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 Berry curvature is integrated over filled bands.

Graphene has two bands who touch at the two high symmetric points(forming the Dirac dispersion). If the Chern number is calculated for either Graphene or the Haldene model over the two filled bands we simple get 0. Since Graphene is topologically non-trivial, we get 0 for one filled band as well, whereas the non-trivial topology of the Haldene model gives rise to a Chern nuber of +1. I shall write a detailed account of the Haldene model soon.

Graphene dispersion relation:

Haldene model dispersion relation:





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