Laughlin's Gauge Argument and Quantum Spin Hall Effect

Question

I'm reading C.L. Kane and E.J. Mele's original paper of QSHE(DOI:https://doi.org/10.1103/PhysRevLett.95.226801). When proving the existence of edge state, they use the Laughlin's gauge argument which I am familiar with. However, they also say that the gauge argument is only true when the Hamiltonian conserves $\sigma_z$. This makes me feel confused. I don't know why gauge argument can only be used when $\sigma_z$ is conserved.

Answer 1

If I remember correctly they use the gauge argument for each spin component separately --- as if spin-up and spin-down both had their own gauge field and U(1) group. This will only work if there really are two conserved charges, so $\sigma_z$ must commute with the hamiltonian.