Studying the Topological Phase of Graphene and Computing Chern Number

Topological phases of matter are pertinent objects of study, with important applications such as in Chern insulators which exhibit the Quantum Anomalous Hall Effect. Working with Graphene we focused on two primary objectives: (i) transforming graphene into a topological phase using the Haldane model, and (ii) exploring the mathematical structure of this transformation, particularly through the calculation and plotting of Berry curvature, a pseudo magnetic field in momentum space which modulates electron motion, and the topological invariant the Chern number. To achieve the first objective we replicated the Haldane model, in which we broke the lattice inversion symmetry via a mass term and time-reversal symmetry via imaginary chiral second-neighbor hopping terms, inducing a topological phase. In developing the algorithm we faced the issues of accounting for an acquired geometric phase known as the Berry phase as well as the critical issue of gauge dependence, which has the potential to destroy topological robustness. Ad rem, in addressing these issues the algorithm discretizes the Brillouin over square plaquettes, calculates eigenvectors over the grid from which the local berry connections are found. Then it approximates local berry curvature over each plaquette via a line integral method, where the Bloch functions are locally continuous and hence avoids the gauge dependence problem. Lastly, it computes the integral of the Berry curvature across the Brillouin zone, which is precisely the Chern number.