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 $U(1)$ second Chern number C₂ 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 nth Chern number Cₙ is also discussed.