Topological bound states in the continuum with controllable multiplicity

Bound states in the continuum (BICs) are spatially localized states embedded in the continuous spectrum without hybridizing with extended bulk modes. Recent advances in topological band theory have greatly enriched the understanding of BICs, which gives rise to boundary-localized topological BICs with extremely high robustness against disorders. However, there remains a challenge in realizing corner-localized topological BICs in a three-dimensional system due to the absence of both realistic theoretical models and effective topological characterization schemes. In particular, how to engineer a controllable number of corner-localized topological BIC is still an open question. Here, we propose that the corner-localized topological BICs can emerge in a class of generalized breathing pyrochlore lattice with general inter-cell hoppings. We further show that the number of BICs at each corner can be arbitrarily adjusted by changing the parameters of inter-cell hoppings. Remarkably, although these corner-localized topological BICs are intertwined with a substantial number of bulk modes, we can accurately characterize them through the polarized topological charges, which are nodal points with topological properties in Brillouin zone and are measurable in experiments. We also reveal three types of topological phase transitions of corner-localized BICs, which are associated with the different ways of closing the bulk energy gap and can be intuitively captured by the polarized topological charges. This work not only promotes the theoretical research of corner-localized topological BICs, but also opens an avenue for their experimental observation in the future.