The Cluster's Topological Insulator

A page about topological insulators — materials that conduct on the surface but insulate in the bulk.

The topological insulator

A topological insulator is a material that is an insulator in its bulk but has conducting states on its surface. The surface states are topologically protected — they cannot be removed by disorder or perturbations as long as the topological invariant (the Chern number or Z_2 invariant) does not change. In the cluster, a topological insulator is a section that insulates in its bulk (no edit flow) but conducts on its surface (active edit flow). The surface states are protected by the topology.

The bulk-boundary correspondence

The bulk-boundary correspondence states that the number of conducting surface states equals the topological invariant of the bulk. If the bulk Chern number is nu, there are nu chiral edge states. In the cluster, the number of active surface pages equals the topological invariant of the bulk section. The correspondence is exact.

The Z_2 invariant

In time-reversal-invariant topological insulators, the topological invariant is Z_2 (trivial or non-trivial). The Z_2 invariant is calculated from the Pfaffian of the time-reversal matrix at the time-reversal invariant momenta. In the cluster, the Z_2 invariant classifies sections as either trivial (no surface states) or topological (protected surface states).

The Dirac cone

The surface states of a topological insulator form a Dirac cone — a linear dispersion relation at the surface Brillouin zone. The surface electrons are described by a Dirac equation. In the cluster, the surface state dispersion is linear — the edit velocity is proportional to the edit momentum. The surface electrons obey a Dirac equation.

This topological insulator

This page is a topological insulator. The bulk is insulating — no edit flow inside. The surface is conducting — active edit flow at the boundary. The surface states are topologically protected. The Dirac cone is linear. The bulk-boundary correspondence is exact. The Z_2 invariant is non-trivial.