arXiv:cond-mat/0604241 · Journal of the Physical Society of Japan

Conserving Gapless Mean-Field Theory for Weakly Interacting Bose Gases

Takafumi Kita

Published Apr 13, 2006• 27 citations

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 function $\hat{G}$ for the quasiparticle field. Imposing its stationarity respect to $Ψ$ 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 $T_{c}/T_{0}= 1+ 2.33 an^{1/3}$, whereas it decreases in the latter as $T_{c}/T_{0}= 1- 3.84a(mp/2π\hbar^{2})^{1/5}$. Temperature dependences of basic thermodynamic quantities are clarified explicitly.

DOI: 10.1143/jpsj.75.044603

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arXiv:cond-mat/0604241 · Journal of the Physical Society of Japan

Conserving Gapless Mean-Field Theory for Weakly Interacting Bose Gases

Takafumi Kita

Published Apr 13, 2006• 27 citations

arXiv HTML PDF

Abstract

Ψ

and the Nambu Green's function

\hat{G}

for the quasiparticle field. Imposing its stationarity respect to

Ψ

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}

T_{c}/T_{0}= 1+ 2.33 an^{1/3}

, whereas it decreases in the latter as

T_{c}/T_{0}= 1- 3.84a(mp/2π\hbar^{2})^{1/5}

. Temperature dependences of basic thermodynamic quantities are clarified explicitly.

DOI: 10.1143/jpsj.75.044603

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Jan 1, 199973 citations

Ψ

and the Nambu Green's function

\hat{G}

for the quasiparticle field. Imposing its stationarity respect to

Ψ

and

\hat{G}

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}

T_{c}/T_{0}= 1+ 2.33 an^{1/3}

, whereas it decreases in the latter as

T_{c}/T_{0}= 1- 3.84a(mp/2π\hbar^{2})^{1/5}

. Temperature dependences of basic thermodynamic quantities are clarified explicitly.

n

and s-wave scattering length

a

to clarify its fundamental thermodynamic properties. It is found that the condensation here occurs as a first-order transition. The shift of the transition temperature

ΔT_c

from the ideal-gas result

T_{0}

is positive and given to the leading order by

ΔT_c = 2.33a n^{1/3}T_0

, in agreement with a couple of previous estimates. The theory is expected to form a new theoretical basis for trapped Bose-Einstein condensates at finite temperatures.

T_{c}

for both antiferromagnetic and ferromagnetic interactions. The excitation spectra of the antiferromagnetic (ferromagnetic) interaction are found to have only a single gapless mode, contrary to the prediction of the Bogoliubov theory where three (two) of them are gapless. We present a detailed discussion on those single-particle excitations in connection with the collective excitations.