Five-dimensional generalization of the topological Weyl semimetal

We generalize the concept of three-dimensional topological Weyl semimetals to a class of five dimensional (5D) gapless solids, where Weyl points are generalized to Weyl surfaces which are two-dimensional closed manifolds in the momentum space. Each Weyl surface is characterized by a $U\left(1\right)$ second Chern number $C_2$ defined on a four-dimensional manifold enclosing the Weyl surface, which is equal to its topological linking number with other Weyl surfaces in 5D. In analogy to the Weyl semimetals, the surface states of the 5D metal take the form of topologically protected Weyl fermion arcs, which connect the projections of the bulk Weyl surfaces. The further generalization of topological metals in $2n+1$ dimensions carrying the $n\mathrm{th}$ Chern number $C_n$ is also discussed.

Figure 1

(a) Weyl surfaces ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ plotted in the subspace $k_4=k_5=0$. (b) Weyl surface ${\mathcal{M}}_3$ plotted in the subspace $k_1=k_2=0$. (c) The Weyl surfaces can be seen linked in the subspace $k_2=k_4=0$. A Weyl fermion arc occurs on the boundary and crosses all the possible $\partial {\mathcal{V}}_2$. (d) Energy spectrum $E\left(\mathbf{k}\right)$ in the subspace $k_1=k_2=0$ calculated with the open boundary condition in the $k_5$ direction, where there are two arcs from the top and bottom boundaries, respectively.