Spontaneous symmetry breaking and Nambu-Goldstone modes in dissipative systems

TL;DR

This paper explores spontaneous symmetry breaking and Nambu-Goldstone modes in dissipative systems, revealing two types of NG modes and their relation to conserved charges, with implications for diffusive and propagating behaviors.

Contribution

It demonstrates the existence of two types of NG modes in dissipative systems and clarifies their relation to conserved Noether charges, extending the understanding of symmetry breaking beyond Hamiltonian systems.

Findings

Type-A NG modes are diffusive in dissipative systems and propagating in Hamiltonian systems.

The conservation of specific Noether charges determines whether NG modes are diffusive or propagating.

Type-B NG modes exhibit different dispersion relations in dissipative systems compared to Hamiltonian systems.

Abstract

We discuss spontaneous breaking of internal symmetry and its Nambu-Goldstone (NG) modes in dissipative systems. We find that there exist two types of NG modes in dissipative systems corresponding to type-A and type-B NG modes in Hamiltonian systems. To demonstrate the symmetry breaking, we consider a $O(N)$ scalar model obeying a Fokker-Planck equation. We show that the type-A NG modes in the dissipative system are diffusive modes, while they are propagating modes in Hamiltonian systems. We point out that this difference is caused by the existence of two types of Noether charges, $Q_R^\alpha$ and $Q_A^\alpha$: $Q_R^\alpha$ are symmetry generators of Hamiltonian systems, which are not conserved in dissipative systems. $Q_A^\alpha$ are symmetry generators of dissipative systems described by the Fokker-Planck equation, which are conserved. We find that the NG modes are propagating modes if $Q_R^\alpha$ are conserved, while those are diffusive modes if they are not conserved. We also consider a $SU(2)\times U(1)$ scalar model with a chemical potential to discuss the type-B NG modes. We show that the type-B NG modes have a different dispersion relation from those in the Hamiltonian systems.