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arXiv:cond-mat/0305138 · Reviews of Modern Physics

Theory of the weakly interacting Bose gas

Jens O. Andersen

Published Jul 1, 2004• 506 citations
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Abstract

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 $T=0.$ The perturbative expansion of these properties is essentially an expansion in the gas parameter $\sqrt{{\mathrm{na}}^{3}},$ where $a$ is the $s$-wave scattering length and $n$ 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 $1/N$ techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.

DOI
10.1103/revmodphys.76.599

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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.
In the framework of the O(N)\mathrm{O}(N)O(N) three-dimensional effective scalar field model for homogeneous dilute weakly interacting Bose gases we use the 1∕N1∕N1∕N expansion, within the large-NNN limit, to evaluate the parameter rc{r}_{c}rc​ which is directly related to the critical chemical potential ${\ensuremath{\mu}}_{c}$. This quantity enters the order-a2n2∕3{a}^{2}{n}^{2∕3}a2n2∕3 coefficient contributing to the critical temperature shift $\ensuremath{\Delta}{T}_{c}$ where aaa represents the sss-wave scattering length and nnn represents the density. Compared to the recent precise numerical lattice simulation results, our calculation suggests that the large-NNN approximation performs rather well even for the physical case N=2N=2N=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-NNN and large-NNN cases.
The low-density expansions for the energy, chemical potential, and condensate depletion of the homogeneous dilute dipolar Bose gas are obtained by regularizing the dipole-dipole interaction at long distances. It is shown that the leading term, proportional to the density, allows a simple physical interpretation and consistently describes the thermodynamic stability of the system. The long-range asymptotics are obtained analytically for the normal and anomalous one-particle correlation functions and the pair distribution function. We discuss the properties of the two-body scattering with zero relative momentum for the dipole-dipole interaction; in particular, we derive the asymptotics of the wave function and a correction to the scattering length for small values of the dipolar range. We show how the density expansions can be derived within the Bogoliubov model of weakly interacting particles without any divergence from the assumption of universality of the expansions at low densities.
The low-density expansion for a homogeneous interacting Bose gas at zero temperature can be formulated as an expansion in powers of $\sqrt{\ensuremath{\rho}{a}^{3}}$, where \ensuremath{\rho} is the number density and a is the S-wave scattering length. Logarithms of \ensuremath{\rho}a3{\mathrm{a}}^{3}a3 appear in the coefficients of the expansion. We show that these logarithms are determined by the renormalization properties of the effective field theory that describes the scattering of atoms at zero density. The leading logarithm is determined by the renormalization of the pointlike 3\ensuremath{\rightarrow}3 scattering amplitude.
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.
Abstract Using the time-dependent Hartree–Fock–Bogoliubov approach, where the condensate is coupled with the thermal cloud and the anomalous density, we study the equilibrium and the dynamical properties of three-dimensional quantum-degenerate Bose gas at finite temperature. Effects of the anomalous correlations on the condensed fraction and the critical temperature are discussed. In uniform Bose gas, useful expressions for the Bogoliubov excitations spectrum, the first and second sound, the condensate depletion and the superfluid fraction are derived. Our results are tested by comparing the findings computed by quantum Monte Carlo simulations. We present also a systematic investigation of the collective modes of a Bose condensate confined in an external trap. Our predictions are in qualitative agreement with previous experimental and theoretical results. We show in particular that our theory is capable of explaining the so-called anomalous behavior of the m = 0 mode.
We use the nonperturbative linear $\ensuremath{\delta}$ expansion method to evaluate analytically the coefficients c1{c}_{1}c1​ 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 T0{T}_{0}T0​ 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 c1{c}_{1}c1​ to order ${\ensuremath{\delta}}^{2}$ with the result c1=3.06.{c}_{1}=3.06.c1​=3.06. Here, we push the calculation to the next two orders obtaining c1=2.45{c}_{1}=2.45c1​=2.45 at order ${\ensuremath{\delta}}^{3}$ and c1=1.48{c}_{1}=1.48c1​=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/N1/N1/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 ${c}_{2}^{\ensuremath{''}}=75.7\ifmmode\pm\else\textpm\fi{}0.4.$