Üçgen-sel optik ağlar üstünde Z2-topolojik yalıtkanlar ve kuantumlu spin Hall etkisi

In this thesis we study Z2 topological insulators and quantum spin Hall effect on triangular optical lattices. In the first part we examine the effect of a strong bichromatic deformation to the Z2 topological insulator in a fermionic ultracold atomic system on triangular lattices. We modify a system proposed by B. Beri and N. R. Cooper, Phys. Rev. Lett. 107, 145301 (2011) to a bichromatic optical lattice. Large insulating gap of this system allows for examination of strong perturbations. The calculations are done in the nearly free electron limit. We conclude that the Z2 topological character of the system is robust against large global perturbations that break the inversion symmetry but preserve the time-reversal symmetry. In the second part, we study the tight binding regime of the same system and compare the band structures and density of states of nearly free electron and tight binding limits. In the final part, we also propose a tight binding model for the quantum spin Hall system on triangular optical lattices. We show that the proposed Hamiltonian corresponding to 1/4 magnetic field flux unit can be realised by a time-periodic driving-like lattice shaking. We determine the edge state spectrum which contains gap traversing states as the hallmark of a Z2 topological insulator.