Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0507391 · Physical Review E

Expansion of the Gibbs potential for quantum many-body systems: General formalism with applications to the spin glass and the weakly nonideal Bose gas

T. Plefka

Published Jan 24, 2006• 11 citations
arXivHTMLPDF

Abstract

For general quantum systems the power expansion of the Gibbs potential and consequently the power expansion of the self-energy is derived in terms of the interaction strength. Employing a generalization of the projector technique, a compact representation of the general terms of the expansion results. The general aspects of the approach are discussed with special emphasis on the effects characteristic for quantum systems. The expansion is systematic and leads directly to contributions beyond the mean field of all thermodynamic quantities. These features are explicitly demonstrated and illustrated for two nontrivial systems, the infinite-range quantum spin glass and the weakly interacting Bose gas. The Onsager terms of both systems are calculated, which represent the first beyond-mean-field contributions. For the spin glass Thouless-Anderson-Palmer-like equations are presented and discussed in the paramagnetic region. The investigation of the Bose gas leads to a beyond-mean-field thermodynamic description. At the Bose-Einstein condensation temperature complete agreement is found with the results presented recently by alternative techniques.

DOI
10.1103/physreve.73.016129

Related papers

Quantum corrections to the thermodynamic potential of trapped bosons

Subhasis Sinha

We calculate the quantum corrections of the thermodynamic quantities of a system of confined bosons at finite temperature. Systematically quantum corrections are written in a series of $\ensuremath{\Elzxh},$ which is convergent when $\mathrm{kT}$ is much larger than the spacing between energy levels of the system. We apply this method to the different physical systems. First we calculate the thermodynamic potential of spin-1 bosons confined in a general power-law potential, and calculate the paramagnetic susceptibility of the gas including the quantum corrections. Local-density approximation for charged bosons in the weak magnetic field is improved by the semiclassical expansion. Diamagnetic susceptibility of a system of confined charged bosons is calculated using this method. Finally we apply this method to calculate analytically the thermodynamic potential of a weakly interacting Bose-gas confined in at three-dimensional harmonic-oscillator potential. For a large number of particles, quantum corrections become small, and contribute to the finite-size corrections to scaling.

arXiv:cond-mat/9801162•Physical Review A•Oct 1, 1998•6 citations

Perturbation theory of spin-1 Bose-Einstein condensates

Shingo Ohtsuka, Susumu Kurihara

We generalize the perturbation theory of weakly interacting bosons to Bose-Einstein condensates with hyperfine spin $F=1.$ Analytical expressions of second-order self-energies beyond the mean-field approximation are derived at zero temperature. We further extend the perturbation theory to the finite-temperature case, and derive the corresponding self-energies. We evaluate these self-energies near the poles of the first-order (mean-field) Green's functions within the on-shell approximation, and expand the results in powers of wave number in the long-wavelength limit. Excitation spectra for each mode are obtained analytically at low temperatures. We also derive the ground-state energy and thermodynamic quantities such as free energy of the system.

Physical Review A•Jul 7, 2003•6 citations

1/N-Expansion and the Dilute Bose Gas Beyond Mean-field Theory

Jens O. Andersen

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.

arXiv:cond-mat/0608265•Aug 11, 2006•0 citations

MEAN-FIELD EXPANSION IN BOSE–EINSTEIN CONDENSATES WITH FINITE-RANGE INTERACTIONS

RALF SCHÜTZHOLD, MICHAEL UHLMANN, Yan Xu +1 more

We present a formal derivation of the mean-field expansion for dilute Bose–Einstein condensates with two-particle interaction potentials which are weak and finite-range, but otherwise arbitrary. The expansion allows for a controlled investigation of the impact of microscopic interaction details (e.g. the scaling behavior) on the mean-field approach and the induced higher-order corrections beyond the s-wave scattering approximation.

arXiv:cond-mat/0505074•International Journal of Modern Physics B•Sep 30, 2006•130 citations

Theory of the weakly interacting Bose gas

Jens O. Andersen

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.

arXiv:cond-mat/0305138•Reviews of Modern Physics•Jul 1, 2004•506 citations

Effective Action Approach to Quantum and Thermal Effects: From One Particle to Bose–Einstein Condensates

Luca Salasnich

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.

arXiv:2511.18020•Atoms•Dec 1, 2025•1 citation
For general quantum systems the power expansion of the Gibbs potential and consequently the power expansion of the self-energy is derived in terms of the interaction strength. Employing a generalization of the projector technique, a compact representation of the general terms of the expansion results. The general aspects of the approach are discussed with special emphasis on the effects characteristic for quantum systems. The expansion is systematic and leads directly to contributions beyond the mean field of all thermodynamic quantities. These features are explicitly demonstrated and illustrated for two nontrivial systems, the infinite-range quantum spin glass and the weakly interacting Bose gas. The Onsager terms of both systems are calculated, which represent the first beyond-mean-field contributions. For the spin glass Thouless-Anderson-Palmer-like equations are presented and discussed in the paramagnetic region. The investigation of the Bose gas leads to a beyond-mean-field thermodynamic description. At the Bose-Einstein condensation temperature complete agreement is found with the results presented recently by alternative techniques.
We calculate the quantum corrections of the thermodynamic quantities of a system of confined bosons at finite temperature. Systematically quantum corrections are written in a series of $\ensuremath{\Elzxh},$ which is convergent when kT\mathrm{kT}kT is much larger than the spacing between energy levels of the system. We apply this method to the different physical systems. First we calculate the thermodynamic potential of spin-1 bosons confined in a general power-law potential, and calculate the paramagnetic susceptibility of the gas including the quantum corrections. Local-density approximation for charged bosons in the weak magnetic field is improved by the semiclassical expansion. Diamagnetic susceptibility of a system of confined charged bosons is calculated using this method. Finally we apply this method to calculate analytically the thermodynamic potential of a weakly interacting Bose-gas confined in at three-dimensional harmonic-oscillator potential. For a large number of particles, quantum corrections become small, and contribute to the finite-size corrections to scaling.
We generalize the perturbation theory of weakly interacting bosons to Bose-Einstein condensates with hyperfine spin F=1.F=1.F=1. Analytical expressions of second-order self-energies beyond the mean-field approximation are derived at zero temperature. We further extend the perturbation theory to the finite-temperature case, and derive the corresponding self-energies. We evaluate these self-energies near the poles of the first-order (mean-field) Green's functions within the on-shell approximation, and expand the results in powers of wave number in the long-wavelength limit. Excitation spectra for each mode are obtained analytically at low temperatures. We also derive the ground-state energy and thermodynamic quantities such as free energy of the system.
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.
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 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.