Topological phases on the real projective plane

We investigate two-dimensional spinless systems in which the fundamental domain in momentum space takes the form of a non-orientable closed manifold known as the real projective plane (RP2), in contrast to the usual case of a torus. We construct Wilson loops on RP2 to define a Z2 invariant that identifies topologically distinct phases. We find that the transition between the trivial and topological phases is mediated by an odd number of Weyl points within the fundamental domain, and that these Weyl points cannot all be annihilated. The topological phase is characterized by the presence of gapless bi-directional edge states, a feature attributed to the Fermi-arc connectivity of the Weyl points. Lastly, we demonstrate that these systems are examples of "momentum quadrupole insulators" that exhibit a linear response of momentum current to a translation gauge field.