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Consider a 3D topological insulator surface state Hamiltonian with mass possibly due to magnetic doping $$\sum_{i,j=x,y}v_{ij} k_i \sigma_j + m\sigma_z.$$ What determines the helicity $X=\mathrm{sgn}[\det(v_{ij})]$ and the sign of mass? Are they tunable or do they vary among different systems or opposite surfaces of one system? I usually see the form $p_x\sigma_y-p_y\sigma_x$ with $X=1$. Is it the same for all cases and why?

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