Non-Hermitian flat-band generator in one dimension

Flat bands---dispersionless bands---have been actively studied thanks to their sensitivity to perturbations, which makes them natural candidates for hosting novel and exotic states of matter. In parallel, non-Hermitian systems have attracted much attention in recent years as a simplified description of open system with gain and loss, motivated by potential applications. In particular, non-Hermitian system with dispersionless energy bands in their spectrum have been studied theoretically and experimentally. In general, flat bands require fine tuning of the Hamiltonian or protection by a symmetry. A number of methods was suggested to construct non-Hermitian flat bands relying either on the presence of symmetries, or specific frustrated geometries, often inspired by Hermitian models featuring flat bands. We introduce a systematic method to construct non-Hermitian flat bands using one-dimensional two-band tight-binding networks as an example, extending the methods used to construct systematically Hermitian flat bands. We show that the non-Hermitian case admits fine-tuned, nonsymmetry protected flat bands and provides more types of flat bands than the Hermitian case.