Quantum-field-theoretic descriptions of interacting condensed bosons have suffered from the lack of self-consistent approximation schemes satisfying Goldstone's theorem and dynamical conservation laws simultaneously. We present a procedure to construct such approximations systematically by using either an exact relation for the interaction energy or the Hugenholtz-Pines relation to express the thermodynamic potential in a Luttinger-Ward form. Inspection of the self-consistent perturbation expansion up to the third order with respect to the interaction shows that the two relations yield a unique identical result at each order, reproducing the conserving-gapless mean-field theory [T. Kita, J. Phys. Soc. Jpn. 74, 1891 (2005)] as the lowest-order approximation. The uniqueness implies that the series becomes exact when infinite terms are retained. We also derive useful expressions for the entropy and superfluid density in terms of Green's function and a set of real-time dynamical equations to describe thermalization of the condensate.
Quantum-field-theoretic descriptions of interacting condensed bosons have suffered from the lack of self-consistent approximation schemes satisfying Goldstone's theorem and dynamical conservation laws simultaneously. We present a procedure to construct such approximations systematically by using either an exact relation for the interaction energy or the Hugenholtz-Pines relation to express the thermodynamic potential in a Luttinger-Ward form. Inspection of the self-consistent perturbation expansion up to the third order with respect to the interaction shows that the two relations yield a unique identical result at each order, reproducing the conserving-gapless mean-field theory [T. Kita, J. Phys. Soc. Jpn. 74, 1891 (2005)] as the lowest-order approximation. The uniqueness implies that the series becomes exact when infinite terms are retained. We also derive useful expressions for the entropy and superfluid density in terms of Green's function and a set of real-time dynamical equations to describe thermalization of the condensate.
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 G^ for the quasiparticle field. Imposing its stationarity respect to Ψ and 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 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+2.33an1/3, whereas it decreases in the latter as Tc/T0=1−3.84a(mp/2πℏ2)1/5. Temperature dependences of basic thermodynamic quantities are clarified explicitly.
We clarify basic properties of an effective action (i.e., self-consistent perturbation expansion) for interacting Bose-Einstein condensates, where field ψ itself acquires a finite thermodynamic average ⟨ψ⟩ besides two-point Green's function G^ to form an off-diagonal long-range order. It is shown that the action can be expressed concisely order by order in terms of the interaction vertex and a special combination of ⟨ψ⟩ and G^ so as to satisfy both Noether's theorem and Goldstone's theorem (I) corresponding to the first proof. The self-energy is predicted to have a one-particle-reducible structure due to ⟨ψ⟩=0 to transform the Bogoliubov mode into a bubbling mode with a substantial decay rate.
Quantum field theory of equilibrium and nonequilibrium Bose-Einstein condensates is formulated so as to satisfy three basic requirements: the Hugenholtz-Pines relation; conservation laws; identities among vertices originating from Goldstone's theorem I. The key inputs are irreducible four-point vertices, in terms of which we derive a closed system of equations for Green's functions, three- and four-point vertices, and two-particle Green's functions. It enables us to study correlated Bose-Einstein condensates with a gapless branch of single-particle excitations without encountering any infrared divergence. The single- and two-particle Green's functions are found to share poles, i.e., the structure of the two-particle Green's functions predicted by Gavoret and Nozières for a homogeneous condensate at T=0 is also shown to persist at finite temperatures, in the presence of inhomogeneity, and also in nonequilibrium situations.
The ground-state energy per particle E/N and condensate density n0 of a dilute Bose gas are studied with a self-consistent perturbation expansion satisfying the Hugenholtz-Pines theorem and conservation laws simultaneously. A class of Feynman diagrams for the self-energy, which has escaped consideration so far, is shown to add an extra constant cip∼O(1) to the expressions reported by Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957) ] as E/N=(2πℏ2an/m)[1+(128/15π+16cip/5)a3n] and n0/n=1−(8/3π+cip)a3n, where a, n, and m are are the s-wave scattering length, particle density, and particle mass, respectively. We present a couple of estimates for cip; the third-order perturbation expansion yields cip=0.412.
We derive Ward-Takahashi identities for correlated Bose-Einstein condensates based on the expressions of the first-order variations (δΨ,δG) due to perturbations obtained in the preceding paper [T. Kita, J. Phys. Soc. Jpn. 90, 024001 (2021)] for the condensate wave function Ψ and Green's function G. They enable us to obtain several exact results on the density and current correlation functions Kνν′, and also express Kνν′ in terms of low-energy Green's functions and vertices. The latter expressions open up the possibility of constructing theory of superfluid Bose liquids in the same way as that for fermions at low temperatures. The vertices are found to have different limits depending on which of frequency ω and wavenumber q is set equal to zero first.