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arXiv:cond-mat/9607188 · Physical Review A

Renormalization group theory of the three-dimensional dilute Bose gas

M. Bijlsma, Henk T. C. Stoof

Published Dec 1, 1996• 127 citations
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Abstract

We study the three-dimensional atomic Bose gas using renormalization group techniques. Using our knowledge of the microscopic details of the interatomic interaction, we determine the correct initial values of our renormalization group equations and thus obtain also information on nonuniversal properties. As a result, we can predict, for instance, the critical temperature of the gas and the superfluid and condensate density of the Bose-Einstein condensed phase in the regime na${\mathrm{\ensuremath{\Lambda}}}_{\mathit{th}}^{2}$\ensuremath{\ll}1, where the average interaction energy is small compared to the average kinetic energy. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.5085

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This article reviews recent advances in the theory of the three-dimensional dilute homogeneous Bose gas at zero and finite temperature. Effective-field-theory methods are used to formulate a systematic perturbative framework that can be used to calculate the properties of a system at T=0.T=0.T=0. The perturbative expansion of these properties is essentially an expansion in the gas parameter na3,\sqrt{{\mathrm{na}}^{3}},na3​, where aaa is the sss-wave scattering length and nnn is the number density. In particular, the leading quantum corrections to the ground-state energy density, the condensate depletion, and long-wavelength collective excitations are rederived in an efficient and economical manner. Nonuniversal effects are also discussed. These effects are higher-order corrections that depend on properties of the interatomic potential other than the scattering length, such as the effective range. The article critically examines various approaches to the dilute Bose gas in equilibrium at finite temperature. These include the Bogoliubov approximation, the Popov approximation, the Hartree-Fock-Bogoliubov approximation, the \ensuremath{\Phi}-derivable approach, optimized perturbation theory, and renormalization-group techniques. The article ends with a look at recent calculations of the critical temperature of the dilute Bose gas, which include 1/N1/N1/N techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.
Due to the observation of Bose-Einstein condensation in dilute atomic 87^{87}87Rb and 23^{23}23Na vapors last year, there is currently a great interest in the properties of a degenerate Bose gas. One aspect that is not yet understood is the effect of the interatomic interactions on the critical temperature of the phase transition. We present here a renormalization group study of the weakly-interacting Bose gas and predict the critical temperature at which Bose-Einstein condensation occurs as a function of the (positive) scattering length aaa of the interatomic potential.
We investigate Bose-Einstein condensation for ultracold bosonic atoms in two-dimensional systems. The functional renormalization group for the average action allows us to follow the effective interactions from molecular scales (microphysics) to the characteristic extension of the probe lll (macrophysics). In two dimensions the scale dependence of the dimensionless interaction strength $\ensuremath{\lambda}$ is logarithmic. Furthermore, for large lll the frequency dependence of the inverse propagator becomes quadratic. We find an upper bound for $\ensuremath{\lambda}$, and for large $\ensuremath{\lambda}$ substantial deviations from the Bogoliubov results for the condensate depletion, the dispersion relation and the sound velocity. The melting of the condensate above the critical temperature Tc{T}_{c}Tc​ is associated to a phase transition of the Kosterlitz-Thouless type. The critical temperature in units of the density Tc∕n{T}_{c}∕nTc​∕n vanishes for $l\ensuremath{\rightarrow}\ensuremath{\infty}$ logarithmically.
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.
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 mmm and the sss-wave scattering length ad{a}_{d}ad​ of the bosons are known (ddd is the space dimension). Here $\ensuremath{\mu}$ and TTT 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 F2{\mathcal{F}}_{2}F2​ 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.
We study the d -dimensional Bose gas at finite temperature using the renormalization-group method. The flow-equations and the free energy are obtained for dimension d , and the cases d < 2 and d = 2 have been analysed in the limit of low and high temperatures. The critical temperature, the coherence length and the specific heat of a two-dimensional Bose gas are obtained using a regular solution for the coupling constant which is obtained from the scaling equations. The relevance of the dilute Bose gas for recent experimental results is also discussed.