In a Bose superfluid, the coupling between transverse (phase) and longitudinal fluctuations leads to a divergence of the longitudinal correlation function, which is responsible for the occurrence of infrared divergences in the perturbation theory and the breakdown of the Bogoliubov approximation. We report a nonperturbative renormalization-group calculation of the one-particle Green’s function of an interacting boson system at zero temperature. We find two regimes separated by a characteristic momentum scale $k_G$ (“Ginzburg” scale). While the Bogoliubov approximation is valid at large momenta and energies, $\left|\mathbf{p}\right|,\left|\omega \right|/c⪢k_G$ (with $c$ as the velocity of the Bogoliubov sound mode), in the infrared (hydrodynamic) regime, $\left|\mathbf{p}\right|,\left|\omega \right|/c⪡k_G$, the normal and anomalous self-energies exhibit singularities reflecting the divergence of the longitudinal correlation function. In particular, we find that the anomalous self-energy agrees with the Bogoliubov result ${\Sigma }_{\text{an}}\left(\mathbf{p},\omega \right)\simeq \text{const}$ at high energies and behaves as ${\Sigma }_{\text{an}}\left(\mathbf{p},\omega \right)\sim {\left(c^2{\mathbf{p}}^2-{\omega }^2\right)}^{\left(d-3\right)/2}$ in the infrared regime (with $d$ as the space dimension), in agreement with the Nepomnyashchii identity ${\Sigma }_{\text{an}}\left(0,0\right)=0$ and the predictions of Popov’s hydrodynamic theory. We argue that the hydrodynamic limit of the one-particle Green’s function is fully determined by the knowledge of the exponent $3-d$ characterizing the divergence of the longitudinal susceptibility and the Ward identities associated to gauge and Galilean invariances. The infrared singularity of ${\Sigma }_{\text{an}}\left(\mathbf{p},\omega \right)$ leads to a continuum of excitations (coexisting with the sound mode) which shows up in the one-particle spectral function.

• Received 16 July 2009

DOI:https://doi.org/10.1103/PhysRevA.80.043627