In systems with a spontaneously broken continuous symmetry, the perturbative loop expansion is plagued by infrared divergences due to the coupling between transverse and longitudinal fluctuations. As a result, the longitudinal susceptibility diverges and the self-energy becomes singular at low energy. We study the crossover from the high-energy Gaussian regime, where perturbation theory remains valid, to the low-energy Goldstone regime characterized by a diverging longitudinal susceptibility. We consider both the classical linear O ($N$) model and interacting bosons at zero temperature, using a variety of techniques: perturbation theory, hydrodynamic approach (i.e., for bosons, Popov’s theory), large-$N$ limit, and nonperturbative renormalization group. We emphasize the essential role of the Ginzburg momentum scale $p_G$, below which the perturbative approach breaks down. Even though the action of (nonrelativistic) bosons includes a first-order time derivative term, we find remarkable similarities in the weak-coupling limit between the classical O($N$) model and interacting bosons at zero temperature.

• Received 22 November 2010

DOI:https://doi.org/10.1103/PhysRevE.83.031120