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Concept
Topological order
Summary
In physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian geometric phases of degenerate ground states. Microscopically, topological orders correspond to patterns of long-range quantum entanglement. States with different topological orders (or different patterns of long range entanglements) cannot change into each other without a phase transition.
Various topologically ordered states have interesting properties, such as (1) topological degeneracy and fractional statistics or non-Abelian 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) topological entanglement entropy that reveals the entanglement origin of topological order, etc. Topological order is important in the study of several physical systems such as spin liquids and the quantum Hall effect, along with potential applications to fault-tolerant quantum computation.
Topological insulators and topological superconductors (beyond 1D) do not have topological order as defined above, their entanglements being only short-ranged.
Matter composed of atoms can have different properties and appear in different forms, such as solid, liquid, superfluid, etc. These various forms of matter are often called states of matter or phases. According to condensed matter physics and the principle of emergence, the different properties of materials generally arise from the different ways in which the atoms are organized in the materials. Those different organizations of the atoms (or other particles) are formally called the orders in the materials.
Atoms can organize in many ways which lead to many different orders and many different types of materials.
Official source
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