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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 ...