Fascinating physical phenomena emerges when stacking one single layer of graphene on the other one with a small twist around 1.1^◦. Moiré potential induced by the small twsit can result in flat bands. The electrons in the flat bands seem to be freezed and it causes strong correlation effect in twist bilayer graphene system. This thesis can be mainly separated into two parts. In the first part, I will review the classical method to cope with the electronic spectrum in this kind of flat band system – continumm model(CM). Based on the observations from CM, we proposed a tight binding planewave scheme to do band structure calculation which can also provide us a flat band picture. Compared with CM, our method has more details and is easy to reutilize for further calculation. In the second pa...[ Read more ]

Fascinating physical phenomena emerges when stacking one single layer of graphene on the other one with a small twist around 1.1^◦. Moiré potential induced by the small twsit can result in flat bands. The electrons in the flat bands seem to be freezed and it causes strong correlation effect in twist bilayer graphene system. This thesis can be mainly separated into two parts. In the first part, I will review the classical method to cope with the electronic spectrum in this kind of flat band system – continumm model(CM). Based on the observations from CM, we proposed a tight binding planewave scheme to do band structure calculation which can also provide us a flat band picture. Compared with CM, our method has more details and is easy to reutilize for further calculation. In the second part, I will introduce two different techniques to help us navigate into the strong correlation physics in TBG by taking advantage of tight binding planewave basis. The first application is using constrained Random Phase Approximation to obtain effective interaction in flat bands. The second one, I try to construct tight binding planewave basis set under weak magnetic field in periodic Landau guage.