Order parameters, broken symmetry, and topology

Abstract

This chapter introduces order parameters -- the reduction of a complex system of interacting particles into a few fields that describe the local equilibrium behavior at each point in the system. It introduces an organized approach to studying a new material system -- identify the broken symmetries, define the order parameter, examine the elementary excitations, and classify the topological defects. It uses order parameters to describe crystals and liquid crystals, superfluids and magnets. It touches upon broken gauge symmetries and the Anderson/Higgs mechanism and an analogue to braiding of non-abelian quantum particles. Exercises explore sound, second sound, and Goldstone’s theorem; fingerprints and soccer balls; Landau theory and other methods for generating emergent theories from symmetries and commutation relations; topological defects in magnets, liquid crystals, and superfluids, and defect entanglement.