# Spontaneous symmetry breaking and Nambu-Goldstone modes in open classical and quantum systems

## TL;DR

This paper explores how spontaneous symmetry breaking in open classical and quantum systems leads to various types of Nambu-Goldstone modes, including diffusive and propagating, and establishes a generalized Nambu-Goldstone theorem.

## Contribution

It extends the Nambu-Goldstone theorem to open systems and classifies the resulting modes into four distinct types based on their dispersion characteristics.

## Key findings

- Identifies four types of Nambu-Goldstone modes in open systems.
- Derives low-energy coefficients governing mode dispersion.
- Establishes a generalized Nambu-Goldstone theorem for open systems.

## Abstract

We discuss spontaneous symmetry breaking of open classical and quantum systems. When a continuous symmetry is spontaneously broken in an open system, a gapless excitation mode appears corresponding to the Nambu-Goldstone mode. Unlike isolated systems, the gapless mode is not always a propagation mode, but it is a diffusion one. Using the Ward-Takahashi identity and the effective action formalism, we establish the Nambu-Goldstone theorem in open systems, and derive the low-energy coefficients that determine the dispersion relation of Nambu-Goldstone modes. Using these coefficients, we classify the Nambu-Goldstone modes into four types: type-A propagation, type-A diffusion, type-B propagation, and type-B diffusion modes.