What is "topological insulator"

n. a material which conducts electrons only along its edges and surfaces, but does not conduct in its interior

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 Z 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).