User:ProbablyOrthogonal/Topological insulator

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A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor, meaning that electrons can only move along the surface of the material.

A topological insulator is an insulator for the same reason a "trivial" (ordinary) 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.

Since this results from a global property of the topological insulator's band structure, local disturbances cannot damage this surface state. This is unique to topological insulators: while ordinary insulators can also support conductive surface states, only the surface states of topological insulators have this robustness property.

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.

The properties of topological insulators and their surface states are highly dependent on both the dimension of the material and its underlying symmetries, and can be classified using the so-called periodic table of topological insulators. Some combinations of dimension and symmetries forbid topological insulators completely. All topological insulators have at least U(1) symmetry from particle number conservation, and often have time-reversal symmetry from the absence of a magnetic field. In this way, topological insulators are an example of symmetry-protected topological order. So-called "topological invariants", taking values in $$\mathbb{Z}_2$$ or $$\mathbb{Z}$$, allow classification of insulators as trivial or topological, and can be computed by various methods.

The surface states of topological insulators can have exotic properties. For example, in time-reversal symmetric 3D topological insulators, 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.

Despite their origin in quantum mechanical systems, analogues of topological insulators can also be found in classical media. There exist photonic, magnetic, and acoustic topological insulators, among others.

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Classification

Bloch's theorem allows a full characterization of the wave propagation properties of a material by assigning a matrix to each wave vector in the Brillouin zone.

Mathematically, this assignment creates a vector bundle. Different materials will have different wave propagation properties, and thus different vector bundles. If we consider all insulators (materials with a band gap), this creates a space of vector bundles. It is the topology of this space (modulo trivial bands) from which the "topology" in topological insulators arises.

Specifically, the number of connected components of the space indicates how many different "islands" of insulators exist amongst the metallic states. Insulators in the connected component containing the vacuum state are identified as "trivial", and all other insulators as "topological". The connected component in which an insulator lies can be identified with a number, referred to as a "topological invariant".

This space can be restricted under the presence of symmetries, changing the resulting topology. Although unitary symmetries are usually significant in quantum mechanics, they have no effect on the topology here. Instead, the three symmetries typically considered are time-reversal symmetry, particle-hole symmetry, and chiral symmetry (also called sublattice symmetry). Mathematically, these are represented as, respectively: an anti-unitary operator which commutes with the Hamiltonian; an anti-unitary operator which anti-commutes with the Hamiltonian; and a unitary operator which anti-commutes with the Hamiltonian. All combinations of the three together with each spatial dimension result in the so-called periodic table of topological insulators.