The Berry phase can only be computed if the Hamiltonian has a gap. For a Hamiltonian \(H(\mathbf{k})\) with many bands \(E_n(\mathbf{k})\), this means that we can compute the Chern number only for an isolated band \(E_n(\mathbf{k})\) which does not touch any other band. If there is a band touching, the Berry phase is undefined.

Now let’s go back to our analogy with electromagnetism. We know that we cannot compute the electric or magnetic flux through a surface if there are electric or magnetic charges sitting exactly on it. That’s because the electric or magnetic fields are not defined at the points where their sources are.

This analogy suggests the following: that the sources for Berry flux in momentum space are points where two bands touch, just like the Dirac points at the \(\mathbf{K}\) and \(\mathbf{K}'\) points of the Brillouin zone in graphene.

This may sound a bit abstract and confusing: where are these points located? We are used to thinking about sources of flux in real space, not in momentum space. In fact, just like you do with a two-dimensional sphere surrounding a charge in three-dimensional space, you can think of the Brillouin zone as lying in a three-dimensional space, with two directions given by \(k_x\) and \(k_y\) and the third given by the magnitude of the energy gap.

The situation is explained by the following sketch, which also gives a bird’s-eye view of the phase diagram of the Haldane model as a function of the ratio \(t_2/M\):

What you see in the sketch above is a schematic illustration of the energy spectrum close to the Dirac points in the Brillouin zone, for some representative values of \(t_2/M\) (for simplicity we drew the Brillouin zone as a square and not a hexagon, but that’s not essential). The two massless Dirac cones appearing for \(t_2=\pm M/(3\sqrt{3})\) are the sources of the Berry curvature, which then “spreads“ along the vertical axis, passing through the Brillouin zones of the gapped phases.

The \(t_2=0\) Brillouin zone is “sandwiched“ between the two gap closings: it has opposite curvature for the two Dirac points, and a total Chern number of zero.

The Brillouin zones for \(|t_2|>M/(3\sqrt{3})\), on the other hand, have Berry curvature with the same sign for both Dirac points, and a total Chern number equal to \(\pm 1\).

To see this more clearly, we can compute the Berry curvature numerically and plot it over the whole Brillouin zone as a function of \(t_2\):

You can see that the Berry curvature is really located around the Dirac points. Around \(t_2=0\), the two Dirac points give canceling contributions. After a gap closing however, the contribution of one of the two Dirac points changes sign, so that the two add to \(\pm 1\) instead of canceling each other.

From both the plots above, you can also infer that each Dirac point always contributes a Berry curvature equal to \(\pm 1/2\), depending on the sign of the mass in the effective Dirac Hamiltonian. We always obtain an integer number because the number of Dirac points in the Brillouin zone is even. It also implies that when the gap changes sign at a Dirac point, the Chern number changes by exactly one!

At the same time it’s important to know that the particular distribution of the Berry curvature depends on all the details of the eigenstates of the Hamiltonian, so it changes a lot from model to model. And in fact, it is a special feature of the Haldane model that the Berry curvature is focused around two distinct points in the Brillouin zone.

For instance, here is a slider plot for the Berry curvature for the quantum Hall lattice model studied in the previous chapter.

You can see that for \(\mu < -2t - 2\gamma\) there is a net curvature, and that when \(\mu = -2t - 2\gamma\) some flux of opposite sign appears at \(k_x = k_y=0\), the Dirac point, which leaves no net curvature and leads to a change in the Chern number. This is the signature of the topological transition seen from the Berry curvature.