Spontaneous Symmetry Breaking and Nambu-Goldstone Bosons in Quantum Many-Body Systems

TL;DR

This paper reviews how spontaneous symmetry breaking manifests in non-Lorentz invariant quantum many-body systems, detailing Goldstone bosons, effective theories, and specific examples across physics domains.

Contribution

It provides a comprehensive, self-contained review of symmetry breaking phenomena in nonrelativistic and nonzero density systems, including counting rules and effective Lagrangians.

Findings

Derived counting rules for Nambu-Goldstone bosons.

Analyzed dispersion relations in various systems.

Constructed effective Lagrangians for broken symmetries.

Abstract

Spontaneous symmetry breaking is a general principle, that constitutes the underlying concept of a vast number of physical phenomena ranging from ferromagnetism and superconductivity in condensed matter physics to the Higgs mechanism in the standard model of elementary particles. I focus on manifestations of spontaneously broken symmetries in systems that are not Lorentz invariant, which include both, nonrelativistic systems as well as relativistic systems at nonzero density, providing a self-contained review of the properties of spontaneously broken symmetries specific to such theories. Topics covered include: (i) Introduction to the mathematics of spontaneous symmetry breaking and the Goldstone theorem. (ii) Minimization of Higgs-type potentials for higher-dimensional representations. (iii) Counting rules for Nambu-Goldstone bosons and their dispersion relations. (iv) Construction of effective Lagrangians. Specific examples in both relativistic and nonrelativistic physics are worked out in detail.