Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9910013 · Physical Review Letters

Density Dependence of the Transition Temperature in a Homogeneous Bose-Einstein Condensate

J. D. Reppy, Benjamin Colburn Crooker, Bernard HÉBRAL, A. D. Corwin, Jizhong He, G. M. Zassenhaus

Published Mar 6, 2000• 67 citations
arXivHTMLPDF

Abstract

Transition temperature data obtained as a function of particle density in the ${}^{4}\mathrm{He}$-Vycor system are compared with recent theoretical calculations for 3D Bose-condensed systems. In the low density dilute Bose gas regime we find, in agreement with theory, a positive shift in the transition temperature of the form $\ensuremath{\Delta}T/{T}_{0}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\gamma}({\mathrm{na}}^{3}{)}^{1/3}$. At higher densities a maximum is found in the ratio of ${T}_{c}/{T}_{0}$ for a value of the interaction parameter, ${\mathrm{na}}^{3}$, that is in agreement with path-integral Monte Carlo calculations.

DOI
10.1103/physrevlett.84.2060

Related papers

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.

arXiv:cond-mat/0404238•Physical Review A•Nov 18, 2004•56 citations

Critical Temperature of Bose-Einstein Condensation of Hard-Sphere Gases

P. Grüter, David M. Ceperley, Franck Laloë

We determine the critical temperature of a 3D homogeneous system of hard-sphere bosons by path-integral Monte Carlo simulations and finite-size scaling. In the low density limit, we find that the critical temperature is increased by the repulsive interactions, as $\ensuremath{\Delta}{T}_{C}{/T}_{0}\ensuremath{\sim}({\mathrm{na}}^{3}{)}^{\ensuremath{\gamma}}$, where $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.34\ifmmode\pm\else\textpm\fi{}0.03$. At high densities the result for liquid helium, namely, a lower critical temperature than in the noninteracting case, is recovered. We give a microscopic explanation for the observed behavior.

arXiv:cond-mat/9707028•Physical Review Letters•Nov 10, 1997•171 citations

Zero-density-limit extrapolation of the superfluid transition temperature in a unitary atomic Fermi gas on a lattice

Qijin Chen

The superfluid transition temperature ${T}_{c}$ of a unitary Fermi gas on a three-dimensional isotropic lattice with an attractive on-site interaction is investigated as a function of density $n$, from half filling down to $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ per unit cell, using a pairing fluctuation theory. We show that except at very low densities (${n}^{1/3}<0.2$), where ${T}_{c}/{E}_{F}$ is linear in ${n}^{1/3}$, ${T}_{c}/{E}_{F}$ exhibits a significant higher-order nonlinear dependence on ${n}^{1/3}$. This calls for extra caution against possible significant error in the zero-density-limit value of ${T}_{c}/{E}_{F}$ from typical quantum Monte Carlo (QMC) simulations, obtained by linearly extrapolating data points at intermediate and high densities. Our result, ${T}_{c}/{E}_{F}=0.256$, at the $n=0$ limit, is close to, and should be compared with, the maximum ${T}_{c}/{E}_{F}$ near the Bose-Einstein condensation regime obtained from QMC calculations.

arXiv:1109.5327•Physical Review A•Aug 9, 2012•8 citations

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

Boris Kastening

We compute the shift of the transition temperature for a homogeneous weakly interacting Bose gas in leading order in the scattering length a for given particle density n. Using variational perturbation theory through six loops in a classical three-dimensional scalar field theory, we obtain $\ensuremath{\Delta}{T}_{c}{/T}_{c}=1.25\ifmmode\pm\else\textpm\fi{}{0.13an}^{1/3},$ in agreement with recent Monte Carlo results.

arXiv:cond-mat/0303486•Physical Review A•Dec 4, 2003•45 citations

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.

