We propose a new kind of bound states in the continuum (BICs) in a class of systems constructed by coupling multiple identical one-dimensional chains, each with inversion symmetry. In such systems, a specific separation of the Hilbert space into a topological and a non-topological subspace exists. Bulk-boundary correspondence in the topological subspace guarantees the existence of a localized interface state which can lie in the continuum of extended states in the non-topological subspace, forming a BIC. We further show non-Hermitian bulk systems with PT-symmetry can be characterized by topological invariants and PT-symmetry is compatible with topologically protected interface states. Consequently topologically protected bound states in the continuum can also be achieved even i...[ Read more ]

We propose a new kind of bound states in the continuum (BICs) in a class of systems constructed by coupling multiple identical one-dimensional chains, each with inversion symmetry. In such systems, a specific separation of the Hilbert space into a topological and a non-topological subspace exists. Bulk-boundary correspondence in the topological subspace guarantees the existence of a localized interface state which can lie in the continuum of extended states in the non-topological subspace, forming a BIC. We further show non-Hermitian bulk systems with PT-symmetry can be characterized by topological invariants and PT-symmetry is compatible with topologically protected interface states. Consequently topologically protected bound states in the continuum can also be achieved even in PT-symmetric non-Hermitian systems. By utilizing the idea of the separation of Hilbert space, and introducing disorder to the system so that only eigenstates in one subspace are affected and become Anderson localized, we achieve the peculiar phenomenon that Anderson localized states and extended states can coexist both spectrally and spatially. Finally, by appropriately coupling multiple Haldane-model layers by intermediate triangular-lattice layers, we show arbitrarily large Chern numbers can be obtained for two invariant Haldane bands. It is found that two phase transitions occur when varying the interlayer hopping: gapped Chern insulator to Weyl semimetal, then to gapped Chern insulator. Further we show that surface states protected by the nontrivial Chern number are in fact also bound states in the continuum, which do not couple with some of the bulk bands.