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arXiv:2101.02389 · Journal of the Physical Society of Japan

Quantum Field Theory of Correlated Bose–Einstein Condensates: I. Basic Formalism

Takafumi Kita

Published Jan 7, 2021• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.7566/jpsj.90.024001

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Starting from the Dyson-Beliaev and generalized Gross-Pitaevskii equations with an extra nonlocal potential, we derive an exact expression of the two-particle Green's function $\underset{̱}{\mathcal{K}}$ for an interacting Bose-Einstein condensate in terms of unambiguously defined self-energies and vertices. The formula can be a convenient basis for approximate calculations of $\underset{̱}{\mathcal{K}}$. It also tells us that poles of $\underset{̱}{\mathcal{K}}$ are not shared with (i.e., shifted from) those of the single-particle Green's function, contrary to the conclusion of previous studies.
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.
We derive Ward-Takahashi identities for correlated Bose-Einstein condensates based on the expressions of the first-order variations (δΨ,δG)(δΨ,δG)(δΨ,δG) due to perturbations obtained in the preceding paper [T. Kita, J. Phys. Soc. Jpn. 90\bf 9090, 024001 (2021)] for the condensate wave function ΨΨΨ and Green's function GGG. They enable us to obtain several exact results on the density and current correlation functions Kνν′K_{νν'}^{}Kνν′​, and also express Kνν′K_{νν'}^{}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 ωωω and wavenumber qqq is set equal to zero first.
We present a topical review of the development of finite-temperature field theories of Bose-Einstein condensation in weakly interacting atomic gases. We highlight the difficulties in obtaining a consistent finite-temperature theory that has a gapless excitation spectrum in accordance with Goldstone's theorem and which is free from both ultraviolet and infrared divergences. We present results from the two consistent theories developed so far. These are the Hartree-Fock-Bogoliubov theory within the Popov approximation and a many-body T -matrix approach which we have termed gapless-Hartree-Fock-Bogoliubov (GHFB). Comparison with the available experimental results is made and the remaining difficulties are highlighted.
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 TcT_{c}Tc​ 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.
We clarify basic properties of an effective action (i.e., self-consistent perturbation expansion) for interacting Bose-Einstein condensates, where field ψψψ itself acquires a finite thermodynamic average ⟨ψ⟩\langle ψ\rangle⟨ψ⟩ besides two-point Green's function G^\hat GG^ 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⟩\langleψ\rangle⟨ψ⟩ and G^\hat GG^ 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\langle ψ\rangle\neq 0⟨ψ⟩=0 to transform the Bogoliubov mode into a bubbling mode with a substantial decay rate.