An introduction to spontaneous symmetry breaking

TLDR

Symmetry in physics can be broken spontaneously, meaning a state need not share the symmetries of its governing theory. The lecture notes aim to introduce symmetry breaking and its consequences, beginning with a precise definition of symmetry in physics. They cover the physics of singular limits, Nambu‑Goldstone modes, quantum corrections, phase transitions, topological defects, gauge fields, and are tailored for graduate students. The notes illustrate symmetry breaking with numerous examples from high‑energy and condensed‑matter physics.

Abstract

Perhaps the most important aspect of symmetry in physics is the idea that a state does not need to have the same symmetries as the theory that describes it. This phenomenon is known as spontaneous symmetry breaking. In these lecture notes, starting from a careful definition of symmetry in physics, we introduce symmetry breaking and its consequences. Emphasis is placed on the physics of singular limits, showing the reality of symmetry breaking even in small-sized systems. Topics covered include Nambu-Goldstone modes, quantum corrections, phase transitions, topological defects and gauge fields. We provide many examples from both high energy and condensed matter physics. These notes are suitable for graduate students.