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arXiv:1903.05230 · Journal of the Physical Society of Japan

A Renormalization-Group Study of Interacting Bose–Einstein Condensates: Absence of the Bogoliubov Mode below Four ( T > 0) and Three ( T = 0) Dimensions

Takafumi Kita

Published Apr 16, 2019• 6 citations
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Abstract

We derive exact renormalization-group equations for the $n$-point vertices ($n=0,1,2,\cdots$) of interacting single-component Bose-Einstein condensates based on the vertex expansion of the effective action. They have a notable feature of automatically satisfying Goldstone's theorem (I), which yields the Hugenholtz-Pines relation $Σ(0)-μ=Δ(0)$ as the lowest-order identity. Using them, it is found that the anomalous self-energy $Δ(0)$ vanishes below $d_{\rm c}=4$ ($d_{\rm c}=3$) dimensions at finite temperatures (zero temperature), contrary to the Bogoliubov theory predicting a finite "sound-wave" velocity $v_{\rm s}\propto [Δ(0)]^{1/2}>0$. It is also argued that the one-particle density matrix $ρ({\bf r})\equiv\langle\hatψ^\dagger({\bf r}_1)\hatψ({\bf r}_1+{\bf r})\rangle$ for $d0$. The anomalous dimension $η$ at finite temperatures is predicted to behave for $d=4-ε$ dimensions ($0

DOI
10.7566/jpsj.88.054003

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A Renormalization-Group Study of Interacting Bose–Einstein Condensates: II. Anomalous Dimension η for d≲4d \lesssim 4d≲4 at Finite Temperatures
We study the anomalous dimension ηηη of homogeneous interacting single-component Bose-Einstein condensates at finite temperatures for d≲4d\lesssim 4d≲4 dimensions. This ηηη is defined in terms of the one-particle density matrix ρ(r)≡⟨ψ^†(r1)ψ^(r1+r)⟩ρ({\bf r})\equiv \langle \hatψ^\dagger({\bf r}_1)\hatψ({\bf r}_1+{\bf r})\rangleρ(r)≡⟨ψ^​†(r1​)ψ^​(r1​+r)⟩ through its asymptotic behavior ρ(r)→N0/V+Cr−d+2−ηρ({\bf r})\rightarrow N_{\bf 0}/V+C r^{-d+2-η}ρ(r)→N0​/V+Cr−d+2−η for r→∞r\rightarrow \inftyr→∞, where N0/VN_{\bf 0}/VN0​/V is the condensate density and CCC is a constant. It is shown that the anomalous dimension is given by η=0.181ε2η=0.181ε^2η=0.181ε2 to the leading order in ε≡d−4ε\equiv d-4ε≡d−4. The change of the prefactor 0.1810.1810.181 from the value 0.020.020.02 at the transition point of the O(2){\rm O}(2)O(2) symmetric ϕ4ϕ^4ϕ4 model is attributed to the emergence of three-point vertices and the anomalous Green's function when N0N_{\bf 0}N0​ acquires a finite value.
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.
Wilson’s renormalization-group approach to the weakly-interacting single-component Bose gas is discussed within the symmetry-broken, condensate phase. Extending upon the work by Bijlsma and Stoof, wave-function renormalization of the temporal derivative contributions to the effective action is included in order to capture sound-like quasiparticle excitations with wave lengths larger than the healing-length scale. By means of a suitable rescaling scheme we achieve convergence of the coupling flows, which serve as a means to determine the condensate depletion in accordance with Bogoliubov theory, as well as the interaction-induced shift of the critical temperature.
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We exploit the renormalization-group approach to establish the exact infrared behavior of an interacting Bose system at zero temperature. The local-gauge symmetry in the broken-symmetry phase is implemented through the associated Ward identities, which reduce the number of independent running couplings to a single one. For this coupling the \ensuremath{\epsilon} expansion can be controlled to all orders in \ensuremath{\epsilon} $(=3\ensuremath{-}d).$ For spatial dimensions 1<d< 31<d<~31<d< 3 the Bogoliubov fixed point is unstable towards a different fixed point characterized by the divergence of the longitudinal correlation function. The Bogoliubov linear spectrum, however, is found to be independent of the critical behavior of this correlation function, being exactly constrained by Ward identities. The new fixed point properly gives a finite value of the coupling among transverse fluctuations, but due to virtual intermediate longitudinal fluctuations, the effective coupling affecting the transverse correlation function flows to zero. As a result, no transverse anomalous dimension is present. This treatment allows us to recover known results for the quantum Bose gas in the context of a unifying framework and also to reveal the nontrivial skeleton structure of its perturbation theory.