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In physics, topological order is a kind of order in zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined/described by robust ground state degeneracy and quantized non-Abelian geometric phases of degenerate ground states〔 (just like superfluid order is defined/described by vanishing viscosity and quantized vorticity). Microscopically, topological order corresponds to patterns of long-range quantum entanglement (just like superfluid order corresponds to boson condensation). States with different topological orders (or different patterns of long range entanglements) cannot change into each other without a phase transition. Topologically ordered states have some scientifically interesting properties, such as ground state degeneracy that cannot be lifted by any local perturbations but depends on the topology of space, quasiparticle fractional statistics and fractional charges, perfect conducting edge states even in presence of magnetic impurities, topological entanglement entropy, etc. Topological order is important in the study of several physical systems such as spin liquids,〔V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett., 59, 2095 (1987), "Equivalence of the resonating-valence-bond and fractional quantum Hall states"〕〔Xiao-Gang Wen, F. Wilczek and A. Zee, Phys. Rev., B39, 11413 (1989), "Chiral Spin States and Superconductivity"〕 the quantum Hall effect, along with potential applications to fault-tolerant quantum computation. We note that topological insulators and topological superconductors (beyond 1D) do not have topological order as defined above (see discussion below). ==Background== Although all matter is formed by atoms, matter can have different properties and appear in different forms, such as solid, liquid, superfluid, magnet, 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 originate 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. Landau symmetry-breaking theory provides a general understanding of these different orders. It points out that different orders really correspond to different symmetries in the organizations of the constituent atoms. As a material changes from one order to another order (i.e., as the material undergoes a phase transition), what happens is that the symmetry of the organization of the atoms changes. For example, atoms have a random distribution in a liquid, so a liquid remains the same as we displace it by an arbitrary distance. We say that a liquid has a ''continuous translation symmetry''. After a phase transition, a liquid can turn into a crystal. In a crystal, atoms organize into a regular array (a lattice). A lattice remains unchanged only when we displace it by a particular distance (integer times of lattice constant), so a crystal has only ''discrete translation symmetry''. The phase transition between a liquid and a crystal is a transition that reduces the continuous translation symmetry of the liquid to the discrete symmetry of the crystal. Such change in symmetry is called ''symmetry breaking''. The essence of the difference between liquids and crystals is therefore that the organizations of atoms have different symmetries in the two phases. Landau symmetry-breaking theory is a very successful theory. For a long time, physicists believed that Landau symmetry-breaking theory describes all possible orders in materials, and all possible (continuous) phase transitions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「topological order」の詳細全文を読む スポンサード リンク
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