Infrared behavior and spectral function of a Bose superfluid at zero temperature

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 non-perturbative renormalization-group (NPRG) calculation of the one-particle Green 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, $|\p|,|\w|/c\gg k_G$ (with $c$ the velocity of the Bogoliubov sound mode), in the infrared (hydrodynamic) regime $|\p|,|\w|/c\ll k_G$ 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 $\Sigan(\p,\w)\simeq\const$ at high-energies and behaves as $\Sigan(\p,\w)\sim (c^2\p^2-\w^2)^{(d-3)/2}$ in the infrared regime (with $d$ the space dimension), in agreement with the Nepomnyashchii identity $\Sigan(0,0)=0$ and the predictions of Popov's hydrodynamic theory. We argue that the hydrodynamic limit of the one-particle Green 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 $\Sigan(\p,\w)$ leads to a continuum of excitations (coexisting with the sound mode) which shows up in the one-particle spectral function.