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

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 $\rho({\bf r})\equiv \langle \hat\psi^\dagger({\bf r}_1)\hat\psi({\bf r}_1+{\bf r})\rangle$ through its asymptotic behavior $\rho({\bf r})\rightarrow N_{\bf 0}/V+C r^{-d+2-\eta}$ 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 $\eta=0.181\epsilon^2$ 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 ${\rm O}(2)$ symmetric $\phi^4$ model is attributed to the emergence of three-point vertices and the anomalous Green's function when $N_{\bf 0}$ acquires a finite value.