Topological order

Topological order is a type of quantum order characterized by global properties and long-range entanglement rather than a local order parameter. It can produce robust ground-state degeneracy and excitations with exotic exchange statistics.

Quantum entanglement: A physical relationship in which a joint quantum state cannot be separated into independent states of its parts. Long-range entanglement is the defining feature distinguishing this order from ordinary local order.
Anyon: A quasiparticle in two dimensions whose exchange statistics can differ from those of bosons and fermions. Anyonic excitations are characteristic of many topologically ordered phases.
Fractional quantum Hall effect: A two-dimensional electron phenomenon in which Hall conductance forms fractional quantized plateaus under strong magnetic fields. Its fractional states host anyons and provide the clearest experimental setting for topological order.
Topological quantum computing: A quantum-computing approach that stores and processes information using topologically protected states and operations. Nonlocal encoding in topological phases can protect quantum information from many local errors.
Landau theory of phase transitions: A framework that describes phase transitions through symmetry breaking and associated order parameters. It captures conventional ordered phases but does not classify intrinsic topological order.
Quantum phase of matter: A stable state of matter whose properties are governed by quantum mechanics, often at low temperature. Topological order classifies phases that cannot be described by symmetry breaking alone.
Topological ground-state degeneracy: A degeneracy of ground states that depends on the topology of space and is robust against local perturbations. Its dependence on spatial topology expresses the global character of the order.
Kitaev honeycomb model: An exactly solvable spin model on a honeycomb lattice with bond-dependent interactions and emergent Majorana excitations. Its gapped phases illustrate how a simple spin Hamiltonian can support topological order.
Topological quantum error correction: Error correction that encodes quantum information in global degrees of freedom associated with a lattice or topology. The toric code turns topological order into a model for protecting encoded information.
Symmetry-protected topological phase: A gapped phase that is nontrivial only while a protecting symmetry is preserved and has no intrinsic anyonic order. It is topological in a broader sense but lacks the intrinsic long-range entanglement of topological order.

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