In physics, topological order[1] describes a state or phase of matter that arises in quantum mechanics.
Various topologically ordered states have interesting properties, such as (1) ground state degeneracy[3] and fractional statistics or non-abelian group statistics that can be used to realize a topological quantum computer; (2) perfect conducting edge states that may have important device applications; (3) emergent gauge field and Fermi statistics that suggest a quantum information origin of elementary particles;[4] (4) topological entanglement entropy that reveals the entanglement origin of topological order, etc.
For a long time, physicists believed that Landau Theory described all possible orders in materials, and all possible (continuous) phase transitions.
In an attempt to explain high temperature superconductivity[14] the chiral spin state was introduced.
[5][6] At first, physicists still wanted to use Landau symmetry-breaking theory to describe the chiral spin state.
This means that the chiral spin states contain a new kind of order that is beyond the usual symmetry description.
[16][17][18] New quantum numbers, such as ground state degeneracy[15] (which can be defined on a closed space or an open space with gapped boundaries, including both Abelian topological orders[19][20] and non-Abelian topological orders[21][22]) and the non-Abelian geometric phase of degenerate ground states,[1] were introduced to characterize and define the different topological orders in chiral spin states.
The fractional quantum Hall (FQH) state was discovered in 1982[9][10] before the introduction of the concept of topological order in 1989.
This kind of topological order corresponds to integral quantum Hall state, which can be characterized by the Chern number of the filled energy band if we consider integer quantum Hall state on a lattice.
Theoretical calculations have proposed that such Chern numbers can be measured for a free fermion system experimentally.
Current research works show that the loop and string like excitations exist for topological orders in the 3+1 dimensional spacetime, and their multi-loop/string-braiding statistics are the crucial signatures for identifying 3+1 dimensional topological orders.
[32] A large class of 2+1D topological orders is realized through a mechanism called string-net condensation.
[34] The condensations of other extended objects such as "membranes",[35] "brane-nets",[36] and fractals also lead to topologically ordered phases[37] and "quantum glassiness".
Quantum operator algebra is a very important mathematical tool in studying topological orders.
For example, ferromagnetic materials that break spin rotation symmetry can be used as the media of digital information storage.
Liquid crystals that break the rotational symmetry of molecules find wide application in display technology.
Crystals that break translation symmetry lead to well defined electronic bands which in turn allow us to make semiconducting devices such as transistors.
The non-locality means that the quantum entanglement in a topologically ordered state is distributed among many different particles.
The existence of topological order appears to indicate that nature is much richer than Landau symmetry-breaking theory has so far indicated.
However, in the presence of symmetry, even short range entangled states are nontrivial and can belong to different phases.
[50] SPT order generalizes the notion of topological insulator to interacting systems.
Some suggest that topological order (or more precisely, string-net condensation) in local bosonic (spin) models has the potential to provide a unified origin for photons, electrons and other elementary particles in our universe.