Emergent non-Hermitian models

The Hatano-Nelson and the non-Hermitian Su-Schrieffer-Heeger model are paradigmatic examples of non-Hermitian systems that host non-trivial boundary phenomena. In this work, we use recently developed graph-theoretical tools to design systems whose isospectral reduction -- akin to an effective Hamiltonian -- has the form of either of these two models. In the reduced version, the couplings and on-site potentials become energy-dependent. We show that this leads to interesting phenomena such as an energy-dependent non-Hermitian skin effect, where eigenstates can simultaneously localize on either ends of the systems, with different localization lengths. Moreover, we predict the existence of various topological edge states, pinned at non-zero energies, with different exponential envelopes, depending on their energy. Overall, our work sheds new light on the nature of topological phases and the non-Hermitian skin effect in one-dimensional systems.