Emergent non-Hermitian models

The Hatano-Nelson and the non-Hermitian Su-Schrieffer-Heeger models are paradigmatic examples of non-Hermitian systems that host nontrivial 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 onsite 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 nonzero energies, with different exponential envelopes, depending on their energy. Overall, our work sheds light on the nature of topological phases and the non-Hermitian skin effect in one-dimensional systems.