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

Abstract

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 $\Sigma (0)-\mu =\Delta (0)$ as the lowest-order identity. Using them, it is found that the anomalous self-energy $\Delta (0)$ 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 $v_{\mathrm{s}}\propto [\Delta (0){]}^{1/2}>0$. It is also argued that the one-particle density matrix $\rho (\mathbf{r})\equiv ⟨{\hat{\psi }}^{†}({\mathbf{r}}_1)\hat{\psi }({\mathbf{r}}_1+\mathbf{r})⟩$ for $d<d_{\mathrm{c}}$ dimensions approaches the off-diagonal-long-range-order value $N_0/V$ asymptotically as $r^{-d+2-\eta }$ with an exponent $\eta >0$. The anomalous 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-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.

19 pages, 5 figures

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• Exact evolution equation for the effective potential
• Functional renormalization for Bose-Einstein Condensation
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Cited by in corpus (1)

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