We show that lattices with higher-order topology can support corner-localized bound states in the continuum (BICs). We propose a method for the direct identification of BICs in condensed matter settings and use it to demonstrate the existence of BICs in a concrete lattice model. Although the onset for these states is given by corner-induced filling anomalies in certain topological crystalline phases, additional symmetries are required to protect the BICs from hybridizing with their degenerate bulk states. We demonstrate the protection mechanism for BICs in this model and show how breaking this mechanism transforms the BICs into higher-order topological resonances. Our work shows that topological states arising from the bulk-boundary correspondence in topological phases are more robust than previously expected, expanding the search space for crystalline topological phases to include those with boundary-localized BICs or resonances.
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