Abstract: The one-loop effective potential for non-relativistic bosons with a delta function repulsive potential is calculated for a given chemical potential using functional methods. After renormalization and at zero temperature it reproduces the standard ground state energy and pressure as function of the particle density. At finite temperatures it is found necessary to include ring corrections to the one-loop result in order to satisfy the Goldstone theorem. It is natural to introduce an effective chemical potential directly related to the order parameter and which uniformly decreases with increasing temperatures. This is in contrast to the the ordinary chemical potential which peaks at the critical temperature. The resulting thermodynamics in the condensed phase at very low temperatures is found to be the same as in the Bogoliubov approximation where the degrees of freedom are given by the Goldstone bosons. At higher temperatures the ring corrections dominate and result in a critical temperature unaffected by the interaction. 1
Abstract: The one-loop effective potential for non-relativistic bosons with a delta function repulsive potential is calculated for a given chemical potential using functional methods. After renormalization and at zero temperature it reproduces the standard ground state energy and pressure as function of the particle density. At finite temperatures it is found necessary to include ring corrections to the one-loop result in order to satisfy the Goldstone theorem. It is natural to introduce an effective chemical potential directly related to the order parameter and which uniformly decreases with increasing temperatures. This is in contrast to the the ordinary chemical potential which peaks at the critical temperature. The resulting thermodynamics in the condensed phase at very low temperatures is found to be the same as in the Bogoliubov approximation where the degrees of freedom are given by the Goldstone bosons. At higher temperatures the ring corrections dominate and result in a critical temperature unaffected by the interaction. 1
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.
We show that the chemical potential of a one-dimensional (1D) interacting Bose gas exhibits a nonmonotonic temperature dependence which is peculiar of superfluids. The effect is a direct consequence of the phononic nature of the excitation spectrum at large wavelengths exhibited by 1D Bose gases. For low temperatures T, we demonstrate that the coefficient in T2 expansion of the chemical potential is entirely defined by the zero-temperature density dependence of the sound velocity. We calculate that coefficient along the crossover between the Bogoliubov weakly interacting gas and the Tonks-Girardeau gas of impenetrable bosons. Analytic expansions are provided in the asymptotic regimes. The theoretical predictions along the crossover are confirmed by comparison with the exactly solvable Yang-Yang model in which the finite-temperature equation of state is obtained numerically by solving Bethe-ansatz equations. A 1D ring geometry is equivalent to imposing periodic boundary conditions and arising finite-size effects are studied in detail. At T=0 we calculated various thermodynamic functions, including the inelastic structure factor, as a function of the number of atoms, pointing out the occurrence of important deviations from the thermodynamic limit.
We derive the equation of state of a Bose gas with contact interactions using relativistic quantum field theory. The calculation accounts for both thermal and quantum corrections up to one-loop order. We work in the Hartree-Fock-Bogoliubov approximation and follow Yukalov's prescription of introducing two chemical potentials, one for the condensed phase and another one for the excited phase, to circumvent the well-known Hohenberg-Martin dilemma. As a check on the formalism, we take the nonrelativistic limit and reproduce the known results. Finally, we translate our results to the hydrodynamical, two-fluid model for finite-temperature superfluids. Our results are relevant for the phenomenology of Bose-Einstein condensate and superfluid dark matter candidates, as well as the color-flavor locking phase of quark matter in neutron stars.
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.
Thermodynamical properties of an interacting boson system at finite temperatures and zero chemical potential are studied within the framework of the Skyrme-like mean-field toy model. It is assumed that the mean field contains both attractive and repulsive terms. Self-consistency relations between the mean field and thermodynamic functions are derived. It is shown that, for sufficiently strong attractive interactions, this system develops a first-order phase transition via the formation of a Bose condensate. An interesting prediction of the model is that the condensed phase is characterized by a constant total density of particles. It is shown that the energy density exhibits a jump at the critical temperature.