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 $\Psi$ and the Nambu Green's function $\hat{G}$ for the quasiparticle field. Imposing its stationarity respect to $\Psi$ and $\hat{G}$ 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 $T_c$ inherent in Bose systems, i.e., the Landau-Ginzburg expansion is not possible here. The transition temperature $T_c$ shows quite a different interaction dependence between the $n$-fixed and $p$-fixed cases. In the former case $T_c$ increases from the ideal gas value $T_0$ as $T_c/T_0=1+2.33a{n}^{1/3}$, whereas it decreases in the latter as $T_c/T_0=1-3.84a(mp/2\pi {\hslash }^2{)}^{1/5}$. Temperature dependences of basic thermodynamic quantities are clarified explicitly.

arXiv comment19 pages, 8 figures

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