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:1203.1788 · Physical Review A

Universal thermodynamics of a two-dimensional Bose gas

Adam Rançon, N. Dupuis

Published Jun 5, 2012• 55 citations
arXivHTMLPDF

Abstract

Using renormalization-group arguments we show that the low-temperature thermodynamics of a three- or two-dimensional dilute Bose gas is fully determined by a universal scaling function ${\mathcal{F}}_{d}(\ensuremath{\mu}/{k}_{B}T,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T))$ once the mass $m$ and the $s$-wave scattering length ${a}_{d}$ of the bosons are known ($d$ is the space dimension). Here $\ensuremath{\mu}$ and $T$ denote the chemical potential and temperature of the gas, and the temperature-dependent dimensionless interaction constant $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T)$ is a function of $m{a}_{d}^{2}{k}_{B}T/{\ensuremath{\hbar}}^{2}$. We compute the scaling function ${\mathcal{F}}_{2}$ using a nonperturbative renormalization-group approach and find that both the $\ensuremath{\mu}/{k}_{B}T$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T)$ dependencies are in very good agreement with recent experimental data obtained for a quasi-two-dimensional Bose gas with or without optical lattice. We also show that the nonperturbative renormalization-group estimate of the Berezinskii-Kosterlitz-Thouless transition temperature compares well with the result obtained from a quantum Monte Carlo simulation of an effective classical field theory.

DOI
10.1103/physreva.85.063607

Related papers

Critical Point of a Weakly Interacting Two-Dimensional Bose Gas

Nikolay V. Prokof'ev, Oliver Ruebenacker, Boris V. Svistunov

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 $T$ is the temperature, $m$ is the mass, and $U$ 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.

arXiv:cond-mat/0106075•Physical Review Letters•Dec 11, 2001•262 citations

Thermodynamics and renormalized quasiparticles in the vicinity of the dilute Bose gas quantum critical point in two dimensions

Jan Krieg, Dominik Straßel, Simon Streib +2 more

We use the functional renormalization group (FRG) to derive analytical expressions for thermodynamic observables (density, pressure, entropy, and compressibility) as well as for single-particle properties (wave-function renormalization and effective mass) of interacting bosons in two dimensions as a function of temperature $T$ and chemical potential $\ensuremath{\mu}$. We focus on the quantum disordered and the quantum critical regime close to the dilute Bose gas quantum critical point. Our approach is based on a truncated vertex expansion of the hierarchy of FRG flow equations and the decoupling of the two-body contact interaction in the particle-particle channel using a suitable Hubbard-Stratonovich transformation. Our analytic FRG results extend previous analytical renormalization-group calculations for thermodynamic observables at $\ensuremath{\mu}=0$ to finite values of $\ensuremath{\mu}$. To confirm the validity of our FRG approach, we have also performed quantum Monte Carlo simulations to obtain the magnetization, susceptibility, and correlation length of the two-dimensional spin-$1/2$ quantum $XY$ model with coupling $J$ in a regime where its quantum critical behavior is controlled by the dilute Bose gas quantum critical point. We find that our analytical results describe the Monte Carlo data for $\ensuremath{\mu}\ensuremath{\le}0$ rather accurately up to relatively high temperatures $T\ensuremath{\lesssim}0.1J$.

arXiv:1607.01015•Physical review. B./Physical review. B•Jan 13, 2017•3 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 Bose gas near the superfluid–Mott-insulator transition

Adam Rançon, N. Dupuis

We study the thermodynamics near the generic (density-driven) superfluid--Mott-insulator transition in the three-dimensional Bose-Hubbard model using the nonperturbative renormalization-group approach. At low energy, the physics is controlled by the Gaussian fixed point and becomes universal. Thermodynamic quantities can then be expressed in terms of the universal scaling functions of the dilute Bose gas universality class while the microscopic physics enters only via two nonuniversal parameters, namely, the effective mass ${m}^{*}$ and the ``scattering length'' ${a}^{*}$ of the elementary excitations at the quantum critical point between the superfluid and Mott-insulating phases. A notable exception is the condensate density in the superfluid phase which is proportional to the quasiparticle weight ${Z}_{\mathrm{qp}}$ of the elementary excitations. The universal regime is defined by ${m}^{*}{a}^{*}{}^{2}T\ensuremath{\ll}1$ and ${m}^{*}{a}^{*}{}^{2}|\ensuremath{\delta}\ensuremath{\mu}|\ensuremath{\ll}1$ or, equivalently, $|\overline{n}\ensuremath{-}{\overline{n}}_{c}|{a}^{*}{}^{3}\ensuremath{\ll}1$, where $\ensuremath{\delta}\ensuremath{\mu}=\ensuremath{\mu}\ensuremath{-}{\ensuremath{\mu}}_{c}$ is the chemical potential shift from the quantum critical point $(\ensuremath{\mu}={\ensuremath{\mu}}_{c},\phantom{\rule{0.28em}{0ex}}T=0)$ and $\overline{n}\ensuremath{-}{\overline{n}}_{c}$ the doping with respect to the commensurate density ${\overline{n}}_{c}$ of the $T=0$ Mott insulator. We compute ${Z}_{\mathrm{qp}}$, ${m}^{*}$, and ${a}^{*}$ and find that they vary strongly with both the ratio $t/U$ between hopping amplitude and onsite repulsion and the value of the (commensurate) density ${\overline{n}}_{c}$. Finally, we discuss the experimental observation of universality and the measurement of ${Z}_{\mathrm{qp}}$, ${m}^{*}$, and ${a}^{*}$ in a cold-atomic gas in an optical lattice.

