A topological insulator is a material 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. Topological insulators have non-trivial symmetry-protected topological order; however, having a conducting surface is not unique to topological insulators, since ordinary band insulators can also support conductive surface states. What is special about topological insulators is that their surface states are symmetry-protected Dirac fermions by particle number conservation and time-reversal symmetry. In two-dimensional (2D) systems, this ordering is analogous to a conventional electron gas subject to a strong external magnetic field causing electronic excitation gap in the sample bulk and metallic conduction at the boundaries or surfaces.The distinction between 2D and 3D topological insulators is characterized by the Z-2 topological invariant, which defines the ground state. In 2D, there is a single Z-2 invariant distinguishing the insulator from the quantum spin-Hall phase, while in 3D, there are four Z-2 invariant that differentiate the insulator from “weak” and “strong” topological insulators.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
2
{\displaystyle \mathbb {Z} _{2}}
topological invariants) similar to the genus in topology.As long as time-reversal symmetry is preserved (i.e., there is no magnetism), the
Z
2
{\displaystyle \mathbb {Z} _{2}}
index cannot change by small perturbations and the conducting states at the surface are symmetry-protected. On the other hand, in the presence of magnetic impurities, the surface states will generically become insulating. Nevertheless, if certain crystalline symmetries like inversion are present, the
Z
2
{\displaystyle \mathbb {Z} _{2}}
index is still well defined. These materials are known as magnetic topological insulators and their insulating surfaces exhibit a half-quantized surface anomalous Hall conductivity.
Photonic topological insulators are the classical-wave electromagnetic counterparts of (electronic) topological insulators, that provide unidirectional propagation of electromagnetic waves.