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In Ref. a nearest-neighbor, tight-binding method was applied with a large rectangular supercell along a sinusoidal potential as basis. Extra touching points at the Fermi level were observed and, together with the group velocity in their vicinity, their influence on the density of states was demonstrated.
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The implicit function theorem cannot be applied if the gradient (Jacobian) of the dispersion relation is zero. This occurs because of the crossing of the bands at these points: from the form of the derivatives, too long to write down, we can see that they are both zero if sin (λ Ww ) =sin (Λ Wb ) =0 while cos (λ Ww ) =cos (Λ Wb ).
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The implicit function theorem cannot be applied if the gradient (Jacobian) of the dispersion relation is zero. This occurs because of the crossing of the bands at these points: from the form of the derivatives, too long to write down, we can see that they are both zero if sin (λ W w) = sin (Λ W b) = 0 while cos (λ W w) = cos (Λ W b).
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