Topological Order

Abstract

Introductory accounts of matter often begin with four familiar phases: solid, liquid, gas, and plasma. At a more advanced level, one encounters many others, including numerous crystalline and liquid-crystalline phases, ferromagnets, antiferromagnets, and superfluids. Landau’s theory of spontaneous symmetry breaking provides a remarkably unified framework for understanding these apparently different forms of matter. Some quantum phases, however, lie beyond that framework. Examples include fractional quantum Hall states, quantum spin liquids, and superconducting states when the electromagnetic field is treated dynamically. These phases exhibit a distinct form of order—topological order. Topologically ordered systems can support quasiparticles with fractional or non-Abelian statistics, emergent gauge interactions, topology-dependent ground-state degeneracy, and robust boundary phenomena. Microscopically, topological order is a pattern of long-range quantum entanglement rather than a pattern of local symmetry breaking. Its systematic description therefore requires concepts such as topological quantum field theory, mapping-class-group representations, and braided fusion categories.