Spontaneous symmetry breaking and Nambu-Goldstone modes in dissipative systems

This chapter addresses the spontaneous breaking of chiral symmetry and Nambu-Goldstone (NG) modes from the vacuum to high-density quark matter. In the vacuum, pions and kaons are NG modes associated with the spontaneous breaking of chiral symmetry. The Nambu-Goldstone theorem states that there is a one-to-one correspondence between the broken symmetries and NG modes. The situation is different in high-density quark matter. It is known that two types of NG modes appear in the kaon-condensed color-flavor locked phase: One is an NG mode with linear dispersion, which is the same property as pions in the vacuum. The other is an NG mode with quadratic dispersion, which is no counterpart in the vacuum. In addition, the number of NG modes differs from the number of broken symmetries. This chapter discusses the generalization of the Nambu-Goldstone theorem to cover the high-density quark matter and gives a unified description of these two NG modes.

Non-Abelian vortices arise when a non-Abelian global symmetry is exact in the ground state but spontaneously broken in the vicinity of their cores. In this case, there appear (non-Abelian) Nambu-Goldstone (NG) modes confined and propagating along the vortex. In relativistic theories, the Coleman-Mermin-Wagner theorem forbids the existence of a spontaneous symmetry breaking, or a long-range order, in 1+1 dimensions: quantum corrections restore the symmetry along the vortex and the NG modes acquire a mass gap. We show that in non-relativistic theories NG modes with quadratic dispersion relation confined on a vortex can remain gapless at quantum level. We provide a concrete and experimentally realizable example of a three-component Bose-Einstein condensate with U(1) x U(2) symmetry. We first show, at the classical level, the existence of S^3 = S^1 |x S^2 (S^1 fibered over S^2) NG modes associated to the breaking U(2) -> U(1) on vortices, where S^1 and S^2 correspond to type I and II NG modes, respectively. We then show, by using a Bethe ansatz technique, that the U(1) symmetry is restored, while the SU(2) symmery remains broken non-pertubatively at quantum level. Accordingly, the U(1) NG mode turns into a c=1 conformal field theory, the Tomonaga-Luttinger liquid, while the S^2 NG mode remains gapless, describing a ferromagnetic liquid. This allows the vortex to be genuinely non-Abelian at quantum level.

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.

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 the expectation values of a charge density and a local operator that break the same symmetry coexist.

We discuss spontaneous symmetry breaking and the Nambu-Goldstone (NG) modes at finite temperature and density. We focus on internal symmetry breaking for general systems including QCD at finite temperature and density. We show that there are two types of NG modes: type-A and type-B NG modes. The difference of these modes can be understood as the different motions: harmonic oscillation and precession motion. We also discuss the counting rule of NG modes and their dispersion relations.

We develop the effective field theory of diffusive Nambu-Goldstone (NG) modes associated with spontaneous internal symmetry breaking taking place in nonequilibrium open systems. The effective Lagrangian describing semi-classical dynamics of the NG modes is derived and matching conditions for low-energy coefficients are also investigated. Due to new terms peculiar to open systems, the associated NG modes show diffusive gapless behaviors in contrast to the propagating NG mode in closed systems. We demonstrate two typical situations relevant to the condensed matter physics and high-energy physics, where diffusive type-A or type-B NG modes appear.

The ground state of QCD with two flavors at a finite baryon chemical potential under rapid rotation is a chiral soliton lattice (CSL) of the η meson, consisting of a stack of sine-Gordon solitons carrying a baryon number, due to the anomalous coupling of the η meson to the rotation. In a large parameter region, the ground state becomes a non-Abelian CSL, in which due to the neutral pion condensation each η soliton decays into a pair of non-Abelian sine-Gordon solitons carrying S2 moduli originated from Nambu-Goldstone (NG) modes localized around it, corresponding to the spontaneously broken vector symmetry SU(2)V. There, the S2 modes of neighboring solitons are anti-aligned, and these modes should propagate in the transverse direction of the lattice due to the interaction between the S2 modes of neighboring solitons. In this paper, we calculate excitations including gapless NG modes and excited modes around non-Abelian and Abelian (η) CSLs, and find three gapless NG modes with linear dispersion relations (type-A NG modes): two isospinons (S2 modes) and a phonon corresponding to the spontaneously broken vector SU(2)V and translational symmetries around the non-Abelian CSL, respectively, and only a phonon for the Abelian CSL because of the recovering SU(2)V. We also find in the deconfined phase that the dispersion relation of the isospinons becomes of the Dirac type, i.e. linear even at large momentum.

We discuss the counting of Nambu-Goldstone (NG) modes associated with the spontaneous breaking of higher-form global symmetries. Effective field theories of NG modes are developed based on symmetry-breaking patterns, using a generalized coset construction for higher-form symmetries. We derive a formula of the number of gapless NG modes, which involves expectation values of the commutators of conserved charges, possibly of different degrees.

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.

In fermionic superfluids that are charge neutral, Nambu-Goldstone (NG) modes also known as Anderson-Bogoliubov modes emerge as a result of spontaneous symmetry breaking. Here, we discuss DC transport properties of such NG modes through a quantum point contact. We show that contrary to a naive view that enhancement of the phase stiffness may suppress transport of the NG modes, there must be an anomalous contribution that survives at low temperature. This contribution originates from the conversion process between the condensate and NG mode. We find that within the BCS regime the anomalous contribution is enhanced with increasing channel transmittance and attractive interaction, and leads to a temperature-dependent Lorenz number and absence of the bunching effect in current noise.

