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Recent years have witnessed a surge of research on the non-Hermitian skineffect (NHSE) in one-dimensional lattices with finite-range couplings. In thiswork, we show that the long-range couplings that decay as $1/l^{\alpha}$ atdistance $l$ can fundamentally modify the behavior of NHSE and the scaling ofquantum entanglement in the presence of nonreciprocity. At $\alpha=0$, thenonlocality of couplings gives rise to the scale-free skin modes, whoselocalization length is proportional to the system size. Increasing the exponent$\alpha$ drives a complex-to-real spectral transition and a crossover from ascale-free to constant localization length. Furthermore, the scaling ofnonequilibrium steady-state entanglement entropy exhibits a subextensive lawdue to the nonlocality and the complex spectrum, in contrast to an area lawarising from NHSE. Our results provide a theoretical understanding on theinterplay between long-range couplings and non-Hermiticity.

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