We compute thermodynamic properties of dilute Bose gases using non-perturbative approximations of the two-particle irreducible (2PI) effective action. It is shown how to systematically renormalise the self-consistent descriptions beyond conventional Gaussian approximations such as Hartree-Fock-Bogoliubov theory. This allows us to determine the condensate depletion from low to high temperatures, including its critical behaviour at the phase transition. While the universal anomalous dimension at criticality is vanishing for Gaussian approximations, we determine its non-zero value at next-to-leading order of a self-consistent expansion in the number of field components.
We compute thermodynamic properties of dilute Bose gases using non-perturbative approximations of the two-particle irreducible (2PI) effective action. It is shown how to systematically renormalise the self-consistent descriptions beyond conventional Gaussian approximations such as Hartree-Fock-Bogoliubov theory. This allows us to determine the condensate depletion from low to high temperatures, including its critical behaviour at the phase transition. While the universal anomalous dimension at criticality is vanishing for Gaussian approximations, we determine its non-zero value at next-to-leading order of a self-consistent expansion in the number of field components.
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 na3, 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.
In this work, we build upon the theoretical framework developed in our previous study [N. V. Thu and D. T. Pham, Phys. Lett. A 523, 129787 (2024)], where the Cornwall–Jackiw–Tomboulis effective action formalism was applied to a dilute Bose gas, to investigate its thermodynamic properties in the condensed phase at sufficiently low temperatures within the improved Hartree–Fock approximation. We focus on the analyses of the chemical potential, pressure, energy density and specific heat at constant volume, which are derived within a unified, self-consistent and gapless framework. The results obtained show excellent agreement with those predicted by the Hartree–Fock–Bogoliubov theory in the appropriate limits. Remarkably, our results demonstrate universal behavior for the reduced thermodynamic quantities, and a comparison with available experimental data for the chemical potential is made.
We investigate Bose-Einstein condensation for interacting bosons at zero and nonzero temperature. Functional renormalization provides us with a consistent method to compute the effect of fluctuations beyond the Bogoliubov approximation. For three-dimensional dilute gases, we find an upper bound on the scattering length a which is of the order of the microphysical scale---typically the range of the van der Waals interaction. In contrast to fermions near the unitary bound, no strong interactions occur for bosons with approximately pointlike interactions, thus explaining the high quantitative reliability of perturbation theory for most quantities. For zero temperature we compute the quantum phase diagram for bosonic quasiparticles with a general dispersion relation, corresponding to an inverse microphysical propagator with terms linear and quadratic in the frequency. We compute the temperature dependence of the condensate and particle density n, and find for the critical temperature Tc a deviation from the free theory, $\ensuremath{\Delta}{T}_{c}/{T}_{c}=2.1a{n}^{1/3}$. For the sound velocity at zero temperature we find very good agreement with the Bogoliubov result, such that it may be used to determine the particle density accurately.
In this paper we develop a gapless theory of Bose-Einstein condensation (BEC) which can be applied to both trapped and homogeneous gases at zero and finite temperature. The starting point for the theory is the second quantized, many-body Hamiltonian for a system of structureless bosons with pairwise interactions. A number-conserving approach is used to rewrite this Hamiltonian in a form which is approximately quadratic with higher-order cubic and quartic terms. The quadratic part of the Hamiltonian can be diagonalized exactly by transforming to a quasiparticle basis, while requiring that the condensate satisfy the Gross-Pitaevskii equation. The non-quadratic terms are assumed to have a small effect and are dealt with using first- and second-order perturbation theory. The conventional treatment of these terms, based on factorization approximations, is shown to be inconsistent. Infrared divergences can appear in individual terms of the perturbation expansion, but we show analytically that the total contribution beyond quadratic order is finite. The resulting excitation spectrum is gapless and the energy shifts are small for a dilute gas away from the critical region, justifying the use of perturbation theory. Ultraviolet divergences can appear if a contact potential is used to describe particle interactions. We show that the use of this potential as an approximation to the two-body T -matrix leads naturally to a high-energy renormalization. The theory developed in this paper is therefore well defined at both low and high energy and provides a systematic description of BEC in dilute gases. It can therefore be used to calculate the energies and decay rates of the excitations of the system at temperatures approaching the phase transition.
We derive and discuss the equations of motion for the condensate and its fluctuations for a dilute, weakly interacting Bose gas in an external potential within the self-consistent Hartree-Fock-Bogoliubov (HFB) approximation. Account is taken of the depletion of the condensate and the anomalous Bose correlations, which are important at finite temperatures. We give a critical analysis of the self-consistent HFB approximation in terms of the Hohenberg-Martin classification of approximations (conserving vs gapless) and point out that the Popov approximation to the full HFB gives a gapless single-particle spectrum at all temperatures. The Beliaev second-order approximation is discussed as the spectrum generated by functional differentiation of the HFB single-particle Green's function. We emphasize that the problem of determining the excitation spectrum of a Bose-condensed gas (homogeneous or inhomogeneous) is difficult because of the need to satisfy several different constraints.
In this thesis we develop a gapless theory of BEC which can be applied to both trapped and homogeneous gases at zero and finite temperature. The starting point for the theory is the second quantized, many-body Hamiltonian for a system of structureless bosons with pairwise interactions. A number conserving approach is used to rewrite this Hamiltonian in a form which is approximately quadratic with higher order cubic and quartic terms. The quadratic part of the Hamiltonian can be diagonalized exactly by transforming to a quasiparticle basis, while requiring that the condensate satisfy the Gross-Pitaevskii equation. The non-quadratic terms are assumed to have a small effect and are dealt with using first and second order perturbation theory. The conventional treatment of these terms, based on factorization approximations, is shown to be inconsistent. Infra-red divergences can appear in individual terms of the perturbation expansion, but we show analytically that the total contribution beyond quadratic order is finite. The resulting excitation spectrum is gapless and the energy shifts are small for a dilute gas away from the critical region, justifying the use of perturbation theory. Ultra-violet divergences can appear if a contact potential is used to describe particle interactions. We show that the use of this potential as an approximation to the two-body T-matrix leads naturally to a high-energy renormalization. The theory developed in this thesis is therefore well-defined at both low and high energy and provides a systematic description of Bose-Einstein condensation in dilute gases. It can be used to calculate the energies and decay rates of the excitations of the system at temperatures approaching the phase transition. We present analytical results for these quantities in the homogeneous limit and describe how they can be calculated numerically for trapped gases. (author)