Nambu-Goldstone (NG) modes for 0-form and higher-form symmetries can become unstable in the presence of background fields. Examples include the instability of a photon with a time-dependent axion background or with a chirality imbalance, known as the chiral plasma instability, and the instability of a dynamical axion with a background electric field. We show that all these phenomena can be universally described by a symmetry algebra for 0-form and higher-form symmetries. We prove a counting rule for the number of unstable NG modes in terms of correlation functions of broken symmetry generators. Based on our unified description, we further give a simple new example where one of the NG modes associated with the spontaneous 0-form symmetry breaking $U(1) \times U(1) \to \{1\}$ becomes unstable.

We study topological defects in the Georgi-Machacek model in a hierarchical symmetry breaking in which extra triplets acquire vacuum expectation values before the doublet. We find a possibility of topologically stable non-Abelian domain walls and non-Abelian flux tubes (vortices) in this model. In the limit of the vanishing $U(1)_{\rm Y}$ gauge coupling in which the custodial symmetry becomes exact, the presence of a vortex spontaneously breaks the custodial symmetry, giving rise to $S^2$ Nambu-Goldstone (NG) modes localized around the vortex corresponding to non-Abelian fluxes. Vortices are continuously degenerated by these degrees of freedom, thereby called non-Abelian. By taking into account the $U(1)_{\rm Y}$ gauge coupling, the custodial symmetry is explicitly broken, the NG modes are lifted, and all non-Abelian vortices fall into a topologically stable $Z$-string. This is in contrast to the SM in which $Z$-strings are non-topological and are unstable in the realistic parameter region.Non-Abelian domain walls also break the custodial symmetry and are accompanied by localized $S^2$ NG modes. Finally, we discuss the existence of domain wall solutions bounded by flux tubes, where their $S^2$ NG modes match. The domain walls may quantum mechanically decay by creating a hole bounded by a flux tube loop, and would be cosmologically safe. Gravitational waves produced from unstable domain walls could be detected by future experiments

Influence of the Nambu-Goldstone (NG) mode on the energy-weighted sum (EWS) of the excitation strengths is analyzed, within the random-phase approximation (RPA). When a certain symmetry is broken at the mean-field level, a NG mode emerges in the RPA, which can be represented by canonical variables forming a two-dimensional Jordan block. A general formula is rederived which separates out the NG-mode contribution to the EWS, via the projection on the subspace directed by the NG mode. As examples, the formula is applied to the $E1$ excitation and the rotational excitations in nuclei, further confirming theoretical consistency of the RPA.

The Nambu-Goldstone modes on the exotic chiral condensed phase with chiral and tensor-type quark-antiquark condensates are investigated by using the two-point vertex functions. It is shown that one of the Nambu-Goldstone modes appears as a result of meson mixing. As is well known, another method to find the Nambu-Goldstone modes is given by the use of the algebraic commutation relations between broken generators and massless modes obtained through the spontaneous symmetry breaking. This method is adopted to the cases of the chiral symmetry breakings due to the tensor-type condensate and the inhomogeneous chiral condensate. The result obtained by the use of the meson two-point vertex functions is obviously reproduced in the case of the tensor-type condensate. Furthermore, we investigate the general rules for determining the broken symmetries and the Nambu-Goldstone modes algebraically. As examples, the symmetry breaking pattern and the Nambu-Goldstone modes due to the tensor-type condensate or the inhomogeneous chiral condensate are shown by adopting the general rules developed in this paper in the algebraic method. Published by the American Physical Society 2025

We consider the scenario in which the light Higgs scalar boson appears as the pseudo-Goldstone boson. We discuss examples in both condensed matter and relativistic field theory. In $^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{B}$ the symmetry breaking gives rise to four Nambu-Goldstone (NG) modes and 14 Higgs modes. At lower energy one of the four NG modes becomes the Higgs boson with a small mass. This is the mode measured in experiments with the longitudinal NMR, and the Higgs mass corresponds to the Leggett frequency ${M}_{\mathrm{H}}=\ensuremath{\hbar}{\mathrm{\ensuremath{\Omega}}}_{B}$. The formation of the Higgs mass is the result of the violation of the hidden spin-orbit symmetry at low energy. In this scenario the symmetry-breaking energy scale $\mathrm{\ensuremath{\Delta}}$ (the gap in the fermionic spectrum) and the Higgs mass scale ${M}_{\mathrm{H}}$ are highly separated: ${M}_{\mathrm{H}}\ensuremath{\ll}\mathrm{\ensuremath{\Delta}}$. On the particle physics side we consider the model inspired by the models of Refs. Cheng et al. [J. High Energy Phys. 08 (014) 095] and Fukano et al. [Phys. Rev. D 90, 055009 (2014)]. At high energies the SU(3) symmetry is assumed which relates the left-handed top and bottom quarks to the additional fermion ${\ensuremath{\chi}}_{L}$. This symmetry is softly broken at low energies. As a result the only $CP$-even Goldstone boson acquires a mass and may be considered as a candidate for the 125 GeV scalar boson. We consider a condensation pattern different from that typically used in top-seesaw models, where the condensate $⟨{\overline{t}}_{L}{\ensuremath{\chi}}_{R}⟩$ is off-diagonal. In our case the condensates are mostly diagonal. Unlike the work of Cheng et al. [J. High Energy Phys. 08 (014) 095] and Fukano et al. [Phys. Rev. D 90, 055009 (2014)], the explicit mass terms are absent and the soft breaking of SU(3) symmetry is given solely by the four-fermion terms. This reveals a complete analogy with $^{3}\mathrm{He}$, where there is no explicit mass term and the spin-orbit interaction has the form of the four-fermion interaction.