# Is topological surface state always tangential to bulk bands?

Think of a topologically nontrivial $D$-dimensional system. Its bulk bands form a $D+1$-dimensional manifold ($+1$ from energy). Its surface/edge bands form a $D$-dimensional one. Is the latter always tangential to the former? If so, why? If not, any counterexample?

For instance, on page 340 Fig.2a of this paper, it is said that in a Weyl metal (Fermi level not at Weyl point), the Fermi arc tangentially connect two Fermi loops around two Weyl points.

Since the velocity $v_\parallel$ parallel to the surface is given by the derivative $v_\parallel = dE/dp_\parallel$, with $p_\parallel$ the parallel momentum component, the velocity would vary discontinuously if the surface bands would not be tangential to the bulk bands. In particular, at the transition point the velocity would not exist. The reason this does not happen is that the transition from a bulk state to a surface state is a smooth crossover: the perpendicular momentum component $p_\perp$ varies continuously from purely imaginary to purely real in the transition from surface state to bulk state. Nothing singular happens with $v_\parallel$ at the transition point where $p_\perp=0$.