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:1105.4932 · Physical Review A

Auxiliary-field approach to dilute Bose gases with tunable interactions

Fred Cooper, Bogdan Mihaila, John F. Dawson, Chih-Chun Chien, Eddy Timmermans

Published May 20, 2011• 22 citations
arXivHTMLPDF

Abstract

We rewrite the Lagrangian for a dilute Bose gas in terms of auxiliary fields related to the normal and anomalous condensate densities. We derive the loop expansion of the effective action in the composite-field propagators. The lowest-order auxiliary field (LOAF) theory is a conserving mean-field approximation consistent with the Goldstone theorem without some of the difficulties plaguing approximations such as the Hartree and Popov approximations. LOAF predicts a second-order phase transition. We give a set of Feynman rules for improving results to any order in the loop expansion in terms of composite-field propagators. We compare results of the LOAF approximation with those derived using the Popov approximation. LOAF allows us to explore the critical regime for all values of the coupling constant, and we determine various parameters in the unitarity limit.

DOI
10.1103/physreva.83.053622

Related papers

Nonperturbative Predictions for Cold Atom Bose Gases with Tunable Interactions

Fred Cooper, Chih-Chun Chien, Bogdan Mihaila +2 more

We derive a theoretical description for dilute Bose gases as a loop expansion in terms of composite-field propagators by rewriting the Lagrangian in terms of auxiliary fields related to the normal and anomalous densities. We demonstrate that already in leading order this nonperturbative approach describes a large interval of coupling-constant values, satisfies Goldstone's theorem, yields a Bose-Einstein transition that is second order, and is consistent with the critical temperature predicted in the weak-coupling limit by the next-to-leading-order large-N expansion.

arXiv:1011.1886•Physical Review Letters•Dec 7, 2010•32 citations

Analytical limits for cold-atom Bose gases with tunable interactions

Bogdan Mihaila, Fred Cooper, John F. Dawson +2 more

We discuss the equilibrium properties of dilute Bose gases using a nonperturbative formalism based on auxiliary fields related to the normal and anomalous densities. We show analytically that for a dilute Bose gas of weakly interacting particles at zero temperature, the leading-order auxiliary field (LOAF) approximation leads to well-known analytical results. Close to the critical point the LOAF predictions are the same as those obtained using an effective field theory in the large-$N$ approximation. We also report analytical approximations for the LOAF results in the unitarity limit, which compare favorably with our numerical results. LOAF predicts that the equation of state for the Bose gas in the unitarity limit is $E/(pV)=1$, unlike the case of the Fermi gas when $E/(pV)=3/2$.

arXiv:1107.3750•Physical Review A•Aug 2, 2011•10 citations

Auxiliary field formalism for dilute fermionic atom gases with tunable interactions

Bogdan Mihaila, John F. Dawson, Fred Cooper +2 more

We develop the auxiliary field formalism corresponding to a dilute system of spin-1/2 fermions. This theory represents the Fermi counterpart of the Bose-Einstein condensation (BEC) theory developed recently by F. Cooper et al. [Phys. Rev. Lett. 105, 240402 (2010)] to describe a dilute gas of Bose particles. Assuming tunable interactions, this formalism is appropriate for the study of the crossover from the regime of Bardeen-Cooper-Schriffer (BCS) pairing to the regime of BEC in ultracold fermionic atom gases. We show that when applied to the Fermi case at zero temperature, the leading-order auxiliary field (LOAF) approximation gives the same equations as obtained in the standard BCS variational picture. At finite temperature, LOAF leads to the theory discussed by S\'a de Melo, Randeria, and Engelbrecht [Phys. Rev. Lett. 71, 3202 (1993); Phys. Rev. B 55, 15153 (1997)]. As such, LOAF provides a unified framework to study the interacting Fermi gas. The mean-field results discussed here can be systematically improved on by calculating the one-particle irreducible action corrections, order by order.

arXiv:1105.4933•Physical Review A•May 31, 2011•16 citations

Josephson relation for the superfluid density and the connection to the Goldstone theorem in dilute Bose atomic gases

