigate the limits of the universal critical behavior in terms of the scattering length alone by using different models for the interatomic potential. We find that this type of universality sets in at small values of the gas parameter na3 < or approximately 10(-4). This value is different from the estimate na3 < or approximately 10(-6) for the validity of the asymptotic expansion in the limit of vanishing na3. In 2D we study the Berezinskii-Kosterlitz-Thouless transition of a gas with hard-core interactions. For this system we find good agreement with the classical lattice |psi|4 model up to very large densities. We also explain the origin of the existing discrepancy between previous studies of the same problem.We calculate the superfluid transition temperature of homogeneous interacting Bose gases in three and two spatial dimensions using large-scale path integral Monte Carlo simulations (with up to N=10;{5} particles). In 3D we investigate the limits of the universal critical behavior in terms of the scattering length alone by using different models for the interatomic potential. We find that this type of universality sets in at small values of the gas parameter na3 < or approximately 10(-4). This value is different from the estimate na3 < or approximately 10(-6) for the validity of the asymptotic expansion in the limit of vanishing na3. In 2D we study the Berezinskii-Kosterlitz-Thouless transition of a gas with hard-core interactions. For this system we find good agreement with the classical lattice |psi|4 model up to very large densities. We also explain the origin of the existing discrepancy between previous studies of the same problem.

arXiv:0801.3424Physical Review LettersApr 9, 200875 citations

With a high-performance Monte Carlo algorithm we study the interaction-induced shift of the critical point in weakly interacting three-dimensional

|ψ|^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:

Δn_c(T) = - C m^3 T^2 U

, the constant

C

found to be equal to

0.0140 \pm 0.0005

. For quantum Bose gas with the scattering length

a

this implies

ΔT_c/T_c = C_0 a n^{1/3}

, with

C_0=1.29 \pm 0.05

{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 ${C}_{0}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.29\ifmmode\pm\else\textpm\fi{}0.05$.

{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.

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.