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The spin-orbit interaction can be written: $$H=\mathscr{E} \hat{L} \cdot \hat{S},$$ where $\hat{L}=\hat{r} \times \hat{p}$. Consider an electron hopping to its next nearest neighbour on the honeycomb lattice. During its hop, the atom closest to it (not counting the atom it's hopping from or toward) has a coulombic potential which affects the hopping ...
A simple method to judge the chirality (or in your words "orientation") of the Hamiltonian is to evaluate the following quantity $$f=\frac{i}{2}\mathrm{Tr}\frac{\partial h}{\partial{q_x}}\frac{\partial h}{\partial{q_y}}\frac{\partial h}{\partial{m}}.$$ The sign of this quantity $f$ gives the chirality of the Hamiltonian. Example: Given the two ...