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In band structures we see avoided crossing when we have degenerate eigenstates (caused by perturbation due to potential energy). However along some direction in first Brillouin zone, even though the energies cross in dispersion relation, they do not split up. I vaguely learned that the reason for this is because of symmetry along those directions in Brillouin zone. Can anybody help me understand why we do not have splitting?

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Splitting happens when the two states have different couplings to the interaction hamiltonian -- such as the classical case where an external magnetic field will differentially couple to different values of the orbital and spin angular momentum of electrons. If the ineteraction Hamiltonian treats both degenerate states identically (say, if you added a uniform Newtonian gravitational field to the hydrogen atom), then you won't get any splitting.

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