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Four years ago, the Nobel Prize in physics was awarded for "for theoretical discoveries of topological phase transitions and topological phases of matter." In line with this, I heard of topological insulators, a new material with the strange property of having an insulating bulk with a conductor surface, having a great interest because they could be useful for many future applications in electronics and quantum computation.

I would like to know what is a topological insulator from a theoretical point of view, i.e., how to recognize one from its band structure and how is the property of bulk-insulator/surface-conductor explained from first principles (at least conceptually).

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    $\begingroup$ Andrés, I was about to post a question too similar to yours. Basically, using first-principles techniques, what do we need to look for as materials modelers? I have a feeling that in order to study TIs we must go beyond DFT and use other tools, however I am nowhere near an expert in this topic. However, I found this review by the late Shou-Cheng Zhang. A true icon in the field. Read "Topological insulators from the perspective of first‐principles calculations" here: onlinelibrary.wiley.com/doi/full/10.1002/pssr.201206414 $\endgroup$
    – Etienne Palos
    Jun 11, 2020 at 22:20
  • $\begingroup$ @EtiennePalos many types of topological material (time-reversal topological insulators, topological crystalline insulators, Weyl semimetals, nodal line semimetals, ...) are well-described within a single-band approximation, so DFT-based calculations are good to study these materials. Of course, like always, there is a choice of functional to make and so on, but there is no fundamental limitation. $\endgroup$
    – ProfM
    Jul 18, 2020 at 14:11

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The bulk band structure of a topological insulator would look just like any other insulator, with the Fermi level in the gap between the valence and conduction bands. If your band structure includes the surface states, then you would see some states in the gap that cross the Fermi level.

There's an example on Wikipedia:

An idealized band structure for a topological insulator. The Fermi level falls within the bulk band gap which is traversed by topologically-protected spin-textured Dirac surface states.

An idealized band structure for a topological insulator. The Fermi level falls within the bulk band gap which is traversed by topologically-protected spin-textured Dirac surface states. By A13ean

As far as whether those surface states will appear in a experimental measurement of the band structure, I am not sure.

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