Twisted topology of non-Hermitian systems induced by long-range coupling

We investigate the twisted topology of the complex eigenspectrum of a one-dimensional non -Hermitian system under the influence of long-range unidirectional coupling. Unlike the complex energy spectrum of the conventional Hatano-Nelson chain, which takes the form of a single loop with a topological winding index of a definite sign, the introduction of long-range unidirectional hopping results in the creation of multiple twisted loops. These twisted loops exhibit opposite signs of the topological winding index, which correlate to alternating clockwise and anticlockwise energy windings. The simultaneous presence of both signs of the winding index translates into a bipolar non -Hermitian skin effect (NHSE), which challenges the conventional wisdom that the NHSE localization is dependent on the direction of the dominant nearest -neighbor interactions. In this bipolar NHSE, the exponents of the complex energy eigenvectors corresponding to clockwise and anticlockwise windings lie inside and outside of the complex unit circle, respectively. Interestingly, at the intersections of oppositely oriented energy loops where the sign of the topological winding index flips, the energy becomes real valued, leading to a suppression of the NHSE. This marks the emergence of Bloch -like contact points, where both the bipolar and conventional NHSE vanish. Based on the non -Hermitian model, we provide analytical insights into the effects of long-range unidirectional coupling to the winding topology of its complex energy spectra and their broader implications for the field of condensed matter physics.