# The surface states and Fermi arcs in Weyl semimetals

I'm confused about surface states in Weyl semimetals. Assume that we have a single pair of Weyl points and the Fermi level turned to this points. In this https://arxiv.org/abs/1301.0330 paper the authors write:

if the Fermi energy is tuned to the Weyl point in the bulk, the bulk states at this energy are only at the projections of the Weyl points to the surface Brillouin zone. Everywhere else, surface states are well defined because they cannot decay into bulk states at the same energy and momentum. As the Weyl point momenta are approached, the surface states penetrate deeper into the bulk.

But I don't understand what they mean, are these surface and bulk states refer to real space or just to momentum space?

Why the bulk states are projected to the surface

In the bulk of a 3D crystal, we have discrete translation symmetry in all 3 directions. As a consequence, we can label states by how they transform under such translations. By Bloch's theorem, it is by just a phase which is determined by a wavevector $$(k_x, k_y, k_z)$$. Each component of the wavevector determines the transformation properties in that direction.

Now when we truncate the crystal to get a surface, we break the translation symmetry normal to the surface. Taking this to be the $$z$$-direction, we can no longer assign a $$k_z$$ to all our eigenstates. Instead, we only have the components parallel to the surface: $$\boldsymbol{k}_\parallel=(k_x, k_y)$$. However as we go deep into the bulk we certainly expect to recover the bulk solutions. So all the bulk states should still be here but we have to project them onto the $$(k_x, k_y)$$ surface since those are the only meaningful labels (i.e. we have to consider the surface Brillouin zone and not the bulk Brillouin zone).

Pure surface states and resonances

While the bulk states are simply projected onto the surface Brillouin zone, the surface also introduces new states to the system. These new states can be roughly categorized into two groups: pure surface states and resonances.

A pure surface state is a state which is in the bulk gap. In other words, there is no bulk state which has the same energy/quasimomentum. These states are isolated at the surface and typically exponentially decay into the bulk.

A resonance is a surface state which lies on top of a projected bulk state. It has the same energy/quasimomentum as a bulk state. For this reason, it costs no energy for this state to penetrate into the bulk. The wavefunction will still have considerable weight at the surface, but it will typically have a non-vanishing oscillatory tail in the bulk.

The Weyl semimetal case

Now for a Weyl semimetal, the Fermi surface in the bulk consists of isolated Weyl points. There is a gap everywhere else in the Brillouin zone. When we project onto a surface Brillouin zone, the projected bulk dispersion must still have that gap everywhere except at the projected Weyl points.

The Fermi arcs connect the projected Weyl points due to the bulk topology. At the Weyl points, these surface states are resonances since they are on top of the bulk Weyl state. However, everywhere in between they are pure surface states.