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arXiv:1512.08446 · Physical Review A

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

Tom Kim, Chih-Chun Chien

Published Mar 21, 2016• 1 citation
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Abstract

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.

DOI
10.1103/physreva.93.033637

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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 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 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.
Predictions from Hartree–Fock (HF), Popov (P), Yukalov–Yukalova (YY) and t -matrix approximations regarding the thermodynamics from the normal to the Bose–Einstein condensation (BEC) phase in weakly interacting Bose gases are considered. By analysing the dependence of the chemical potential μ on the temperature T and particle density ρ, we show that none of them predicts a second-order phase transition as required by symmetry-breaking general considerations. In this work, we find that the isothermal compressibility κ T predicted by these theories does not diverge at criticality as expected in a true second-order phase transition. Moreover, the isotherms μ = μ(ρ, T ) typically exhibit non-single-valued behaviour in the vicinity of the BEC transition, a feature forbidden by general thermodynamic principles. This behaviour can be avoided if a first-order phase transition is appealed. The facts described above show that although these mean field approximations give correct results near zero temperature, they are endowed with thermodynamic anomalies in the vicinity of the BEC transition. We address the implications of these results in the interpretation of current experiments with ultracold trapped alkali gases.
We analyze the results of a recent experiment with bosonic rubidium atoms harmonically confined in a quasi-two-dimensional (2D) geometry. In this experiment a well-defined critical point was identified, which separates the high-temperature normal state characterized by a single component density distribution, and the low-temperature state characterized by a bimodal density distribution and the emergence of high-contrast interference between independent 2D clouds. We first show that this transition cannot be explained in terms of conventional Bose–Einstein condensation of the trapped ideal Bose gas. Using the local density approximation (LDA), we then combine the mean-field (MF) Hartree–Fock theory with the prediction for the Berezinskii–Kosterlitz–Thouless (BKT) transition in an infinite uniform system. If the gas is treated as a strictly 2D system, the MF predictions for the spatial density profiles significantly deviate from those of a recent quantum Monte Carlo (QMC) analysis. However, when the residual thermal excitation of the strongly confined degree of freedom is taken into account, excellent agreement is reached between the MF and the QMC approaches. For the interaction strength corresponding to the experiment, we predict a strong correction to the critical atom number with respect to the ideal gas theory (factor ∼2). Quantitative agreement between theory and experiment is reached concerning the critical atom number if the predicted density profiles are used for temperature calibration.
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.