Interpolating relativistic and non-relativistic Nambu-Goldstone and Higgs modes

TL;DR

This paper develops a framework to interpolate between relativistic and non-relativistic Nambu-Goldstone modes, revealing their nature and associated Higgs modes across different regimes, with applications to condensed matter systems.

Contribution

It introduces a unified Lagrangian framework that connects relativistic and non-relativistic NG modes, clarifying their spectrum and associated Higgs modes in various limits.

Findings

Type-I and II NG modes are accompanied by gapful Higgs and chiral partners.

In the ultra-relativistic limit, NG modes and partners behave as expected from relativistic theories.

In the non-relativistic limit, certain modes become infinitely massive and disappear.

Abstract

When a continuous symmetry is spontaneously broken in non-relativistic theories, there appear Nambu-Goldstone (NG) modes, whose dispersion relations are either linear (type-I) or quadratic (type-II). We give a general framework to interpolate between relativistic and non-relativistic NG modes, revealing a nature of type-I and II NG modes in non-relativistic theories. The interpolating Lagrangians have the nonlinear Lorentz invariance which reduces to the Galilei or Schrodinger invariance in the non-relativistic limit. We find that type-I and type-II NG modes in the interpolating region are accompanied with a Higgs mode and a chiral NG partner, respectively, both of which are gapful. In the ultra-relativistic limit, a set of a type-I NG mode and its Higgs partner remains, while a set of type-II NG mode and gapful NG partner turns to a set of two type-I NG modes. In the non-relativistic limit, the both types of accompanied gapful modes become infinitely massive, disappearing from the spectrum. The examples contain a phonon in Bose-Einstein condensates, a magnon in ferromagnets, and a Kelvon and dilaton-magnon localized around a skyrmion line in ferromagnets.