arXiv:1207.2053•Physical Review A•Oct 24, 2012•23 citations

Universal behavior of two-dimensional bosonic gases at Berezinskii-Kosterlitz-Thouless transitions

Giacomo Ceccarelli, Jacopo Nespolo, Andrea Pelissetto C +1 more

We study the universal critical behavior of two-dimensional (2D) lattice bosonic gases at the Berezinskii-Kosterlitz-Thouless (BKT) transition, which separates the low-temperature superfluid phase from the high-temperature normal phase. For this purpose, we perform quantum Monte Carlo simulations of the hard-core Bose-Hubbard (BH) model at zero chemical potential. We determine the critical temperature by using a matching method that relates finite-size data for the BH model with corresponding data computed in the classical $XY$ model. In this approach, the neglected scaling corrections decay as inverse powers of the lattice size $L$, and not as powers of $1/\mathrm{ln}L$, as in more standard approaches, making the estimate of the critical temperature much more reliable. Then, we consider the BH model in the presence of a trapping harmonic potential, and we verify the universality of the trap-size dependence at the BKT critical point. This issue is relevant for experiments with quasi-2D trapped cold atoms.

arXiv:1306.2510•Physical Review B•Jul 30, 2013•18 citations

Strongly Interacting Two-Dimensional Bose Gases

Li-Chung Ha, Chen-Lung Hung, Xibo Zhang +3 more

We prepare and study strongly interacting two-dimensional Bose gases in the superfluid, the classical Berezinskii-Kosterlitz-Thouless (BKT) transition, and the vacuum-to-superfluid quantum critical regimes. A wide range of the two-body interaction strength 0.05 < g < 3 is covered by tuning the scattering length and by loading the sample into an optical lattice. Based on the equations of state measurements, we extract the coupling constants as well as critical thermodynamic quantities in different regimes. In the superfluid and the BKT transition regimes, the extracted coupling constants show significant down-shifts from the mean-field and perturbation calculations when g approaches or exceeds one. In the BKT and the quantum critical regimes, all measured thermodynamic quantities show logarithmic dependence on the interaction strength, a tendency confirmed by the extended classical-field and renormalization calculations.

