I have been thinking about so-called valley Chern numbers $C_v$ and associated topological phenomena. To my knowledge, they are usually applicable when inter-valley scattering is suppressed, leaving the Berry curvature peaked at a high symmetry point ($K,K',\Gamma,M,...$). Following are few of several examples involving such Hamiltonians:

In theoretical models, people usually use Hamiltonians (sometimes starting from lattice Hamiltonians), to derive a $k\cdot p$ Hamiltonian that is centered at the $K$ point. Then, they sum the Berry curvature of filled bands in the limit $k\rightarrow \infty$ (usually a very large $k$-space numerically) to get the Chern number at each valley $C_K,C_{K'}$. This is then used to calculate $C_v=C_K - C_{K'}$. Numerically, $C_K$ are oftentimes not perfectly quantized, but are very close (ex: $0.999 \rightarrow 1$).

On top of models like these, there are lattice models (such as Haldane's model for the QAHE), that can yield similar quantized results. In these cases, one could do a low-energy expansion at the $K$ point to potentially see something similar (idea of 'topological charge').

My confusion is: What makes these valley-type Hamiltonians special enough to give quantized Chern numbers in these limits? I am used to thinking of topological quantities as properties of the entire Bruillouin zone (ex: Chern number of QAHE), half-Brillouin zone (ex: $\mathbb{Z}_2$ index), or just a loop around gapless points (ex: $\pi$-phase in graphene). So, what makes these valley-type Hamiltonians correctly give the same/accurate information about topological phenomena even though they are simpler and localized to special points? I think that $k\cdot p$ models are diagonal at the $K$ point, and off-diagonals kick in as we move farther away from the chosen point. While I think that some information is lost (not sure what exactly), perhaps there might be some special symmetry that isn't lost? I am not convinced that all this is due to potential linear Dirac dispersions...

I am asking because I messed around with some $k$-space effective Hamiltonians derived from DFT calculations, and they don't seem to behave as nicely/quantizedly compared to aforementioned theoretical/lattice/$k\cdot p$ models. Perhaps, this question might also be related to the question of how a $k\cdot p$ Hamiltonian about a high-symmetry/$K$ point might reflect valley topology accurately whereas a fictitious $k\cdot p$ Hamiltonian about a faraway, non-special, "low-symmetry point" might not.

I apologize this question might not have been phrased properly, but I appreciate your patience.

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    $\begingroup$ In these systems, are there other degenerate or non-degenerate points? Generally, the Berry curvature peaks near band crossings and avoided crossings. Symmetry often enforces crossings at high-symmetry points. $\endgroup$
    – Anyon
    Jul 2 at 20:52
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    $\begingroup$ @Anyon, it's usually been only $K$ and $K'$ (with potential band crossings). However, some models (ex: monolayer $MoS_2$ model using a $3d$ orbital basis) have Berry curvature peaked at the $\Gamma$ point even without crossings. That said, do you have any (ideally not-too-difficult) resources to recommend, that explain the idea of Berry curvature peaking near (band/avoided) crossings? I mean, I see why this would be the case due to the energy denominator ($E_n-E_m$; for bands $n,m$), but I think it can peak due to non-energy-related Hamiltonian parameters/wavefunctions as well. $\endgroup$ Jul 2 at 21:35
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    $\begingroup$ I'm afraid I don't really have such a resource, but you basically have the argument already - small energy differences in the denominator can cause large $\Omega$. The opposite perspective is to look for regions where the wave function (or properties) vary rapidly with $\mathbf{k}$. Often that occurs where the gap is small (example), but you're probably correct that it's not necessarily the case. $\endgroup$
    – Anyon
    Jul 2 at 22:26

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