Scaling laws for non-Hermitian skin effect with long-range couplings

Non-Hermitian systems are sensitive to the boundary conditions. Recent years have seen various studies on non-Hermitian skin effect (NHSE), where extensive bulk eigenstates pile at open boundary. Yet, little is known about the role of long-range coupling in a non-Hermitian system. Here we study the non-Hermitian skin effect in a one-dimensional lattice model with long-range couplings. We show that the non-locality gives rise to the scale-free localization, where the localization length of eigenstate is proportional to the system size. By tuning the coupling range, the competition between the nearest-neighbor coupling and non-locality results in the real-to-complex spectral transition and a crossover in the size dependence of localization length. Furthermore, we study the scaling of nonequilibrium steady-state entanglement entropy, where complex spectrum and NHSE contribute to subextensive and area laws, respectively. Our results shed light on the interplay between long-range coupling and non-Hermiticity.