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arXiv:0709.1293

Weakly interacting Bose gases below the critical temperature

Kovalchuk, E., R. Kobes

Published Sep 10, 2007• 0 citations
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Abstract

We consider a homogeneous non-ideal Bose gas at nonzero temperature in equilibrium below the critical temperature $T_C$ in the framework of finite temperature field theory. An algorithm is described in which a manageable subset of diagrams can be considered which, coupled with a self--consistent condition related to the minimization of the free energy, can be used to obtain physically reasonable results.

DOI
10.48550/arxiv.0709.1293

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We consider a homogeneous non-ideal Bose gas at nonzero temperature in equilibrium below the critical temperature TCT_CTC​ in the framework of finite temperature field theory. An algorithm is described in which a manageable subset of diagrams can be considered which, coupled with a self--consistent condition related to the minimization of the free energy, can be used to obtain physically reasonable results.
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.
We consider the nonrelativistic interacting Bose gas at zero and finite temperature. Using the 1/N-expansion, we derive expressions for the free energy density and the number density to next-to-leading order in 1/N. Outside the critical region and at weak coupling, our calculations reduce to the well-known mean-field results for the dilute Bose gas. We also rederive the nonperturbative critical density for Bose condensation which was first calculated by Baym, Blaizot, Holzmann, Lalöe, and Vautherin.
Final experimental achievement of Bose-Binstein condensation of trapped al- kali atoms boosted a renewed interest in the properties of weakly interacting Bose gases.The low-temperature behavior of such systems has been established years ago and became a standard subject of numerous textbooks.However, attempts to consistently extend the treatment to finite-temperature domain are still rare.In order to bridge this gap, we developed an approach, which allows one to estimate the physical parameters of weakly interacting Bose gases from first principles.We show that our method captures all essential physics in the whole range of tem- peratures except of those close to the critical.The source of the latter drawback turns out to be common to all schemes which rely on perturbative expansions.The issue can supposedly be dealt with by the use of the alternative framework, based on the n-particle irreducible effective action functional techniques.This approach attracts a lot of attention nowadays due to its unique resummation properties and its ability to deal with the non-equilibrium problems consistently.However, it is known to violate the gauge invariance when applied to high energy physics problems.We resolve the question by constructing a new modificcl effcctive theory, which not only preserves all the merits of the base formalism, but is also free from the gauge-dependence issues, thus making the entire approach internally consistent.Breakdown of perturbation expansions 63 Renormalization group approach 63 1.10.5 Calculations of the critical temperature [ 63 ot Jt, 38
We discuss certain specific features of the calculation of the critical temperature of a dilute repulsive Bose gas. Interactions modify the critical temperature in two different ways. First, for gases in traps, temperature shifts are introduced by a change of the density profile, arising itself from a modification of the equation of state of the gas (reduced compressibility); these shifts can be calculated simply within mean field theory. Second, even in the absence of a trapping potential (homogeneous gas in a box), temperature shifts are introduced by the interactions; they arise from the correlations introduced in the gas, and thus lie inherently beyond mean field theory – in fact, their evaluation requires more elaborate, non-perturbative, calculations. One illustration of this non-perturbative character is provided by the solution of self-consistent equations, which relate together non-linearly the various energy shifts of the single particle levels k . These equations predict that repulsive interactions shift the critical temperature (at constant density) by an amount which is positive, and simply proportional to the scattering length a ; nevertheless, the numerical coefficient is difficult to compute. Physically, the increase of the temperature can be interpreted in terms of the reduced density fluctuations introduced by the repulsive interactions, which facilitate the propagation of large exchange cycles across the sample.
A lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density and temperature T. In the dilute regime, i.e., when a 3 ≪ 1, where a denotes the scattering length of the pair-interaction potential, our bound differs to leading order from the expression for non-interacting particles by the term 4πa(2 2 −[−c] 2 +). Here, c(T) denotes the critical density for Bose-Einstein condensation (for the non-interacting gas), and [ ·]+ = max { · , 0} denotes the positive part. Our bound is uniform in the temperature up to temperatures of the order of the critical temperature, i.e., T ∼ 2/3 or smaller. One of the key ingredients in the proof is the use of coherent states to extend the method introduced in [17] for estimating correlations to temperatures below the critical one. 1 Introduction and Main Result The advance of experimental techniques for studying ultra-cold atomic gases