The effective action formalism of quantum field theory is used to study the properties of the non-relativistic interacting Bose gas. The Gaussian approximation is formulated by calculating the effective action to the first order of the optimized expansion. In the homogeneous limit the method respects the Hughenholz-Pines theorem, leading to the gapless spectrum both for excitations and for density fluctuations. Renormalization is carried out by adopting dimensional regularization. The results for critical temperature are compared with that obtained in the loop expansion.
The effective action formalism of quantum field theory is used to study the properties of the non-relativistic interacting Bose gas. The Gaussian approximation is formulated by calculating the effective action to the first order of the optimized expansion. In the homogeneous limit the method respects the Hughenholz-Pines theorem, leading to the gapless spectrum both for excitations and for density fluctuations. Renormalization is carried out by adopting dimensional regularization. The results for critical temperature are compared with that obtained in the loop expansion.
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.
A simple pedagogical introduction to the effective action method of quantum field theory is given at a level suitable for beginning postgraduate students. It is shown how to obtain the effective potential at zero temperature from a regularized zero-point energy. The results are applicable to curved as well as to flat space. The generalization to finite temperatures is also given. It is shown how to obtain high temperature expansions of the thermodynamic potential for the neutral free Bose gas and the charged Bose gas in both the relativistic and nonrelativistic limits. The results are obtained for an arbitrary spatial dimension and in curved space. Results are also obtained for the self-interacting relativistic gas in three spatial dimensions. A detailed discussion of how the formalism may be applied to study Bose-Einstein condensation is given. The interpretation of Bose-Einstein condensation as symmetry breaking is discussed. Application is given to the study of charged bosons, both relativistic and nonrelativistic, in a constant magnetic field. The Meissner effect is obtained for the nonrelativistic gas in three spatial dimensions. The final application is to the study of nonrelativistic bosons in a harmonic oscillator confing potential trap. A number of analytical approaches to this are discussed.
We present a detailed derivation of the quantum and quantum–thermal effective action for non-relativistic systems, starting from the single-particle case and extending to the Gross–Pitaevskii (GP) field theory for weakly interacting bosons. In the single-particle framework, we introduce the one-particle-irreducible (1PI) effective action formalism, taking explicitly into account the choice of the initial quantum state, its saddle-point plus Gaussian-fluctuation approximation, and its finite-temperature extension via Matsubara summation, yielding a clear physical interpretation in terms of zero-point and thermal contributions to the Helmholtz free energy. The formalism is then applied to the GP action, producing the 1PI effective potential at zero and finite temperature, including beyond-mean-field Lee–Huang–Yang and thermal corrections. We discuss the gapless and gapped Bogoliubov spectra, their relevance to equilibrium and non-equilibrium regimes, and the role of regularization. Applications include the inclusion of an external potential within the local density approximation, the derivation of finite-temperature Josephson equations, and the extension to D-dimensional systems, with particular attention to the zero-dimensional limit. This unified approach provides a transparent connection between microscopic quantum fluctuations and effective macroscopic equations of motion for Bose–Einstein condensates.
We consider a massive relativistic Bose gas with N complex scalars at finite density. At zero temperature, we calculate the pressure, charge density and the speed of sound in the one-loop approximation. In the nonrelativistic limit, we obtain the classic results for the dilute Bose gas. We also discuss finite-temperature effects. In particular, we consider the problem of calculating the critical temperature for Bose-Einstein condensation. Dimensional reduction and effective-field-theory methods are used to perturbatively calculate the effects of the nonstatic Matsubara modes. Calculations of Tc in the effective 3d theory require nonperturbative methods. Using the Monte Carlo simulations of X. Sun [Phys. Rev. {\bf E67}, 066702 (2003)] and the seven-loop variational perturbation theory (VPT) calculations of B. Kastening [Phys. Rev. {\bf A70}, 043621 (2004)], we obtain Tc for N=2 to second order in the interaction.
The effective-action formalism is applied to a gas of bosons. The equations describing the condensate and the excitations are obtained using the loop expansion for the effective action. For a homogeneous gas, the expansion in terms of the diluteness parameter is identified in terms of the loop expansion. The loop expansion and the limits of validity of the well-known Bogoliubov [J. Phys. (Moscow) 11, 23 (1947)] and Popov, (Zh. \'Eksp. Teor. Fiz. 47, 1759 (1964) [Sov. Phys. JETP 20, 1185 (1965)]) equations are examined analytically for a homogeneous dilute Bose gas and numerically for a gas trapped in a harmonic-oscillator potential. The expansion to one-loop order, and hence the Bogoliubov equation, is shown to be valid for the zero-temperature trapped gas as long as the characteristic length of the trapping potential exceeds the s-wave scattering length.
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.