Come for the bandstructure, stay for the topology
So, our latest obsession is with topological insulators. They are not hard to find, but are hard to identify. Plotting the band-structure will not give them away, they can hide in plain sight. However, calculating the topological invariant can reveal their secret identity. There is another tell-tale sign. If we put them next to a topoogically trivial material and diagonalize the Hamiltonian, we find eigen-energies at the middle of the gaps. The corresponding eigen-vectors are found to be concentrated close to the edge; the reason behind their names. As shown in my last blog post, a simple SSH model can be a topological insulator when the inter-hopping terms are larger than the intra-hopping energies. But, we focus on the Haldene Chern insulator, the prototypical model that was honored by the 2016 Nobel prize. Haldene model is the graphene model adorned with a complex hopping parameter which gives rise to effective magnetic fields in unit cells and non-zero chern numbers when the
