Dispersion relations of Nambu-Goldstone modes at finite temperature and density

TL;DR

This paper analyzes the dispersion relations of Nambu-Goldstone modes at finite temperature and density, revealing their propagation characteristics and effects of explicit symmetry breaking.

Contribution

It provides a comprehensive derivation of dispersion relations for NG modes at finite temperature/density, including gap formulas and coexistence of different NG modes.

Findings

Type-A NG modes have linear dispersion with quadratic imaginary parts.

Type-B NG modes have quadratic dispersion with quartic imaginary parts.

NG modes can propagate far when momentum is small.

Abstract

We discuss the dispersion relations of Nambu-Goldstone (NG) modes associated with spontaneous breaking of internal symmetries at finite temperature and/or density. We show that the dispersion relations of type-A (I) and type-B (II) NG modes are linear and quadratic in momentum, whose imaginary parts are quadratic and quartic, respectively. In both cases, the real parts of the dispersion relations are larger than the imaginary parts when the momentum is small, so that the NG modes can propagate far away. We derive the gap formula for NG modes in the presence of a small explicit breaking term. We also discuss the gapped partners of type-B NG modes, when type-A and type-B NG modes coexist.