  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 anti-clockwise
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 NHSE and the traditional 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.