John F. Dawson, Bogdan Mihaila, Fred Cooper

We derive the Josephson relation for a dilute Bose gas in the framework of an auxiliary-field (AF) resummation of the theory in terms of the normal- and anomalous-density condensates. In the leading order of the AF loop expansion (LOAF) we find two critical temperatures in the phase diagram: the critical temperature ${T}_{c}$, where the atom Bose-Einstein condensate (BEC) appears first, and the temperature ${T}^{\ensuremath{\star}}>{T}_{c}$, which indicates the onset of superfluidity and the appearance of a diatom condensate in the system. In this context, the Josephson relation shows that the superfluid density is related to the square of a second-order parameter, the anomalous density, $\ensuremath{\langle}A\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\varphi}\ensuremath{\varphi}\ensuremath{\rangle}$, which can be thought of as a diatom condensate analogous to the Cooper-pair condensate discussed in the BCS approach to dilute Fermi gases. This is in contrast with the corresponding result in the Bose gas theory without a diatom condensate, which predicts that the superfluid density is proportional to the square of the usual atom BEC condensate, $\ensuremath{\langle}\ensuremath{\varphi}\ensuremath{\rangle}$. In the region between ${T}_{c}$ and ${T}^{\ensuremath{\star}}$ the anomalous-density Green's function contains a zero energy-momentum state corresponding to a Goldstone state. This is a two-particle state with U(1) charge two. Our findings are consistent with the prediction that in the temperature range between ${T}_{c}$ and ${T}^{\ensuremath{\star}}$ a fraction of the system is in the superfluid state in the absence of the usual atom BEC condensate. This situation is similar to the case of dilute Fermi gases, where the superfluid density is proportional to the square of the gap parameter. The Josephson relation relies on the existence of zero energy and momentum excitations showing the intimate relationship between superfluidity and the Goldstone theorem.

arXiv:1112.2233•Physical Review A•Jul 5, 2012•7 citations

Leading-order auxiliary-field theory of the Bose-Hubbard model

John F. Dawson, Fred Cooper, Chih-Chun Chien +1 more

We discuss the phase diagram of the Bose-Hubbard (BH) model in the leading-order auxiliary field (LOAF) theory. LOAF is a conserving nonperturbative approximation that treats on equal footing the normal and anomalous density condensates. The mean-field solutions in LOAF correspond to first-order and second-order phase transition solutions with two critical temperatures corresponding to a vanishing Bose-Einstein condensate ${T}_{c}$ and a vanishing diatom condensate ${T}^{\ensuremath{\star}}$. The second-order phase transition solution predicts the correct order of the transition in continuum Bose gases. For either solution, the superfluid state is tied to the presence of the diatom condensate related to the anomalous density in the system. In ultracold Bose atomic gases confined on a three-dimensional lattice, the critical temperature ${T}_{c}$ exhibits a quantum phase transition, where ${T}_{c}$ goes to zero at a finite coupling. The BH phase diagram in LOAF features a line of first-order transitions ending in a critical point beyond which the transition is second order while approaching the quantum phase transition. We identify a region where a diatom condensate is expected for temperatures higher than ${T}_{c}$ and less than ${T}_{0}$, the critical temperature of the noninteracting system. The LOAF phase diagram for the BH model compares qualitatively well with existing experimental data and results of ab initio Monte Carlo simulations.

arXiv:1304.4826•Physical Review A•Aug 12, 2013•4 citations

Renormalised thermodynamics for Bose gases from low to critical temperatures

Michael H. Heinrich, Alexander Wowchik, Jürgen Berges

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.

