An Introduction to Spontaneous Symmetry Breaking

TL;DR

This paper provides an accessible introduction to spontaneous symmetry breaking, explaining its fundamental concepts, physical implications, and examples across high energy and condensed matter physics, emphasizing the role of singular limits.

Contribution

It offers a comprehensive, graduate-level overview of spontaneous symmetry breaking, including recent developments and diverse applications in physics.

Findings

Clarifies the concept of symmetry breaking in finite systems

Explores the physics of Nambu-Goldstone modes and topological defects

Connects symmetry breaking phenomena across different physics domains

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.