In condensed matter systems, integrating Berry curvature within an adiabatic loop in parameter space over a Dirac point of some model usually gives one its chirality ($\pm 1$, for instance).

In the literature, the various types of Chern numbers (charge, spin, valley, spin-valley) all take this chirality into account. For instance, the spin Chern number is the difference between the chiralities of each Dirac point for each spin (up/down) of a two-level system.

Also, what is so treasured about topological invariance is the robustness of these models against gauge/impurities. For example, a trivial state of a topological insulator will have its Dirac point chiralities in the filled band sum to 0.

However, there can be models that do not explicitly highlight the existence of those Dirac chiralities. For instance, a trivial state of the Haldane model can have a Dirac point with +1 chirality and the other point with -1 chirality on the same level. The Chern number of the filled band will then be $= (+1)+(-1)=0$, as expected.

My question is, what is the absolute number of Dirac singularities on a level good for, ignoring chirality at individual points? In the example just discussed, the 'absolute number' is $=2$, because that's the number of Dirac points on it that exist as a so-called magnetic monopole.

I could not find any resources that discuss the significance of the total number of Dirac points on a level (even in literature for various types of Chern number).

I am particularly interested in understanding any consequences of this absolute number of chiral points in interband transitions. For example, in the Qi-Wu-Zhang model, the trivial state has Chern number 0, but the top level can have 2 pairs of Dirac monopoles that cancel each other's chirality out, whereas the bottom level has none. The charge Chern number is still = 0, but I am wondering whether the fact that the top level having a different number of Berry curvature singularities matters at all.


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