The discovery of the classical Hall effect marked the beginning of serious attention to the role of topological concepts in condensed matter physics. It revealed that the phase transition between an ordinary insulator an‎d a topological insulator, unlike conventional phase transitions, is neither accompanied by spontaneous symmetry breaking nor describable by a local order parameter. This phenomenon indicated that a novel class of matter states exists, which cannot be encompassed within the Ginzburg-‎-Lan‎dau theory. In this thesis, the Chern insulators an‎d Z2 insulators, as the main examples of topological insulators, are investigated an‎d their prominent features are analyzed. Subsequently, methods for calculating topological invariants in these systems are introduced an‎d explained. In the final part, by employing the Kane-‎-Mele model for graphene an‎d taking advantage of the symmetries present in this material, the Z2invariant is computed. Moreover, suitable numerical tools are presented to facilitate the determination of topological invariants using computational codes in a simpler an‎d more efficient manner

پيمان صاحب سرا

حميده شاكري پور

حميده شاكري پور , اسماعيل عبدالحسيني سارسري