Abstract
The low-energy and low-momentum dynamics of systems with a spontaneously broken continuous symmetry is dominated by the ensuing Nambu-Goldstone bosons. It can be conveniently encoded in a model-independent effective field theory whose structure is fixed by symmetry up to a set of effective coupling constants. We construct the most general effective Lagrangian for the Nambu-Goldstone bosons of spontaneously broken global internal symmetry up to the fourth order in derivatives. Rotational invariance and spatial dimensionality of one, two or three are assumed in order to obtain compact explicit expressions, but our method is completely general and can be applied without modifications to condensed matter systems with a discrete space group as well as to higher-dimensional theories. The general low-energy effective Lagrangian for relativistic systems follows as a special case. We also discuss the effects of explicit symmetry breaking and classify the corresponding terms in the Lagrangian. Diverse examples are worked out in order to make the results accessible to a wide theoretical physics community.
Highlights
The methods of effective field theory (EFT) have proven invaluable across a range of disciplines as a tool for simplifying practical calculations in systems with two or more widely separated energy or length scales [1, 2]
There is a large class of physical systems where this task can be accomplished at once: whenever a global continuous symmetry is spontaneously broken, the spectrum of the theory contains gapless excitations, the Nambu-Goldstone (NG) bosons
Provided there are no other soft modes in the spectrum, not associated with symmetry, the low-energy dynamics is dominated by the NG bosons alone
Summary
The methods of effective field theory (EFT) have proven invaluable across a range of disciplines as a tool for simplifying practical calculations in systems with two or more widely separated energy or length scales [1, 2]. Physical observables at long distances can be determined using an EFT that respects the correct symmetries and only includes the low-energy degrees of freedom. A major problem in the construction of an EFT is the choice of the appropriate degrees of freedom. There is a large class of physical systems where this task can be accomplished at once: whenever a global continuous symmetry is spontaneously broken, the spectrum of the theory contains gapless excitations, the Nambu-Goldstone (NG) bosons. Provided there are no other soft modes in the spectrum, not associated with symmetry, the low-energy dynamics is dominated by the NG bosons alone. This assumption will be implicit throughout the remainder of the paper
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