5
$\begingroup$

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. enter image description here

$\endgroup$
5
+50
$\begingroup$

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 $p_\perp=0$.

This is discussed in more detail, with a worked-out example, by Duncan Haldane in Attachment of Surface "Fermi Arcs" to the Bulk Fermi Surface: "Fermi-Level Plumbing" in Topological Metals: Close to the boundary, the surface state is extremely weakly-bound, and its properties approach those of the bulk electronic band from which it evolves at the termination point. In particular, its group velocity tangent to the surface will approach that of the bulk band edge at the termination point from which it evolves.

Trivia: The need for tangential connection was not initially appreciated on Wikipedia, see this image which has the surface bands merge with the bulk bands at an angle. It is now corrected.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.