arXiv:1211.3456•Physical Review Letters•Apr 2, 2013•76 citations
Using renormalization-group arguments we show that the low-temperature thermodynamics of a three- or two-dimensional dilute Bose gas is fully determined by a universal scaling function Fd(\ensuremathμ/kBT,g\ifmmode~\else˜\fi(T)){\mathcal{F}}_{d}(\ensuremath{\mu}/{k}_{B}T,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T))Fd​(\ensuremathμ/kB​T,g\ifmmode~\else˜\fi​(T)) once the mass mmm and the sss-wave scattering length ad{a}_{d}ad​ of the bosons are known (ddd is the space dimension). Here \ensuremathμ\ensuremath{\mu}\ensuremathμ and TTT denote the chemical potential and temperature of the gas, and the temperature-dependent dimensionless interaction constant g\ifmmode~\else˜\fi(T)\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T)g\ifmmode~\else˜\fi​(T) is a function of mad2kBT/\ensuremathℏ2m{a}_{d}^{2}{k}_{B}T/{\ensuremath{\hbar}}^{2}mad2​kB​T/\ensuremathℏ2. We compute the scaling function F2{\mathcal{F}}_{2}F2​ using a nonperturbative renormalization-group approach and find that both the \ensuremathμ/kBT\ensuremath{\mu}/{k}_{B}T\ensuremathμ/kB​T and g\ifmmode~\else˜\fi(T)\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}(T)g\ifmmode~\else˜\fi​(T) dependencies are in very good agreement with recent experimental data obtained for a quasi-two-dimensional Bose gas with or without optical lattice. We also show that the nonperturbative renormalization-group estimate of the Berezinskii-Kosterlitz-Thouless transition temperature compares well with the result obtained from a quantum Monte Carlo simulation of an effective classical field 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.
We use the functional renormalization group (FRG) to derive analytical expressions for thermodynamic observables (density, pressure, entropy, and compressibility) as well as for single-particle properties (wave-function renormalization and effective mass) of interacting bosons in two dimensions as a function of temperature TTT and chemical potential $\ensuremath{\mu}$. We focus on the quantum disordered and the quantum critical regime close to the dilute Bose gas quantum critical point. Our approach is based on a truncated vertex expansion of the hierarchy of FRG flow equations and the decoupling of the two-body contact interaction in the particle-particle channel using a suitable Hubbard-Stratonovich transformation. Our analytic FRG results extend previous analytical renormalization-group calculations for thermodynamic observables at $\ensuremath{\mu}=0$ to finite values of $\ensuremath{\mu}$. To confirm the validity of our FRG approach, we have also performed quantum Monte Carlo simulations to obtain the magnetization, susceptibility, and correlation length of the two-dimensional spin-1/21/21/2 quantum XYXYXY model with coupling JJJ in a regime where its quantum critical behavior is controlled by the dilute Bose gas quantum critical point. We find that our analytical results describe the Monte Carlo data for $\ensuremath{\mu}\ensuremath{\le}0$ rather accurately up to relatively high temperatures $T\ensuremath{\lesssim}0.1J$.
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 study the thermodynamics near the generic (density-driven) superfluid--Mott-insulator transition in the three-dimensional Bose-Hubbard model using the nonperturbative renormalization-group approach. At low energy, the physics is controlled by the Gaussian fixed point and becomes universal. Thermodynamic quantities can then be expressed in terms of the universal scaling functions of the dilute Bose gas universality class while the microscopic physics enters only via two nonuniversal parameters, namely, the effective mass m∗{m}^{*}m∗ and the ``scattering length'' a∗{a}^{*}a∗ of the elementary excitations at the quantum critical point between the superfluid and Mott-insulating phases. A notable exception is the condensate density in the superfluid phase which is proportional to the quasiparticle weight Zqp{Z}_{\mathrm{qp}}Zqp​ of the elementary excitations. The universal regime is defined by ${m}^{*}{a}^{*}{}^{2}T\ensuremath{\ll}1$ and ${m}^{*}{a}^{*}{}^{2}|\ensuremath{\delta}\ensuremath{\mu}|\ensuremath{\ll}1$ or, equivalently, $|\overline{n}\ensuremath{-}{\overline{n}}_{c}|{a}^{*}{}^{3}\ensuremath{\ll}1$, where $\ensuremath{\delta}\ensuremath{\mu}=\ensuremath{\mu}\ensuremath{-}{\ensuremath{\mu}}_{c}$ is the chemical potential shift from the quantum critical point $(\ensuremath{\mu}={\ensuremath{\mu}}_{c},\phantom{\rule{0.28em}{0ex}}T=0)$ and $\overline{n}\ensuremath{-}{\overline{n}}_{c}$ the doping with respect to the commensurate density n‾c{\overline{n}}_{c}nc​ of the T=0T=0T=0 Mott insulator. We compute Zqp{Z}_{\mathrm{qp}}Zqp​, m∗{m}^{*}m∗, and a∗{a}^{*}a∗ and find that they vary strongly with both the ratio t/Ut/Ut/U between hopping amplitude and onsite repulsion and the value of the (commensurate) density n‾c{\overline{n}}_{c}nc​. Finally, we discuss the experimental observation of universality and the measurement of Zqp{Z}_{\mathrm{qp}}Zqp​, m∗{m}^{*}m∗, and a∗{a}^{*}a∗ in a cold-atomic gas in an optical lattice.
We study the universal critical behavior of two-dimensional (2D) lattice bosonic gases at the Berezinskii-Kosterlitz-Thouless (BKT) transition, which separates the low-temperature superfluid phase from the high-temperature normal phase. For this purpose, we perform quantum Monte Carlo simulations of the hard-core Bose-Hubbard (BH) model at zero chemical potential. We determine the critical temperature by using a matching method that relates finite-size data for the BH model with corresponding data computed in the classical XYXYXY model. In this approach, the neglected scaling corrections decay as inverse powers of the lattice size LLL, and not as powers of 1/lnL1/\mathrm{ln}L1/lnL, as in more standard approaches, making the estimate of the critical temperature much more reliable. Then, we consider the BH model in the presence of a trapping harmonic potential, and we verify the universality of the trap-size dependence at the BKT critical point. This issue is relevant for experiments with quasi-2D trapped cold atoms.
We prepare and study strongly interacting two-dimensional Bose gases in the superfluid, the classical Berezinskii-Kosterlitz-Thouless (BKT) transition, and the vacuum-to-superfluid quantum critical regimes. A wide range of the two-body interaction strength 0.05 < g < 3 is covered by tuning the scattering length and by loading the sample into an optical lattice. Based on the equations of state measurements, we extract the coupling constants as well as critical thermodynamic quantities in different regimes. In the superfluid and the BKT transition regimes, the extracted coupling constants show significant down-shifts from the mean-field and perturbation calculations when g approaches or exceeds one. In the BKT and the quantum critical regimes, all measured thermodynamic quantities show logarithmic dependence on the interaction strength, a tendency confirmed by the extended classical-field and renormalization calculations.