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 $d_{\\rm c}=4$ ($d_{\\rm c}=3$) 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 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 $d 0$. 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.