  Many complex magnetic structures in a high-symmetry lattice can arise from a
superposition of well-defined magnetic wave vectors. These "multi-q" structures
have garnered much attention because of interesting real-space spin textures
such as skyrmions. However, the role multi-q structures play in the topology of
electronic bands in momentum space has remained rather elusive. Here we show
that the type-I anti-ferromagnetic 1q, 2q and 3q structures in an face-centered
cubic sublattice with band inversion, such as NdBi, can induce unconventional
surface state pairs inside the band-folding hybridization bulk gap. Our density
functional theory calculations match well with the recent experimental
observation of unconventional surface states with hole Fermi arc-like features
and electron pockets below the Neel temperature. We further show that these
multi-q structures have Dirac and Weyl nodes. Our work reveals the special role
that band-folding from anti-ferromagnetism and multi-q structures can play in
developing new types of surface states.
