Topological bands in metals

In crystalline systems with a superstructure, the electron dispersion can form a nontrivial covering of the Brillouin zone. It is proved that the number of sheets in this covering and its monodromy are topological invariants under ambient isotopy. As a concrete manifestation of this nontrivial topology, we analyze three-sublattice models for 120${}^{\circ }$-ordered helimagnets in one, two, and three dimensions. The two-dimensional system exhibits unconventional $f$-wave magnetism and a specific topological metal state characterized by a spin-textured, one-sheeted Fermi surface. The observable transport signatures of the topological metal and its potential experimental realization are briefly discussed.