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A study of topological insulators in two and three dimensions Weeks, William Conan


In this work we introduce a new model for a topological insulator in both two and three dimensions, and then discuss the possibility of creating a fractional topological insulator in the three dimensional version. We also explore the possibility of engineering a quantum spin Hall phase in Graphene through the application of heavy metal adatoms. We show that the two dimensional model on the Lieb lattice, and its three dimensional counterpart, the so called edge centered cubic or Perovskite lattice, possess non-trivial Z₂ invariants, and gapless edge modes, which are the signatures of the topological insulating state. Having established that these lattices can become topological insulators, we then tune several short range hoppings in the model and show that it is possible to flatten the lowest energy bands in each case. After flattening the bands we add in a Hubbard term and then use a mean field decoupling to show that there is a portion of phase space where the system remains non-magnetic, and then conjecture that the many body ground state in three dimensions could become a fractional topological insulator. For the model on graphene, we start by using density functional theory (DFT) to find a pair of suitable heavy elements that are non-magnetic, have a strong spin orbit coupling and prefer to sit at the center of the hexagonal lattice. We then establish that for adatoms distributed in a periodic configuration, again with DFT, that a gap will open at the Dirac point in the presence of spin orbit coupling. To prove the gap is topologically non-trivial, we show that it is possible to adiabatically connect this model to the original Kane-Mele model, a known topological insulator. Lastly, we show that for adatoms distributed randomly the effect survives.

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