Abstract
In this thesis, we explore two methods that may generate Quantum Anomalous Hall effect in room temperature on a honey comb lattice. In the first case, we consider the honeycomb lattice in external magnetic fields. It is shown that the distribution of Chern numbers over bands in Hofstadter butterfly obeys the Diophantine equation. Furthermore, we find that the distribution of Chern numbers can be altered by including the spin orbital interaction when its strength exceeds some threshold. It is argued that by appropriate using the spin orbit interaction, the Quantum Anomalous Hall in a subband of a honeycomb lattice in a magnetic field can be engineered. In the second work, we propose to induce magnetization in the honeycomb lattice by placing the honeycomb lattice in proximity to a ferromagnetic material. We find that under strain, pseudo magnetic fields generate flat bands similar to Landau levels in the honeycomb lattice. These flat bands possess non-vanishing Chern number and can exhibit Quantum Anomalous Hall effects in room temperatures.