cond-mat.quant-gasCross-listedarXiv:2604.16290•Apr 20, 2026•0 citations
We rewrite the Lagrangian for a dilute Bose gas in terms of auxiliary fields related to the normal and anomalous condensate densities. We derive the loop expansion of the effective action in the composite-field propagators. The lowest-order auxiliary field (LOAF) theory is a conserving mean-field approximation consistent with the Goldstone theorem without some of the difficulties plaguing approximations such as the Hartree and Popov approximations. LOAF predicts a second-order phase transition. We give a set of Feynman rules for improving results to any order in the loop expansion in terms of composite-field propagators. We compare results of the LOAF approximation with those derived using the Popov approximation. LOAF allows us to explore the critical regime for all values of the coupling constant, and we determine various parameters in the unitarity limit.
We derive a theoretical description for dilute Bose gases as a loop expansion in terms of composite-field propagators by rewriting the Lagrangian in terms of auxiliary fields related to the normal and anomalous densities. We demonstrate that already in leading order this nonperturbative approach describes a large interval of coupling-constant values, satisfies Goldstone's theorem, yields a Bose-Einstein transition that is second order, and is consistent with the critical temperature predicted in the weak-coupling limit by the next-to-leading-order large-N expansion.
We discuss the equilibrium properties of dilute Bose gases using a nonperturbative formalism based on auxiliary fields related to the normal and anomalous densities. We show analytically that for a dilute Bose gas of weakly interacting particles at zero temperature, the leading-order auxiliary field (LOAF) approximation leads to well-known analytical results. Close to the critical point the LOAF predictions are the same as those obtained using an effective field theory in the large-NNN approximation. We also report analytical approximations for the LOAF results in the unitarity limit, which compare favorably with our numerical results. LOAF predicts that the equation of state for the Bose gas in the unitarity limit is E/(pV)=1E/(pV)=1E/(pV)=1, unlike the case of the Fermi gas when E/(pV)=3/2E/(pV)=3/2E/(pV)=3/2.
We develop the auxiliary field formalism corresponding to a dilute system of spin-1/2 fermions. This theory represents the Fermi counterpart of the Bose-Einstein condensation (BEC) theory developed recently by F. Cooper et al. [Phys. Rev. Lett. 105, 240402 (2010)] to describe a dilute gas of Bose particles. Assuming tunable interactions, this formalism is appropriate for the study of the crossover from the regime of Bardeen-Cooper-Schriffer (BCS) pairing to the regime of BEC in ultracold fermionic atom gases. We show that when applied to the Fermi case at zero temperature, the leading-order auxiliary field (LOAF) approximation gives the same equations as obtained in the standard BCS variational picture. At finite temperature, LOAF leads to the theory discussed by S\'a de Melo, Randeria, and Engelbrecht [Phys. Rev. Lett. 71, 3202 (1993); Phys. Rev. B 55, 15153 (1997)]. As such, LOAF provides a unified framework to study the interacting Fermi gas. The mean-field results discussed here can be systematically improved on by calculating the one-particle irreducible action corrections, order by order.
We derive the Josephson relation for a dilute Bose gas in the framework of an auxiliary-field (AF) resummation of the theory in terms of the normal- and anomalous-density condensates. In the leading order of the AF loop expansion (LOAF) we find two critical temperatures in the phase diagram: the critical temperature Tc{T}_{c}Tc​, where the atom Bose-Einstein condensate (BEC) appears first, and the temperature ${T}^{\ensuremath{\star}}>{T}_{c}$, which indicates the onset of superfluidity and the appearance of a diatom condensate in the system. In this context, the Josephson relation shows that the superfluid density is related to the square of a second-order parameter, the anomalous density, $\ensuremath{\langle}A\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\varphi}\ensuremath{\varphi}\ensuremath{\rangle}$, which can be thought of as a diatom condensate analogous to the Cooper-pair condensate discussed in the BCS approach to dilute Fermi gases. This is in contrast with the corresponding result in the Bose gas theory without a diatom condensate, which predicts that the superfluid density is proportional to the square of the usual atom BEC condensate, $\ensuremath{\langle}\ensuremath{\varphi}\ensuremath{\rangle}$. In the region between Tc{T}_{c}Tc​ and ${T}^{\ensuremath{\star}}$ the anomalous-density Green's function contains a zero energy-momentum state corresponding to a Goldstone state. This is a two-particle state with U(1) charge two. Our findings are consistent with the prediction that in the temperature range between Tc{T}_{c}Tc​ and ${T}^{\ensuremath{\star}}$ a fraction of the system is in the superfluid state in the absence of the usual atom BEC condensate. This situation is similar to the case of dilute Fermi gases, where the superfluid density is proportional to the square of the gap parameter. The Josephson relation relies on the existence of zero energy and momentum excitations showing the intimate relationship between superfluidity and the Goldstone theorem.
We discuss the phase diagram of the Bose-Hubbard (BH) model in the leading-order auxiliary field (LOAF) theory. LOAF is a conserving nonperturbative approximation that treats on equal footing the normal and anomalous density condensates. The mean-field solutions in LOAF correspond to first-order and second-order phase transition solutions with two critical temperatures corresponding to a vanishing Bose-Einstein condensate Tc{T}_{c}Tc​ and a vanishing diatom condensate ${T}^{\ensuremath{\star}}$. The second-order phase transition solution predicts the correct order of the transition in continuum Bose gases. For either solution, the superfluid state is tied to the presence of the diatom condensate related to the anomalous density in the system. In ultracold Bose atomic gases confined on a three-dimensional lattice, the critical temperature Tc{T}_{c}Tc​ exhibits a quantum phase transition, where Tc{T}_{c}Tc​ goes to zero at a finite coupling. The BH phase diagram in LOAF features a line of first-order transitions ending in a critical point beyond which the transition is second order while approaching the quantum phase transition. We identify a region where a diatom condensate is expected for temperatures higher than Tc{T}_{c}Tc​ and less than T0{T}_{0}T0​, the critical temperature of the noninteracting system. The LOAF phase diagram for the BH model compares qualitatively well with existing experimental data and results of ab initio Monte Carlo simulations.
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.