Tentaclelike spectra and bound states in the Hatano-Nelson chain with long-range impurity coupling

In non-Hermitian systems, the energy spectra and eigenstates exhibit high sensitivity to boundary conditions, lattice geometries, and local impurities. In this paper, we study the effect of long-range impurity coupling, located far from the boundaries, on the paradigmatic non-Hermitian Hatano-Nelson model. Through exact analytical treatment, we reveal the intriguing tentaclelike spectral structures that emerge from the otherwise Bloch or non-Bloch spectra under periodic or open boundary conditions, respectively. We show that these spectral tentacles are associated with emergent bound states near the impurity, with their number determined by the coupling range. We further determine the localization length of these tentacled states using the transfer matrix. Our work indicates that the long-range impurity coupling cannot be treated as a mere perturbative effect and holds promise for state manipulations in non-Hermitian systems.