arXiv:0801.3424•Physical Review Letters•Apr 9, 2008•75 citations

Thermodynamics of a dilute Bose gas: A path-integral Monte Carlo study

G. Spada, Sebastiano Pilati, S. Giorgini

We present precise path-integral Monte Carlo results for the thermodynamics of a homogeneous dilute Bose gas. Pressure and energy are calculated as a function of temperature both below and above the Bose-Einstein transition. Specifically, we address interaction effects, focusing on deviations from the ideal-gas law in the thermodynamic limit. We also calculate the isothermal compressibility and the contact parameter, which provide a clear signature of the role played by interactions. In particular, we obtain indications of a discontinuity of the compressibility at the transition point. To gain physical insight, numerical results are systematically compared with the predictions of first-order Hartree-Fock and second-order Popov theories, both giving an approximate description of the gas thermodynamics. The comparison shows the extension of the critical region around the transition point, where the inaccuracies of the perturbative expansions are more pronounced.

arXiv:2110.00283•Physical Review A•Jan 31, 2022•11 citations
Transition temperature data obtained as a function of particle density in the 4He{}^{4}\mathrm{He}4He-Vycor system are compared with recent theoretical calculations for 3D Bose-condensed systems. In the low density dilute Bose gas regime we find, in agreement with theory, a positive shift in the transition temperature of the form \ensuremathΔT/T0=\ensuremathγ(na3)1/3\ensuremath{\Delta}T/{T}_{0}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\gamma}({\mathrm{na}}^{3}{)}^{1/3}\ensuremathΔT/T0​=\ensuremathγ(na3)1/3. At higher densities a maximum is found in the ratio of Tc/T0{T}_{c}/{T}_{0}Tc​/T0​ for a value of the interaction parameter, na3{\mathrm{na}}^{3}na3, that is in agreement with path-integral Monte Carlo calculations.
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 determine the critical temperature of a 3D homogeneous system of hard-sphere bosons by path-integral Monte Carlo simulations and finite-size scaling. In the low density limit, we find that the critical temperature is increased by the repulsive interactions, as $\ensuremath{\Delta}{T}_{C}{/T}_{0}\ensuremath{\sim}({\mathrm{na}}^{3}{)}^{\ensuremath{\gamma}}$, where $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.34\ifmmode\pm\else\textpm\fi{}0.03$. At high densities the result for liquid helium, namely, a lower critical temperature than in the noninteracting case, is recovered. We give a microscopic explanation for the observed behavior.
The superfluid transition temperature Tc{T}_{c}Tc​ of a unitary Fermi gas on a three-dimensional isotropic lattice with an attractive on-site interaction is investigated as a function of density nnn, from half filling down to $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ per unit cell, using a pairing fluctuation theory. We show that except at very low densities (n1/3<0.2{n}^{1/3}<0.2n1/3<0.2), where Tc/EF{T}_{c}/{E}_{F}Tc​/EF​ is linear in n1/3{n}^{1/3}n1/3, Tc/EF{T}_{c}/{E}_{F}Tc​/EF​ exhibits a significant higher-order nonlinear dependence on n1/3{n}^{1/3}n1/3. This calls for extra caution against possible significant error in the zero-density-limit value of Tc/EF{T}_{c}/{E}_{F}Tc​/EF​ from typical quantum Monte Carlo (QMC) simulations, obtained by linearly extrapolating data points at intermediate and high densities. Our result, Tc/EF=0.256{T}_{c}/{E}_{F}=0.256Tc​/EF​=0.256, at the n=0n=0n=0 limit, is close to, and should be compared with, the maximum Tc/EF{T}_{c}/{E}_{F}Tc​/EF​ near the Bose-Einstein condensation regime obtained from QMC calculations.
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 present precise path-integral Monte Carlo results for the thermodynamics of a homogeneous dilute Bose gas. Pressure and energy are calculated as a function of temperature both below and above the Bose-Einstein transition. Specifically, we address interaction effects, focusing on deviations from the ideal-gas law in the thermodynamic limit. We also calculate the isothermal compressibility and the contact parameter, which provide a clear signature of the role played by interactions. In particular, we obtain indications of a discontinuity of the compressibility at the transition point. To gain physical insight, numerical results are systematically compared with the predictions of first-order Hartree-Fock and second-order Popov theories, both giving an approximate description of the gas thermodynamics. The comparison shows the extension of the critical region around the transition point, where the inaccuracies of the perturbative expansions are more pronounced.