Effective Lagrangians for quantum many-body systems

TL;DR

This paper develops a comprehensive, model-independent effective field theory for Nambu-Goldstone bosons arising from spontaneously broken symmetries, applicable across various physical systems and dimensions.

Contribution

It constructs the most general effective Lagrangian up to fourth order in derivatives for Nambu-Goldstone bosons, including explicit symmetry breaking effects, with broad applicability.

Findings

Derived explicit Lagrangian expressions for different dimensions

Classified symmetry-breaking terms in the effective theory

Provided diverse examples for practical application

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.