Additive rule of real and reciprocal space topologies at disclinations

Topological materials are renowned for their ability to harbor states localized at their peripheries, such as surfaces, edges, and corners. Accompanying these states, fractional charges appear on peripheral unit cells. Recently, topologically bound states and fractional charges at disclinations of crystalline defects have been theoretically predicted. This so-called bulk-disclination correspondence has been experimentally confirmed in artificial crystalline structures, such as microwave-circuit arrays and photonic crystals. Here, we demonstrate an additive rule between the real-space topological invariant $\mathbf{s}$ (related to the Burgers vector $\mathbf{B}$) and the reciprocal-space topological invariant $\mathbf{p}$ (vectored Zak's phase of bulk wave functions). The bound states and fractional charges concur at a disclination center only if $\mathbf{s}+\mathbf{p}/2\pi$ is topologically nontrivial; otherwise, no bound state forms even if fractional charges are trapped. Besides the dissociation of fractional charges from bound states, the additive rule also dictates the existence of half-bound states extending over only half of a sample and ultra-stable bound states protected by both real-space and reciprocal-space topologies. Our results add another dimension to the ongoing study of topological matter and may germinate interesting applications.