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arXiv:1012.0239 · Journal of Physics B Atomic Molecular and Optical Physics

On the order of BEC transition in weakly interacting gases predicted by mean-field theory

Luis Olivares‐Quiroz, Vı́ctor Romero-Rochı́n

Published Oct 4, 2010• 24 citations
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Abstract

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.

DOI
10.1088/0953-4075/43/20/205302

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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.
Multiple phases occurring in a Bose gas with finite-range interaction are investigated. In the vicinity of the onset of Bose-Einstein condensation (BEC), the chemical potential and the pressure show a van der Waals--like behavior indicating a first-order phase transition although there is no long-range attraction. Furthermore, the equation of state becomes multivalued near the BEC transition. For a Hartree-Fock or Popov (Hartree-Fock-Bogoliubov) approximation, such a multivalued region can be avoided by the Maxwell construction. For sufficiently weak interaction, the multivalued region can also be removed using a many-body TTT-matrix approximation. However, for strong interactions there remains a multivalued region even for the TTT-matrix approximation and after the Maxwell construction, which is interpreted as a density hysteresis. This unified treatment of normal and condensed phases becomes possible due to the recently found scheme to eliminate self-interaction in the TTT-matrix approximation, which allows one to calculate properties below and above the critical temperature.
We discuss the thermal gas to Bose-Einstein Condensation (BEC) transition of an ultracold Bose gas in a linear quadrupolar trap with contact interatomic interactions within the Hartree-Fock (HF) approximation. We briefly review the theoretical framework of the thermodynamics of a gas confined by a non-uniform potential to show how mechanical equilibria must be formulated in term of generalized volume and pressure variables V and P that replace the usual volume and hydrostatic pressure of a uniform system. We solve for the density profile within HF as a function of temperature T and molar volume v = V/N. With this information, we are able to calculate all the thermodynamic properties of the system and analyze the behavior of the gas through the BEC (superfluid) transition. We find that the transition is completely smooth, showing no sign of critical behavior. We emphasize how these predictions can be readily measured with the current experimental setups.
Abstract The critical properties of the phase transition from a normal gas to a Bose–Einstein condensate (superfluid) of a harmonically confined Bose gas are addressed with the knowledge of an equation of state of the underlying homogeneous Bose fluid. It is shown that while the presence of the confinement trap arrests the usual divergences of the isothermal compressibility and heat capacities, the critical behavior manifests itself now in the divergence of derivatives of the mentioned susceptibilities. This result is illustrated with a mean-field like model of an equation of state for the homogeneous particle density as a function of the chemical potential and temperature of the gas. The model assumes the form of an ideal Bose gas in the normal fluid while in the superfluid state a function is proposed such that, both, asymptotically reaches the Thomas–Fermi solution of a weakly interacting Bose gas at large densities and low temperatures and, at the transition, matches the critical properties of the ideal Bose gas. With this model we obtain the global thermodynamics of the harmonically confined gas, from which we analyze its critical properties. We discuss how these properties can be experimentally tested.
A Hartree–Fock mean-field theory of a weakly interacting Bose-gas in a quenched white noise disorder potential is presented. A direct continuous transition from the normal gas to a localized Bose-glass phase is found which has localized short-lived excitations with a gapless density of states and vanishing superfluid density. The critical temperature of this transition is as for an ideal gas undergoing Bose–Einstein condensation. Increasing the particle-number density a first-order transition from the localized state to a superfluid phase perturbed by disorder is found. At intermediate number densities both phases can coexist.
In this work we revisit the concept of chemical potential μμμ in both classical and quantum gases from a perspective of Equilibrium Statistical Mechanics (ESM). Two new results regarding the equation of state μ=μ(n,T)μ=μ(n,T)μ=μ(n,T), where nnn is the particle density and TTT the absolute temperature, are given for the classical interacting gas and for the weakly-interacting quantum Bose gas. In order to make this review self-contained and adequate for a general reader we provide all the basic elements in an advanced-undergraduate or graduate statistical mechanics course required to follow all the calculations. We start by presenting a calculation of μ(n,T)μ(n,T)μ(n,T) for the classical ideal gas in the canonical ensemble. After this, we consider the interactions between particles and compute the effects of them on μ(n,T)μ(n,T)μ(n,T) for the van der Waals gas. For quantum gases we present an alternative approach to calculate the Bose-Einstein (BE) and Fermi-Dirac (FD) statistics. We show that this scheme can be straightforwardly generalized to determine what we have called Intermediate Quantum Statistics (IQS) which deal with ideal quantum systems where a single-particle energy can be occupied by at most jjj particles with 0⩽j⩽N0 \leqslant j \leqslant N0⩽j⩽N with NNN the total number of particles. In the final part we address general considerations that underlie the theory of weakly interacting quantum gases. In the case of the weakly interacting Bose gas, we focus our attention to the equation of state μ=μ(n,T)μ=μ(n,T)μ=μ(n,T) in the Hartree-Fock mean-field approximation (HF) and the implications of such results in the elucidation of the order of the phase transitions involved in the BEC phase for non-ideal Bose gases.
The phase transition of the one-dimensional Bose gas, interacting weakly with a repulsive potential and trapped in an external parabolic potential, is studied in a mean field approximation. The Bose–Einstein condensed (BEC) state is identified by the non-vanishing thermal average of a bose field. Free energies between the BEC state and normal state are compared. A crossing of the free energies occurring at low temperature suggests the first order phase transition in an interacting one-dimensional bose system. From fitting of numerical results, the relation among the transition temperature T c , the total number of bosons N and the interaction V is also provided as a simple power law, T c ∝( N 2 V ) 2/3 .