Bound states in the continuum are waves that, defying conventional wisdom, remain localized even though they coexist with a continuous spectrum of radiating waves that can carry energy away. Their existence was first proposed in quantum mechanics and, being a general wave phenomenon, later identified in electromagnetic, acoustic, and water waves. They have been studied in a wide variety of material systems such as photonic crystals, optical waveguides and fibers, piezoelectric materials, quantum dots, graphene, and topological insulators. This Review describes recent developments in this field with an emphasis on the physical mechanisms that lead to these unusual states across the seemingly very different platforms. We discuss recent experimental realizations, existing applications, and directions for future work.

⇤ These authors contributed equally to this work.The ability to confine light is important both scientifically and technologically. Many light confinement methods exist, but they all achieve confinement with materials or systems that forbid outgoing waves. Such systems can be implemented by metallic mirrors, by photonic band-gap materials 1 , by highly disordered media (Anderson localization 2 ) and, for a subset of outgoing waves, by translational symmetry (total internal reflection 1 ) or rotation/reflection symmetry 3, 4 . Exceptions to these examples exist only in theoretical proposals [5][6][7][8] . Here we predict and experimentally demonstrate that light can be perfectly confined in a patterned dielectric slab, even though outgoing waves are allowed in the surrounding medium. Technically, this is an observation of an "embedded eigenvalue" 9 -namely a bound state in a continuum of radiation modes-that is not due to symmetry incompatibility [5][6][7][8][10][11][12][13][14][15][16] . Such a bound 1 state can exist stably in a general class of geometries where all of its radiation amplitudes vanish simultaneously due to destructive interference. This method to trap electromagnetic waves is also applicable to electronic 12 and mechanical waves 14,15 .The propagation of waves can be easily understood from the wave equation, but the localization of waves (creation of bound states) is more complex. Typically, wave localization can only be achieved when suitable outgoing waves either do not exist or are forbidden due to symmetry incompatibility. For electromagnetic waves, this is commonly implemented with metals, photonic bandgaps, or total internal reflections; for electron waves, this is commonly achieved with potential barriers. In 1929, von Neumann and Wigner proposed the first counterexample 10 , in which they designed a quantum potential to trap an electron whose energy would normally allow coupling to outgoing waves. However, such artificially designed potential does not exist in reality. Furthermore, the trapping is destroyed by any generic perturbation to the potential. More recently, other counterexamples have been proposed theoretically in quantum systems 11-13 , photonics 5-8 , acoustic and water waves 14,15 , and mathematics 16 ; the proposed systems in refs. 6 and 14 are most closely related to what is demonstrated here. While no general explanation exists, some cases have been interpreted as two interfering resonances that leaves one resonance with zero width 6,11,12 . Among these many proposals, most cannot be readily realized due to their inherent fragility. A different form of embedded eigenvalue has been realized in symmetry-protected systems 3, 4 , where no outgoing wave exists for modes of a particular symmetry.To show that an optical bound state is feasible even when it is surrounded by symmetrycompatible radiation modes, we consider a practical structure: a dielectric slab with a square array 2 of cylindrical holes (Fig. 1a), an example of photonic crystal (PhC) slab 1 . The periodic geomet...

We develop the topological band theory for systems described by non-Hermitian Hamiltonians, whose energy spectra are generally complex. After generalizing the notion of gapped band structures to the non-Hermitian case, we classify "gapped" bands in one and two dimensions by explicitly finding their topological invariants. We find nontrivial generalizations of the Chern number in two dimensions, and a new classification in one dimension, whose topology is determined by the energy dispersion rather than the energy eigenstates. We then study the bulk-edge correspondence and the topological phase transition in two dimensions. Different from the Hermitian case, the transition generically involves an extended intermediate phase with complex-energy band degeneracies at isolated "exceptional points" in momentum space. We also systematically classify all types of band degeneracies.