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We derive exact renormalization-group equations for the $n$-point vertices($n=0,1,2,\cdots$) of interacting single-component Bose-Einstein condensatesbased on the vertex expansion of the effective action. They have a notablefeature of automatically satisfying Goldstone's theorem (I), which yields theHugenholtz-Pines relation $\Sigma(0)-\mu=\Delta(0)$ as the lowest-orderidentity. Using them, it is found that the anomalous self-energy $\Delta(0)$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 $v_{\rm s}\propto [\Delta(0)]^{1/2}>0$. It is also arguedthat the one-particle density matrix $\rho({\bfr})\equiv\langle\hat\psi^\dagger({\bf r}_1)\hat\psi({\bf r}_1+{\bf r})\rangle$for $d<d_{\rm c}$ dimensions approaches the off-diagonal-long-range-order value$N_{\bf 0}/V$ asymptotically as $r^{-d+2-\eta}$ with an exponent $\eta>0$. Theanomalous dimension $\eta$ at finite temperatures is predicted to behave for$d=4-\epsilon$ dimensions ($0<\epsilon\ll 1$) as $\eta\propto\epsilon^2$. Thus,the interacting Bose-Einstein condensates are subject to long-rangefluctuations similar to those at the second-order transition point, and theirexcitations in the one-particle channel are distinct from the Nambu-Goldstonemode with a sound-wave dispersion in the two-particle channel.

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