Topological Insulators#
Last update: 2026-04-27
Topological band theory, which was founded in the 1980s and substantially developed in the 1990s, experienced another tremendous surge of interest in the 2000s. It enabled researchers not only to predict but also to experimentally discover a remarkable class of materials known as topological insulators (TIs) [Bernevig2013]. Both normal insulators (NIs) and TIs possess valence and conduction bands, which are filled and empty, respectively, at temperatures much lower than the energy gap separating them. However, the key difference lies in the structure of these bands: in a topological insulator (TI), the bands are reversed compared to those in a normal insulator (NI), meaning that the usual ordering of electronic states surrounding the band gap (their orbital character) is swapped. As a result, one cannot continuously transform the band structure of a NI into that of a TI without closing the energy gap. In other words, the two materials are separated by a topological phase transition, because no continuous transformation, whcih preserves symmetry and gap, connects them; they belong to different topological classes.
Particularly interesting are the properties of interfaces between TIs and NIs (or vacuum). At these interfaces, special states inevitably emerge as a consequence of the topological difference between the materials. These states can carry electric charge along the boundary and are protected against scattering from material imperfections, i.e., they can support ballistic transport. These protected edge states are considered highly promising for engineering nanodevices, including components of quantum computers.
Below, you can find tutorials that address two-dimensional (2D) time-reversal-invariant TIs and relevant properties of the corresponding materials. Such TIs are often referred to as Quantum Spin Hall systems, where spin-orbit interaction is typically present and plays a crucial role. Note that the edge states in 2D TIs are one-dimensional modes.
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