Abstract
Topological materials host quantum states that have been gathering a large amount of interest
in the past few years. These states are resistant to perturbations that do not break symmetry.
In general, topological properties are analysed for very simple models so that the topological
phase of the electronic states brought about in these materials can be predicted. In this work,
the effect of both symmetry preserving and symmetry breaking perturbations on simple 1D and
2D models is investigated. Once the topological properties of the models are established, their
applicability to real materials is studied. By the end of the investigation, a simple two, three
and four band model are proposed, which can describe the unique quantum state and represent
the band structure of 2D materials with a set of symmetries for real world applications.