A Renormalization-Group Study of Interacting Bose-Einstein Condensates: II. Anomalous Dimension $\eta$ for $d\lesssim 4$ at Finite Temperatures

Abstract

We study the anomalous dimension $\eta$ of homogeneous interacting single-component Bose-Einstein condensates at finite temperatures for $d\lesssim 4$ dimensions. This $\eta$ is defined in terms of the one-particle density matrix $\rho (\mathbf{r})\equiv ⟨{\hat{\psi }}^{†}({\mathbf{r}}_1)\hat{\psi }({\mathbf{r}}_1+\mathbf{r})⟩$ through its asymptotic behavior $\rho (\mathbf{r})\to N_0/V+C{r}^{-d+2-\eta }$ for $r\to \infty$, where $N_0/V$ is the condensate density and $C$ is a constant. It is shown that the anomalous dimension is given by $\eta =0.181{\epsilon }^2$ to the leading order in $\epsilon \equiv d-4$. The change of the prefactor $0.181$ from the value $0.02$ at the transition point of the $\mathrm{O}(2)$ symmetric ${\varphi }^4$ model is attributed to the emergence of three-point vertices and the anomalous Green's function when $N_0$ acquires a finite value.

22 pages, 9 figures

References in corpus (3)

• Exact evolution equation for the effective potential
• 3/2-Body Correlations and Coherence in Bose-Einstein Condensates
• A Renormalization-Group Study of Interacting Bose-Einstein condensates: Absence of the Bogoliubov Mode below Four ($T>0$) and Three ($T=0$) Dimensions

Cited by in corpus (2)

• Quantum Field Theory of Correlated Bose-Einstein condensates: I. Basic Formalism
• Quantum Field Theory of Correlated Bose-Einstein condensates: II. Ward-Takahashi Identities and Correlation Functions