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Counting Rules of Nambu-Goldstone Modes
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 formulas for NGMs associated with
broken internal symmetries that work equally for relativistic and
nonrelativistic systems. We also discuss complexities of NGMs originating from
space-time symmetry breaking. In the process we cover many illuminating
examples from various context. We also present a complementary point of view
from the Lieb-Schultz-Mattis theorem.Comment: 14 pages, 1 figure. Invited review for the Annual Review of Condensed
Matter Physics; Title change
Effective Lagrangian for Nonrelativistic Systems
The effective Lagrangian for Nambu-Goldstone bosons (NGBs) in systems without
Lorentz invariance has a novel feature that some of the NGBs are canonically
conjugate to each other, hence describing dynamical degree of freedom by
two NGB fields. We develop explicit forms of their effective Lagrangian up to
the quadratic order in derivatives. We clarify the counting rules of NGB
degrees of freedom and completely classify possibilities of such canonically
conjugate pairs based on the topology of the coset spaces. Its consequence on
the dispersion relations of the NGBs is clarified. We also present simple
scaling arguments to see whether interactions among NGBs are marginal or
irrelevant, which justifies a lore in the literature about the possibility of
symmetry breaking in dimensions.Comment: 34 pages, 5 tables, 2 figures; v2: published version (to appear in
Physical Review X
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