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

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 $\mathrm{\Sigma }(0)-\mu =\mathrm{\Delta }(0)$ as the lowest-order identity. Using them, it is found that the anomalous self-energy $\mathrm{\Delta }(0)$ vanishes below $d_{\mathrm{c}}=4$ ($d_{\mathrm{c}}=3$) dimensions at finite temperatures (zero temperature), contrary to the Bogoliubov theory predicting a finite “sound-wave” velocity $v_{\mathrm{s}}\propto [\mathrm{\Delta }(0){]}^{1\mathrm{/}2}>0$. It is also argued that the one-particle density matrix $\rho({\bf r})\equiv\langle\hat\psi^\dagger({\bf r}1)\hat\psi({\bf r}1+{\bf r})\rangle$for$d<d{\rm c}$dimensionsapproachestheoff-diagonal-long-range-ordervalue$N{\bf 0}/V$asymptoticallyas$r^{-d+2-\eta}$withanexponent$\eta>0$. The anomalous dimension $\eta$ at finite temperatures is predicted to behave for $d=4-ϵ$ dimensions ($0<ϵ\ll 1$) as $\eta \propto {ϵ}^2$. Thus, the interacting Bose-Einstein condensates are subject to long-range fluctuations similar to those at the second-order transition point, and their excitations in the one-particle channel are distinct from the Nambu-Goldstone mode with a sound-wave dispersion in the two-particle channel.