Topological insulators are strongly spin-orbit-coupled materials that have seemingly mutually exclusive properties: they are both insulating and metallic at the same time. In their interior (bulk), electrons cannot propagate, whereas their surfaces are highly conducting. To get an insight into the complicated theory of these exotic materials, let us recall that electrons in an insulator fully occupy a certain number of allowed energies (bands) in such a way that the highest occupied state is separated from the lowest empty one by a gap of forbidden energies.
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