paper

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

arXiv:0907.2779 · doi:10.1103/PhysRevA.80.043627

Abstract

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 ("Ginzburg" scale). While the Bogoliubov approximation is valid at large momenta and energies, $|\p|,|\w|/c\gg k_G$ (with 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 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 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.

v1) 23 pages, 11 figures. v2) Changes following referee's comments. To appear in Phys. Rev.A. v3) Typos corrected

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