Chern dartboard insulator: sBZ topology and skyrmion multipoles

Topology has played a crucial rule in many physical systems, often associated with the nontrivial gapless surface states. The paradigmatic example is the Chern number defined in the Brillouin zone (BZ) which leads to the robust gapless edge states. In this work, we extend the notion of Chern number to the reduced Chern number defined in a fraction of the total BZ. By introducing the idea of sub-Brillouin zone (sBZ) topology, we explicitly construct a family of models with trivial Chern number, yet hosting quantized reduced Chern number in the sBZ. These Chern dartboard insulators are protected by mirror symmetries and belong to a new class of delicate topology. Interestingly, by going into the pseudospin textures, the (anti)skyrmions appear inside the sBZs which are directly related to the sBZ topology. In addition, we observe the gapless edge states in all these systems, in particular with the Möbius fermions and midgap corner states in certain cases. Remarkably, Chern dartboard insulators are noncompact atomic insulators if C₂ symmetry is present. We suggest the photonic systems to realize the Chern dartboard phases and the corresponding gapless edge states. Our work opens the fruitful area to explore the sBZ topology and its nontrivial surface responses in topological systems.