topological order

The fractional quantum Hall effect, discovered in 1982 by Tsui, Störmer, and Gossard, provided the first experimental evidence of topologically ordered states of matter. The excitations in these systems carry fractional electric charge and exhibit anyonic statistics—neither bosonic nor fermionic—which cannot be explained by the conventional Landau symmetry-breaking framework.

The concept was formally introduced by physicist Xiao-Gang Wen in the late 1980s and early 1990s. Wen used the term to describe states of matter, such as fractional quantum Hall states, that are distinguished not by local symmetries but by global topological properties of their ground states.

Topological quantum computing aims to use anyons—quasiparticles found in topologically ordered systems—to encode and manipulate quantum information. Because the information is stored non-locally in the topological properties of the system, it is theoretically immune to local noise and decoherence, potentially solving one of the biggest challenges in building practical quantum computers.

Conventional phases of matter are classified by Landau's symmetry-breaking theory, where phase transitions involve changes in local symmetries. Topologically ordered phases cannot be distinguished by any local measurement—two phases can have identical local symmetries but different global topological properties, meaning the distinction is encoded in the system's overall structure rather than at any specific point.