Topological classification of solids with momentum-nonsymmorphic symmetries

Conventionally studied symmetries such as time-reversal or particle-hole symmetries map the Brillouin wavevector k to -k. Here we study a class of symmetries that map k to k+2pi/n, with n an integer and 2pi the Brillouin period. These so-called projective symmetries may be found in quasimomentum sub-manifolds of the Brillouin zones of Bravais lattices as projective mirror lines and planes as well as projective rotation lines. We classify such quasi-momentum submanifolds for the tenfold symmetry classes as well as identify the lines and planes in the Brillouin zones of Bravais lattices which have such symmetries. For Chern insulators, for quasimomentum sub-manifolds invariant under projective mirror, a topological invariant can be seen in the form of twists in the eigen-spectrum of Wilson loops. We discuss the properties and robustness of these twists.