Abstract
The topological phase of non-interacting electronic bandstructure can be classified by calculating integer invariants. In this chapter, we introduce the Chern invariant that classifies 2D materials in the absence of symmetry. We then show that this invariant can be used as the building block for the classification of topological insulators, semimetals, and symmetry-protected topological phases. We show how this classification is performed in practice by introducing Z2Pack, a tool which allows calculating topological invariants from \({\mathbf {k}}\cdot \mathbf {p}\) and tight-binding models, as well as first-principles calculations.
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- 1.
This is not always possible, for example in the case of semimetals where the occupation number changes with \({\mathbf {k}}\). In these cases, one often picks the N lowest energy bands instead.
- 2.
For simplicity, we consider the total Berry phase of all bands. The Berry phase can also be defined for a single band, in which case the sum over bands is dropped.
- 3.
To see this, try calculating the Berry phase for \(\left| u_k\right\rangle = e^{i k / 2} \begin{pmatrix}\cos (k)\\ \sin (k)\end{pmatrix}\), for \(k \in [0, 2\pi ]\).
- 4.
TBmodels was initially developed as part of Z2Pack, but later separated because it can be used outside of the scope of calculating topological invariants.
- 5.
Figures and Text in This Section Are Partly Copied from Previous Work of the Authors [28].
- 6.
Note that this gap in the HWCC spectrum is not related to the band gap of the energy spectrum.
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Acknowledgements
The authors were supported by Microsoft Research, the Swiss National Science Foundation through the National Competence Centers in Research MARVEL and QSIT, and the ERC Advanced Grant SIMCOFE.
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Gresch, D., Soluyanov, A. (2018). Calculating Topological Invariants with Z2Pack. In: Bercioux, D., Cayssol, J., Vergniory, M., Reyes Calvo, M. (eds) Topological Matter. Springer Series in Solid-State Sciences, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-76388-0_3
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