An Introduction to Topological Materials: SSH Model, Graphene, and the Haldane Model
##article.authors##
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Katsunori Kubo
Advanced Science Research Center, Japan Atomic Energy Agency
https://orcid.org/0000-0001-8531-900X
https://researchmap.jp/kubo_katsunori/
DOI:
https://doi.org/10.51094/jxiv.1620Keywords:
topological insulator, topological semimetal, Dirac point, edge state, winding numberAbstract
We review three representative models of topological materials: the Su--Schrieffer--Heeger (SSH) model, graphene, and the Haldane model. In this note, we use the term graphene to refer to the tight-binding model with nearest-neighbor hopping on a honeycomb lattice. The SSH model describes a one-dimensional topological insulator. When regarded as a two-dimensional model, it has a point defect of a two-component vector. This point defect underlies the realization of the topological insulator, giving rise to edge states in a system with edges. Graphene is a model of a two-dimensional topological semimetal. As in the SSH model, point defects of a two-component vector appear, leading to the emergence of edge states for systems with edges. By extending graphene, we obtain the Haldane model, a model of a two-dimensional topological insulator. Viewed as a three-dimensional model, it has point defects of a three-component vector. These defects are responsible for the realization of the topological insulator and the appearance of edge states in systems with edges. The topology of these models and their point defects is characterized by the winding number.
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Submitted: 2025-10-07 22:53:48 UTC
Published: 2025-10-08 08:23:55 UTC
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Katsunori Kubo
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