Infrared behavior in systems with a broken continuous symmetry: classical O(N) model vs interacting bosons

TL;DR

This paper investigates the infrared divergences and crossover behavior in systems with broken continuous symmetry, comparing classical O(N) models and interacting bosons using various theoretical techniques.

Contribution

It provides a detailed analysis of the infrared behavior and crossover from Gaussian to Goldstone regimes in both classical and quantum systems, highlighting the role of the Ginzburg scale.

Findings

Infrared divergences cause divergence in longitudinal susceptibility.

The Ginzburg scale marks the breakdown of perturbation theory.

Classical O(N) model and interacting bosons show similar behavior at weak coupling.

Abstract

In systems with a spontaneously broken continuous symmetry, the perturbative loop expansion is plagued with 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 non-perturbative 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 (non-relativistic) 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.