Non-Hermitian skin effect in an extended Su-Schrieffer-Heeger model with balanced gain and loss and nonreciprocal next-nearest-neighbor hopping phase

Nonreciprocal hopping and balanced gain and loss are two regular approaches to revolutionize new physics individually within the scope of a non-Hermitian system. In stark contrast to prior studies, wherein non-Hermitian skin effects (NHSEs) typically arise from asymmetry nearest-neighboring (NN) hopping, here we show that NHSEs can also be enabled and steered in a modified Su-Schrieffer-Heeger (SSH) model with nonreciprocal long-range coupling, i.e., next-nearest-neighboring (NNN) hopping in the presence of balanced gain and loss. It can be found that NHSEs survive in most of the parameter spaces spanned by NN hopping ratio, non-Hermitian strength, and NNN hopping phase. The skin modes vary greatly in various schemes with NHSEs whether in amount or the concentrated degree. For fixed NNN hopping and non-Hermitian strength, the localization length of skin modes can be amplified by increasing the NN hopping amplitude. Beyond critical NN hopping ratio, all bulk states become localized along with the highest concentration degree at the same time. Our paradigm presents an alternative proposal to effectively design and manipulate NHSEs, highlighting the intricate interplay between nonreciprocal hopping and non-Hermitian strengths, which holds promise for future applications in topological non-Hermitian devices.