# Counting Rules of Nambu-Goldstone Modes

## TL;DR

This paper reviews recent formulas for counting Nambu-Goldstone modes resulting from spontaneous symmetry breaking, applicable to both relativistic and nonrelativistic systems, and discusses their complexities and examples.

## Contribution

It provides a comprehensive overview of formulas for counting NGMs in diverse systems, including space-time symmetry breaking, and relates them to the Lieb-Schultz-Mattis theorem.

## Key findings

- Formulas applicable to both relativistic and nonrelativistic systems
- Discussion of NGMs from space-time symmetry breaking
- Examples illustrating the counting rules

## Abstract

When global continuous symmetries are spontaneously broken, there appear gapless collective excitations called Nambu-Goldstone modes (NGMs) that govern the low-energy property of the system. The application of this famous theorem ranges from high-energy, particle physics to condensed matter and atomic physics. When a symmetry breaking occurs in systems that lack the Lorentz invariance to start with, as is usually the case in condensed matter systems, the number of resulting NGMs can be fewer than that of broken symmetry generators, and the dispersion of NGMs is not necessarily linear. In this article, we review recently established formulas for NGMs associated with broken internal symmetries that work equally for relativistic and nonrelativistic systems. We also discuss complexities of NGMs originating from space-time symmetry breaking. In the process we cover many illuminating examples from various context. We also present a complementary point of view from the Lieb-Schultz-Mattis theorem.