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A topological insulator is a material with non-trivial topological order that behaves as an insulator in its interior but whose surface contains conducting states, meaning that electrons can only move along the surface of the material. However, the conducting surface is not the unique character of topological insulator, since the ordinary band insulators can also support conductive surface states. What is special is that the surface states of topological insulators are symmetry protected by particle number conservation and time reversal symmetry.
In the bulk of a non-interacting topological insulator, the electronic band structure resembles an ordinary band insulator, with the Fermi level falling between the conduction and valence bands. On the surface of a topological insulator there are special states that fall within the bulk energy gap and allow surface metallic conduction. Carriers in these surface states have their spin locked at a right-angle to their momentum (spin-momentum locking). At a given energy the only other available electronic states have different spin, so the "U"-turn scattering is strongly suppressed and conduction on the surface is highly metallic. Non-interacting topological insulators are characterized by an index (known as Z2 topological invariants) similar to the genus in topology.
The "protected" conducting states in the surface are required by time-reversal symmetry and the band structure of the material. The states cannot be removed by surface passivation if it does not break the time-reversal symmetry, which does not happen with potential and/or spin-orbit scattering, but happens in case of true magnetic impurities (e.g. spin-cattering).
Prediction and discovery
Time-reversal symmetry protected edge states were predicted in 1987 to occur in quantum wells (very thin layers) of mercury telluride sandwiched between cadmium telluride and were observed in 2007. In 2007, they were predicted to occur in three-dimensional bulk solids of binary compounds involving bismuth. A 3D "strong topological insulator" exists which cannot be reduced to multiple copies of the quantum spin Hall state. The first experimentally realized 3D topological insulator state (symmetry protected surface states) was discovered in bismuth-antimony. Shortly thereafter symmetry protected surface states were also observed in pure antimony, bismuth selenide, bismuth telluride and antimony telluride using ARPES. Many semiconductors within the large family of Heusler materials are now believed to exhibit topological surface states. In some of these materials the Fermi level actually falls in either the conduction or valence bands due to naturally occurring defects, and must be pushed into the bulk gap by doping or gating. The surface states of a 3D topological insulator is a new type of 2DEG (two-dimensional electron gas) where the electron's spin is locked to its linear momentum.
Fully bulk insulating or intrinsic 3D topological insulator states exist in Bi-based materials. 
In 2012 several groups released preprints which suggest that samarium hexaboride has the properties of a topological insulator in accordance with theoretical predictions. Since samarium hexaboride is an established Kondo insulator, i.e. a strongly correlated electron material, the existence of a topological surface state in this material would lead to a topological insulator with strong electronic correlations.
Properties and applications
The spin-momentum locking in the topological insulator allows symmetry protected surface states to host Majorana particles if superconductivity is induced on the surface of 3D topological insulators via proximity effects. (Note that Majorana zero-mode can also appear without topological insulators.) The non-trivialness of topological insulators is encoded in the existence of a gas of helical Dirac fermions. Helical Dirac fermions, which behave like massless relativistic particles, have been observed in 3D topological insulators.
Note that the gapless surface states of topological insulator differ from those in the Quantum Hall effect: the gapless surface states of topological insulator are symmetry protected (i.e. not topological), while the gapless surface states in Quantum Hall effect are topological (i.e. robust against any local perturbations that can break all the symmetries).
The Z2 topological invariants cannot be measured using traditional transport methods, such as spin Hall conductance, and the transport is not quantized by the Z2 invariants. An experimental method to measure Z2 topological invariants was demonstrated which provide a measure of the Z2 topological order. (Note that the term Z2 topological order has also been used to describe the topological order with emergent Z2 gauge theory discovered in 1991.)
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