Counting Rules of Nambu–Goldstone Modes

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 formulae for NGMs associated with broken internal symmetries that work equally for relativistic and nonrelativistic systems. We also discuss complexities of NGMs originating from spacetime symmetry breaking. Along the way we cover many illuminating examples from various context. We also present a complementary point of view from the Lieb-Schultz-Mattis theorem.

Magnetophonons & type-B Goldstones from hydrodynamics to holography

“…(4.14). Moreover, we have verified numerically that 18) which comes simply from the fact that our system is invariant under time-reversal and it is imposed by so-called Onsager constraints. Following [24], one can derive simple formulas for the parametersγ andσ q :…”

“…For an introductory review about this generalized counting criterion see [18]. The most famous example in this category is that of the ferromagnet, in contrast to the antiferromagnet.…”

An introduction to spontaneous symmetry breaking

“…It cannot yet be found in any of the standard text books, but is readily accessible through either the original literature in Refs [36][37][38][39][40][41][42][43], or in the short review of Ref. [44].…”

Gapped dilatons in scale invariant superfluids

“…It is an interesting open question to study whether and how these phonons would coexist with a dilatonic mode [49]. The relevance of the question is threefold: it relates to the counting problem of NG modes for spacetime symmetries [9,16]; it concerns condensed matter systems where a critical scaling and the breaking of translations are intertwined 10 ; it provides insight regarding holographic models where scaling and translation symmetries are broken together [51][52][53][54]. 11 An example of inhomogeneous breaking of spatial translations in field theory was studied in [48].…”