Research Projects

Among the most recently discovered materials, it is hard to find something more influential than graphene. Graphene was the first material in a series of discoveries of 2d Materials, including sister materials of graphene (such a few layers of graphene, many different moir\'e systems, magic-angle twisted bilayer graphene, twisted double bilayer graphene, metastable ABC graphene). Using these in terms of creating new and exciting materials by stacking these different 2D layers nonuniformly (like a set of legos), we made a lot of progress. However, there is a lot that we do not understand about the properties and behaviour of these systems. I am interested in the quantum Hall behaviour in graphene and its sister materials. Even at the half filling the graphene ($\nu=0$) quantum Hall has been predicted to be a canted anti-ferromagnetic phase. However, the transport measurement and the local measurement show a completely different picture. This also includes the edge of these systems as being topological insulators, they are bound by the bulk boundary condition. In the two different parts of the project, we will focus on two main aspects: (i) Fractional Phase Diagram for non-ultra-short range assumption and (ii) Graphene quantum Hall Edge Physics. As pointed out before, even in the integer case, we do not understand the phase diagram. In the case of Fractional, the progress is even less. I would like to expand the interaction model in terms of corrections to the Haldane pseudo-potential to all orders and by exact diagonalization. This method goes beyond the previous attempt [PRL 112, 126804 (2014)] as more Haldane pseudo-potentials are included in the calculation. After building the model, we would like to understand in terms of the wavefunction how they are represented. This will allow us to understand the system with a large number of particles, which is beyond what exact diagonalization can do. The other part of the problem is away from the bulk and near the boundary. The topological nature of the bulk guarantees some properties of the edge. However, this does not predict everything about the system. Some properties can be very hard to measure (there has been a long history of confusing results for the 5/2 state in two-dimensional electron gas). Rather, there are a lot of uncertainties about the boundary modes for different reasons like edge reconstruction, equilibration, and many more. For the case of integer filling, we can do further in terms of mean-field approximation to find edge reconstruction. In the fractional regime, this becomes difficult, but either using composite fermion states to minimize energy or using the Starling approximation to calculate the lowest energy states via Monte Carlo are both possible techniques to find the energy. Recently, the sign-free quantum Monte Carlo method has been developed for $\nu=0$ graphene. We hope to use that also to find the different edge configurations with the experimental consequences.