arXiv:cond-mat/0608472 · Journal of the Physical Society of Japan

Conserving Gapless Mean-Field Theory of a Multi-Component Bose–Einstein Condensate

Yoshiyuki Kondo, Takafumi Kita

Published Feb 13, 2007• 1 citation

Abstract

We develop a mean-field theory for Bose-Einstein condensation of spin-1 atoms with internal degrees of freedom. It is applicable to nonuniform systems at finite temperatures with a plausible feature of satisfying the Hugenholtz-Pines theorem and various conservation laws simultaneously. Using it, we clarify thermodynamic properties and the excitation spectra of a uniform gas. The condensate is confirmed to remain in the same internal state from T=0 up to $T_{c}$ 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.

DOI: 10.1143/jpsj.76.024008

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arXiv:cond-mat/0608472 · Journal of the Physical Society of Japan

Conserving Gapless Mean-Field Theory of a Multi-Component Bose–Einstein Condensate

Yoshiyuki Kondo, Takafumi Kita

Published Feb 13, 2007• 1 citation

arXiv HTML PDF

Abstract

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.

DOI: 10.1143/jpsj.76.024008

Related papers

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T_{c}

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.

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=0

is also shown to persist at finite temperatures, in the presence of inhomogeneity, and also in nonequilibrium situations.

N

and magnetization

M

. For ferromagnetic interactions, however, triple condensations can occur. Our results can be conveniently understood in terms of the interplay of three factors: (anti)ferromagnetic atom-atom interactions,

M

conservation, and the miscibilities between and among different condensed components.