Applications
The materials described by Landau symmetry-breaking theory have had a substantial impact on technology. For example, Ferromagnetic materials that break spin rotation symmetry can be used as the media of digital information storage. A hard drive made of ferromagnetic materials can store gigabytes of information. Liquid crystals that break the rotational symmetry of molecules find wide application in display technology; nowadays one can hardly find a household without a liquid crystal display somewhere in it. Crystals that break translation symmetry lead to well defined electronic bands which in turn allow us to make semiconducting devices such as transistors. Different types of topological orders are even richer than different types of symmetry-breaking orders. This suggests their potential for exciting, novel applications.
One theorized application would be to use topologically ordered states as media for quantum computing in a technique known as topological quantum computing. A topologically ordered state is a state with complicated non-local quantum entanglement. The non-locality means that the quantum entanglement in a topologically ordered state is distributed among many different particles. As a result, the pattern of quantum entanglements cannot be destroyed by local perturbations. This significantly reduces the effect of decoherence. This suggests that if we use different quantum entanglements in a topologically ordered state to encode quantum information, the information may last much longer. The quantum information encoded by the topological quantum entanglements can also be manipulated by dragging the topological defects around each other. This process may provide a physical apparatus for performing quantum computations. Therefore, topologically ordered states may provide natural media for both quantum memory and quantum computation. Such realizations of quantum memory and quantum computation may potentially be made fault tolerant.
Topologically ordered states in general have a special property that they contain non-trivial boundary states. In many cases, those boundary states become perfect conducting channel that can conduct electricity without generating heat. This can be another potential application of topological order in electronic devices.
Similar to topological order, topological insulators also have gapless boundary states. The boundary states of topological insulators play a key role in the detection and the application of topological insulators. This observation naturally leads to a question: are topological insulators examples of topologically ordered states? In fact topological insulators are different from topologically ordered states defined in this article. Topological insulators only have short-ranged entanglements, while the topological order defined in this article is a pattern of long-range entanglement. Topological order is robust against any perturbations. It has emergent gauge theory, emergent fractional charge and fractional statistics. In contrast, topological insulators are robust only against perturbations that respect time-reversal and U(1) symmetries. Their quasi-particle excitations have no fractional charge and fractional statistics. Strictly speaking, topological insulator is an example of SPT order, where the first example of SPT order is the Haldane phase of spin-1 chain.
Read more about this topic: Topological Order