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arXiv:1909.11787 · Journal of the Physical Society of Japan

A Renormalization-Group Study of Interacting Bose–Einstein Condensates: II. Anomalous Dimension η for \(d \lesssim 4\) at Finite Temperatures

Takafumi Kita

Published Sep 25, 2019• 3 citations
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Abstract

We study the anomalous dimension $η$ of homogeneous interacting single-component Bose-Einstein condensates at finite temperatures for $d\lesssim 4$ dimensions. This $η$ is defined in terms of the one-particle density matrix $ρ({\bf r})\equiv \langle \hatψ^\dagger({\bf r}_1)\hatψ({\bf r}_1+{\bf r})\rangle$ through its asymptotic behavior $ρ({\bf r})\rightarrow N_{\bf 0}/V+C r^{-d+2-η}$ for $r\rightarrow \infty$, where $N_{\bf 0}/V$ is the condensate density and $C$ is a constant. It is shown that the anomalous dimension is given by $η=0.181ε^2$ to the leading order in $ε\equiv d-4$. The change of the prefactor $0.181$ from the value $0.02$ at the transition point of the ${\rm O}(2)$ symmetric $ϕ^4$ model is attributed to the emergence of three-point vertices and the anomalous Green's function when $N_{\bf 0}$ acquires a finite value.

DOI
10.7566/jpsj.88.104003

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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 derive exact renormalization-group equations for the nnn-point vertices (n=0,1,2,⋯n=0,1,2,\cdotsn=0,1,2,⋯) 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)Σ(0)-μ=Δ(0)Σ(0)−μ=Δ(0) as the lowest-order identity. Using them, it is found that the anomalous self-energy Δ(0)Δ(0)Δ(0) vanishes below dc=4d_{\rm c}=4dc​=4 (dc=3d_{\rm c}=3dc​=3) dimensions at finite temperatures (zero temperature), contrary to the Bogoliubov theory predicting a finite "sound-wave" velocity vs∝[Δ(0)]1/2>0v_{\rm s}\propto [Δ(0)]^{1/2}>0vs​∝[Δ(0)]1/2>0. It is also argued that 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)⟩ for d0d0d0. The anomalous dimension ηηη at finite temperatures is predicted to behave for d=4−εd=4-εd=4−ε dimensions ($0
Through exact numerical solutions we show Bose-Einstein condensation (BEC) for the one-dimensional (1D) bosons with repulsive short-range interactions at zero temperature by taking a particular large size limit. Following the Penrose-Onsager criterion of BEC, we define condensate fraction by the fraction of the largest eigenvalue of the one-particle reduced density matrix. % We show the finite-size scaling behavior such that condensate fraction is given by a scaling function of one-variable: interaction parameter multiplied by a power of particle number. Condensate fraction is nonzero and constant for any large value of particle number or system size, if the interaction parameter is proportional to the negative power of particle number. %Here the interaction parameter is defined by the coupling constant of the delta-function potentials devided by the density. %With the scaling behavior we derive various themodynamic limits where condensate fraction is constant for any large system size; for instance, it is the case even in the system of a finite particle number.
We calculate the temperature-dependent condensate density rho0(T) of interacting bosons in three dimensions using the functional renormalization group (FRG). From the numerical solution of suitably truncated FRG flow equations for the irreducible vertices we obtain rho0(T) for arbitrary temperatures. We carefully extrapolate our numerical results to the critical point and determine the order parameter exponent beta approximately 0.32 in reasonable agreement with the expected value 0.345 associated with the XY -universality class. We also calculate the condensate density in two dimensions at zero temperature using a truncation of the FRG flow equations based on the derivative expansion including cubic and quartic terms in the expansion of the effective potential in powers of the density. As compared with the widely used quadratic approximation for the effective potential, the coupling constants associated with the cubic and quartic terms lead to small corrections of the condensate density.
An exact expression for the single-particle reduced density matrix of a translationally invariant system of bosons is obtained, as an average of a functional of the density and current operators. An approximation scheme, based on small fluctuations in density and current, is used in its lowest order, to approximate the single-particle reduced density matrix and the condensate density by a functional of the structure factor, which is a measurable quantity. This is done for the ground state and for finite temperatures. The resulting expressions for the condensate density are discussed. For the ideal gas the result is correct below the condensation temperature. For the case of an interacting liquid, Bose-Einstein condensation exists, in the ground states for two- and three-dimensional systems, and for finite temperatures it does exist only in three dimensions. The present experimental knowledge of the structure factor enables only a crude estimate of the condensed fraction which is found to be 0.1-0.2.
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The theory of ultracold, dilute Bose gases is the subject of intensive studies, driven by new experimental applications, which also motivate the study of Bose-Einstein condensation (BEC) in low dimensions. From the theoretical point of view there are few, quite special, models in which we are able to prove BEC for interacting bosons. With the aim of studying the condensation problem I considered a simplified model for a three and two dimensional system of bosons interacting with a repulsive short range potential, obtained by introducing an ultraviolet momentum cutoff. I proved that the interacting theory is well defined at all orders in terms of one (in three dimensions) or two (in two dimensions) effective parameters, related to the intensity of the interaction. Moreover the correlations do not exhibit anomalous dimensions, i.e. the model is in the same universality class of the exactly soluble Bogoliubov model. The result is obtained by using Rigorous Renormalization Group techniques and implementing local Ward identities (WI), which reduce the number of independent running couplings, this fact being crucial for the control of the two dimensional theory, where the effective interactions are relevant. Since the ultraviolet momentum cutoff breaks local gauge invariance, the corrections to the formal WI have been considered.