When we use a tight-binding approach in order to calculate the band structure of electrons on a 2D honeycomb lattice such as Graphene we find that there are two energy bands touching in six points, called Dirac points. Interestingly, the dispersion relation for electrons with momenta near the Dirac points is linear and you can show that the Hamiltonian in the vicinity of those points is given by $$ H \propto \vec{\sigma}\cdot\vec{p} $$ were $\vec{\sigma} = (\sigma_1, \sigma_2)$ and $\sigma_i$ are Pauli matrices. Knowing that helicity is defined as $h = \frac{\vec{\sigma}\cdot\vec{p}}{|\vec{p}|}$, we see that helicity is a conserved quantity for small $\vec{p}$.

What I'm interested in is what the conserved quantity is for generic $\vec{p}$ when the Hamiltonian is not given by the above expression. I know there is something called the $\textit{Chern number}$ which could be important here.

So, knowing very little about topological condensed matter physics but being willing to read about relevant things if somebody has some good literature recommendations, my question is if there is a way to compute the Chern number directly from the band structure. The band structure of Graphene is known and the Chern number is a property of the bands, so maybe there is a way to combine the two in a single equation.


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