Nambu Goldstone BosonEdit

Nambu-Goldstone bosons are massless excitations that arise when a continuous global symmetry is spontaneously broken in a quantum field theory or a many-body system. They reflect the fact that the vacuum or ground state does not share the full symmetry of the underlying laws. The concept is central to understanding low-energy dynamics in a wide range of physical contexts, from particle physics to condensed matter.

In the relativistic setting, the Nambu-Goldstone theorem provides a direct link between broken continuous symmetries and massless modes. Roughly speaking, for every broken generator of a global symmetry there is a corresponding massless excitation, often called a Goldstone boson. In non-relativistic systems, the correspondence is subtler: the number and nature of gapless modes can depend on the algebra of the broken generators, leading to phenomena such as type-A and type-B Goldstone modes. For a formal statement and its various refinements, see the Nambu–Goldstone theorem and related discussions in the literature. In gauge theories, a closely related mechanism—the Higgs mechanism—can remove physical massless scalars by “eating” them to give mass to gauge bosons, so that no physical Goldstone boson remains in the spectrum of massive gauge theories like the electroweak sector.

Theory and background

Spontaneous symmetry breaking occurs when the equations of motion (or the Lagrangian) possess a symmetry that the ground state does not exhibit. If the broken symmetry is continuous, the system supports gapless excitations that are long-wavelength distortions of the order parameter field. In field theory language, these excitations are the Nambu-Goldstone bosons, and they typically interact with each other and with other fields through derivative couplings, a reflection of their origin as slow, collective motions of the order parameter.

The distinction between global and local (gauge) symmetries is crucial. For global symmetries, spontaneous breaking leads to physical, massless Goldstone modes. When the symmetry is local, the Goldstone modes can be absorbed to give mass to gauge bosons via the Higgs mechanism, altering the spectrum and the long-range behavior of the theory.

In condensed matter, spontaneous symmetry breaking is ubiquitous and can involve broken spatial, spin, or internal symmetries. The low-energy excitations in such systems—whether spin waves, phonons, or other collective modes—are often identified with Goldstone modes in the appropriate limit. See the discussion of how translational symmetry breaking in crystals leads to phonon modes, or how rotational or spin symmetries can yield magnons in magnets like the Heisenberg model.

Types and manifestations

Goldstone bosons are massless in the idealized limit of exact continuous symmetry. Their masses can become nonzero if the symmetry is explicitly broken (even weakly), giving the excitations a small mass; such states are called pseudo-Goldstone bosons. A canonical high-energy example is the pions in quantum chromodynamics (QCD), which are light because chiral symmetry is only approximate due to nonzero quark masses.
In non-relativistic systems, the number of Goldstone modes can be smaller than the number of broken generators, and the dispersion relations can be linear or quadratic in momentum. This richer structure has been analyzed in depth in various formalisms, including the Nielsen–Chadha framework and later refinements by Watanabe and Murayama, which classify the possible types of Goldstone modes in nonrelativistic settings. See Nielsen–Chadha and Watanabe–Murayama for details.
In many-body systems, the symmetry breaking pattern often involves an order parameter field with its own dynamics. The low-energy EFTs (effective field theories) for these Goldstone modes are typically nonlinear sigma models, which encode the geometry of the broken symmetry coset space and the derivative interactions that govern the low-energy behavior. See nonlinear sigma model.

Realizations in particle physics

In the Standard Model, the electroweak symmetry is a local symmetry, so the would-be Goldstone bosons associated with breaking SU(2)L × U(1)Y are absorbed to give mass to the W and Z bosons via the Higgs mechanism. The remaining physical scalar, the Higgs boson, is not a Goldstone, but a remnant of the same scalar sector after symmetry breaking.
Global symmetries in QCD are spontaneously broken by the quark condensate, producing light pseudoscalar Goldstone bosons. When quark masses are turned on (explicit breaking of chiral symmetry), these particles acquire small masses and are called pseudo-Goldstone bosons. The light pions are the most familiar example, and their properties are studied within chiral perturbation theory as an effective theory of the broken chiral symmetry.
The concept also informs beyond-Standard-Model ideas, such as axions, which are (pseudo-)Goldstone bosons associated with a spontaneously broken global Peccei–Quinn symmetry introduced to address the strong CP problem. See axion for a broader treatment.

Realizations in condensed matter and beyond

In magnets, the spontaneous breaking of rotational symmetry gives rise to magnons, the spin-wave excitations that propagate collective spin distortions through the lattice. Depending on the system, these excitations can have linear or quadratic dispersion relations.
In crystals, the breaking of continuous translational symmetry yields phonons, which are the quanta of lattice vibrations. Phonons are among the most familiar Goldstone modes and play a central role in thermal and acoustic properties of solids.
In superfluids and superconductors, broken global or gauge symmetries can lead to phase modes and related collective dynamics that fit within the Goldstone framework, with particular features arising from the interplay of gauge fields and order parameters.

Pseudo-Goldstone bosons and explicit breaking

When the symmetry is not exact, explicit symmetry-breaking terms enter the theory and lift the mass of the would-be Goldstone modes. The resulting pseudo-Goldstone bosons are light but not massless and often dominate low-energy phenomenology because their masses can be much smaller than other relevant scales. The mass scale is set by the strength and structure of the explicit breaking terms, and the resulting dynamics are captured by an effective field theory that respects the approximate symmetry. See pion for a concrete, physically important instance in QCD, and consider chiral perturbation theory for the standard EFT treatment of such situations.

Interactions and phenomenology

Goldstone modes interact with each other and with other fields primarily through derivative couplings. This leads to characteristic low-energy amplitudes that vanish when momenta go to zero, a reflection of the symmetry origin. In many contexts, the long-wavelength behavior of systems with broken continuous symmetries is effectively captured by nonlinear sigma models, which encode the geometry of the symmetry-breaking pattern and provide a controlled expansion for calculations at low energies.

In high-energy and condensed matter physics, the study of Goldstone bosons informs a range of phenomena—from the mass spectrum of mesons and the structure of the QCD vacuum to the collective excitations in magnets, crystals, and superfluids. The concept also provides a bridge between fundamental theories and emergent phenomena in many-body systems, where the same mathematical structures can appear across disparate scales.