Take Haldane's Hamiltonian, as quoted from Fruchart et al.'s An Introduction to Topological Insulators:
3.5.3. Haldane's Hamiltonian
The first quantized Hamiltonian of Haldane's model can be written as: $$ \hat H = t \sum_{⟨i,j⟩} |i⟩⟨j| + t_2 \sum_{⟪i,j⟫} |i⟩⟨j| + M \left[ \sum_{i\in A} |i⟩⟨i| - \sum_{j\in B}|j⟩⟨j|\right] \tag{30} $$ where $|i⟩$ represents an electronic state localized at site $i$ (atomic orbital), $⟨i,j⟩$ represents nearest neighbors lattice sites $i$ and $j$, $⟪i,j⟫$ represents second nearest neighbors sites $i$ and $j$, $i\in A$ represents sites in the sublattice $A$ (resp. $i\in B$ in the sublattice $B$). This Hamiltonian is composed of a first nearest neighbors hopping term with a hopping amplitude $t$, a second neighbors hopping term with a hopping parameter $t_2$, and a last sublattice symmetry breaking term with on-site energies $+M$ for sites of sublattice $A$, and $-M$ for sublattice $B$, which thus breaks inversion symmetry. Moreover, the Aharonov-Bohm phases due to the time-reversal breaking local magnetic fluxes are taken into account through the Peierls substitution: $$ t_{ij} \to t_{ij} \exp\mathopen{}\left( -i \frac{e}{\hbar} \int_{\Gamma_{ij}} \vec A \cdot \mathrm d\vec \ell \right)\mathclose{} \tag{31} $$ where $t_{ij}$ is the hopping parameter between sites $i$ and $j$, and where $\Gamma_{ij}$ is the hop trajectory from site $i$ to site $j$ and $\vec A$ is a potential vector accounting for the presence of the magnetic flux.
How do the hopping parameters $t$ and $t_2$ of the model over the honeycomb lattice relate to the local Berry curvature of the unit cell, if at all? I do not think it is obvious, but the Berry connection and Hamiltonian depend on each other. This post discusses involving the Berry connection in the Hamiltonian.
I am more concerned about physical consequences of any potential relationship between the two. To take a wild guess as an example, does the local Berry curvature in k-space that corresponds to next-nearest-neighbor hopping paths in real space = 0? On a semi-related note, must the energy of the particle concerned be minimum on their hopping paths? Any references?
I would appreciate any advice or resources! Thanks!