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:1107.3750 · Physical Review A

Analytical limits for cold-atom Bose gases with tunable interactions

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

Published Aug 2, 2011• 10 citations
arXivHTMLPDF

Abstract

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 $E/(pV)=3/2$.

DOI
10.1103/physreva.84.023603

Related papers

Auxiliary-field approach to dilute Bose gases with tunable interactions

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

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.

arXiv:1105.4932•Physical Review A•May 20, 2011•22 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

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

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

Critical temperature of trapped interacting bosons from large- N -based theories

Tom Kim, Chih-Chun Chien

Ultracold atoms provide clues to an important many-body problem regarding the dependence of the Bose-Einstein condensation transition temperature ${T}_{c}$ on interactions. However, cold atoms are trapped in harmonic potentials and theoretical evaluations of the ${T}_{c}$ shift of trapped interacting Bose gases are challenging. While previous predictions of the leading-order shift have been confirmed, more recent experiments exhibit higher-order corrections beyond available mean-field theories. By implementing two large-$\mathcal{N}$-based theories with the local density approximation (LDA), we extract next-order corrections of the ${T}_{c}$ shift. The leading-order large-$\mathcal{N}$ theory produces results quantitatively different from the latest experimental data. The leading-order auxiliary-field (LOAF) theory, containing both normal and anomalous density fields, captures the ${T}_{c}$ shift accurately in the weak-interaction regime. However, the LOAF theory shows incompatible behavior with the LDA, and forcing the LDA leads to density discontinuities in the trap profiles. We present a phenomenological model based on the LOAF theory, which repairs the incompatibility and provides a prediction of the ${T}_{c}$ shift in the stronger-interaction regime.

arXiv:1512.08446•Physical Review A•Mar 21, 2016•1 citation

Relativistic Bose gases at finite density

Jens O. Andersen

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 $T_c$ 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 $T_c$ for N=2 to second order in the interaction.

arXiv:hep-ph/0501094•Jan 12, 2005•3 citations
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 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 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 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 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.
Ultracold atoms provide clues to an important many-body problem regarding the dependence of the Bose-Einstein condensation transition temperature Tc{T}_{c}Tc​ on interactions. However, cold atoms are trapped in harmonic potentials and theoretical evaluations of the Tc{T}_{c}Tc​ shift of trapped interacting Bose gases are challenging. While previous predictions of the leading-order shift have been confirmed, more recent experiments exhibit higher-order corrections beyond available mean-field theories. By implementing two large-N\mathcal{N}N-based theories with the local density approximation (LDA), we extract next-order corrections of the Tc{T}_{c}Tc​ shift. The leading-order large-N\mathcal{N}N theory produces results quantitatively different from the latest experimental data. The leading-order auxiliary-field (LOAF) theory, containing both normal and anomalous density fields, captures the Tc{T}_{c}Tc​ shift accurately in the weak-interaction regime. However, the LOAF theory shows incompatible behavior with the LDA, and forcing the LDA leads to density discontinuities in the trap profiles. We present a phenomenological model based on the LOAF theory, which repairs the incompatibility and provides a prediction of the Tc{T}_{c}Tc​ shift in the stronger-interaction regime.
We consider a massive relativistic Bose gas with NNN 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 TcT_cTc​ 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 TcT_cTc​ for N=2 to second order in the interaction.