Conserving Gapless Mean-Field Theory for Weakly Interacting Bose Gases

Abstract

This paper presents a conserving gapless mean-field theory for weakly interacting Bose gases. We first construct a mean-field Luttinger-Ward thermodynamic functional in terms of the condensate wave function Ψ and the Nambu Green's functionĜ for the quasiparticle field. Imposing its stationarity respect to Ψ andĜ yields a set of equations to determine the equilibrium for general non-uniform systems. They have a plausible property of satisfying the Hugenholtz-Pines theorem to provide a gapless excitation spectrum. Also, the corresponding dynamical equations of motion obey various conservation laws. Thus, the present mean-field theory shares two important properties with the exact theory: "conserving" and "gapless." The theory is then applied to a homogeneous weakly interacting Bose gas with s-wave scattering length a and particle mass m to clarify its basic thermodynamic properties under two complementary conditions of constant density n and constant pressure p. The superfluid transition is predicted to be first-order because of the non-analytic nature of the order-parameter expansion near Tc inherent in Bose systems, i.e., the Landau-Ginzburg expansion is not possible here. The transition temperature Tc shows quite a different interaction dependence between the n-fixed and p-fixed cases. In the former case Tc increases from the ideal gas value T0 as Tc/T0 = 1+2.33an 1/3 , whereas it decreases in the latter as Tc/T0 = 1−3.84a(mp/2π 2 ) 1/5 . Temperature dependences of basic thermodynamic quantities are clarified explicitly.

Representative statistical ensembles for Bose systems with broken gauge symmetry

“…This was already noticed in the early works analysing the Hartree-Fock-Bogolubov approximation having a gap in the spectrum [18,[24][25][26][27]. The generality of such a change of the phase-transition order from second to first in different mean-field approximations was emphasized in a detailed discussion by Baym and Grinstein [28] and recently by Kita [23]. As is clear, the thermodynamics of a system with a firstorder phase transition is rather different from that of a system displaying a second-order transition.…”

“…This happens because of the internal inconsistencies in the description. The disruption of the phase-transition order is a common feature of inconsistent approximations, which either do not satisfy the stability conditions or possess a spectrum gap [18,[21][22][23]. This was already noticed in the early works analysing the Hartree-Fock-Bogolubov approximation having a gap in the spectrum [18,[24][25][26][27].…”

“…Bogolubov already mentioned [4] that such a mismatch can really influence the appearance or disappearance of the spectrum gap, but renders the theory not self-consistent and the system unstable. One can also kill the spectrum gap by adding to the self-energy phenomenological terms, such that to cancel the gap [21][22][23]. This way is, clearly, equivalent to the previous one, since changing the self-energy by invoking some higher-order approximations is not unique and, hence, is also phenomenological.…”

A Dynamical Self-Consistent Finite-Temperature Kinetic Theory: The ZNG Scheme

“…Despite explicitly accounting for pair anomalous averages, and providing a lower total energy for the system, this approach is problematic as its homogeneous limit leads to a gap in the energy spectrum at low momenta, which violates the Goldstone theorem [39]. This inconsistency can be avoided by neglecting the anomalous average altogether (or using other tricks [40][41][42][43][44]), as discussed by Griffin [32,45] and implemented numerically in Refs. [46,47].…”

Cold bosons in optical lattices

“…Not self-consistent approaches render the system unstable, spoil thermodynamic relations, and disrupt the Bose-Einstein condensation phase transition from the second-order to the incorrect firstorder transition [89][90][91][92]. A detailed analysis of this problem has been done in Ref.…”