We traditionally classify order in many body systems through the lens of Landau theory where the symmetry at the microscopic level is broken to a lower symmetry in the ground states, e.g. ordering of spins along one direction in the Ising model. In topological systems the opposite occurs, the ground states have emergent symmetries not present in the microscopic equations of motion. This has enormous consequences amoung them being that this state of matter is characterized by more than just the energy and few body correlation functions but also by the entropy of subsystems in the ground state.

Early on TO states were proposed as ground states for high temperature superconductors and for fractional quantum Hall states. It was also realized that such states could support excitations that behave as anyons in two dimensions. In normal three dimensional space (and in fact in any higher dimensional space) particles come in two types bosons and fermions according to whether the wavefunction accumulates a +1 or -1 phase under exchange of a pair of identical particles. In two dimensions more possibilities arise such as the accumulation of an arbitrary phase for the case of Abelian anyons, or a matrix valued action for non-Abelian anyons. In a landmark 1997 paper A. Kitaev showed that by manipulating the anyons one could perform quantum computation in an essentially error free way. Such a physically protected quantum computer would be a tremendous boon for experimental realizations because it would avoid some of the large resource overheads of more standard techniques that use active error correction.

There are a variety of physical media that generate topologically ordered systems and anyons including: two dimensional electron gases in the fractional quantum Hall regime, and 2D and 3D spin lattices realized with Josephson junction arrays or trapped atoms/molecules in optical lattices.

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