The existence of mid-gap edge states is a distinct feature of topological electronic states. However, in the presence of vacancies, impurity band also arises inside the gap. The degeneracy of the impurity band with the edge states thus calls for an examination of the stability of topological states
in the presence of the impurity band.
In this thesis, we examine the stability of topological electronic states in honeycomb lattice due to the presence of the impurity band. We shall first investigate the topological indices and their evaluations based on the computation of the Chern number and the spin Chern number. A simple picture will be presented and demonstrates how Z2 topological index arises. Useful numerical algorithms that were developed from lattice gauge theory will be presented to calculate the Chern number and the spin Chern number of a given system that doesn't have translation invariant in real space and time-reversal symmetry.
Using the developed numerical methods, we analyze effects on the topological indices due to vacancies in the 2D Kane-Mele model and its extensions that
are applicable to new developed materials such as silicene and germanene.
A consistent check of the presence of edge states is also performed by computing the spectral function A(k,w) and local density of states.
It is shown that in consistency with our simple picture, changing of the spin Chern number from non-trivial to trivial values is accompanied by the destruction of Dirac cones. When Dirac cones are destroyed, topological insulators are turned into trivial insulators. The transition occurs in a first-order transition fashion at vacancy concentration being around 7 % when vacancies reside only on one sublattice. While if vacancies reside in both sublattices, Dirac cones are destroyed in a continuous fashion so that the transition is continuous. These results are useful for characterizing topological insulators.

1 Introduction 1
2 Theoretical models, topological invariants and their evaluation 5
2.1 Tight-Binding models on honeycomb lattice . . . . . . . . . . . . . . . . . . 5
2.2 Berry phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Chern number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Relation between Berry connection, Berry curvature,and Chern number 12
2.3.2 Edge state and Chern number . . . . . . . . . . . . . . . . . . . . . . 12
2.3.3 Relation between time-reversal symmetry and Chern number . . . . . 14
2.4 Time reversal invariance and Z2 topological order: a simple picture . . . . . 15
2.5 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5.1 Direct Numerical Integration . . . . . . . . . . . . . . . . . . . . . . 18
2.5.2 Numerics based on lattice gauge . . . . . . . . . . . . . . . . . . . . 20
2.5.3 Lattice gauge formulation for Z2 computation . . . . . . . . . . . . . 24
3 Results and discussion 25
3.1 General Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Stability of Z2 topological orders in the presence of impurity band . . . . . . 33
3.3 Half hydrogenated Silicene and Germanene . . . . . . . . . . . . . . . . . . . 42
4 Summary and Conclusion 48

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