User:ProbablyOrthogonal/topological insulators

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.

A topological insulator is an insulator for the same reason a "trivial" insulator is: there exists an energy gap between the valence and conduction bands of the material. But in a topological insulator, these bands are, in an informal sense, "twisted", relative to a trivial insulator. The topological insulator cannot be continuously transformed into a trivial one without untwisting the bands, which closes the band gap and creates a conducting state. Thus, due to the continuity of the underlying field, the border of a topological insulator with a trivial insulator (including vacuum, which is topologically trivial) is forced to support a conducting state.

This leads to a more formal definition of a topological insulator: an insulator which cannot be adiabatically transformed into an ordinary insulator without passing through an intermediate conducting state. In other words, topological insulators and trivial insulators are separate regions in the phase diagram, connected only by conducting phases. In this way, topological insulators provide an example of a state of matter not described by the Landau symmetry-breaking theory that defines ordinary states of matter. This definition is slightly more robust, since ordinary insulators can also support conductive surface states.

The properties of topological insulators and their surface states are highly dependent on both the dimension of the material and its underlying symmetries. Some combinations of dimension and symmetries forbid topological insulators completely. If a topological insulator requires the presence of symmetries to distinguish it from trivial insulators, then it has non-trivial symmetry-protected topological order.

Traditional topological insulators are 2 or 3 dimensional, with time-reversal symmetry guaranteed by the absence of a magnetic field. In these materials, 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 "U"-turn scattering is strongly suppressed and conduction on the surface is highly metallic.

However, despite their origin in quantum mechanical systems, topological insulators are an entirely classical phenomenon. The mathematical properties behind topological insulators can be found in most systems that support wave propagation, meaning that there exist photonic, magnetic, and acoustic topological insulators. The medium may have different underlying symmetries that lead to different behavior (e.g. of surface states), but the principles remain the same.

References

https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.88.035005

https://arxiv.org/pdf/0901.2686.pdf

https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.83.1057

https://arxiv.org/abs/1002.3895