Size-Dependent Skin Effect Transitions in Weakly Coupled Non-Reciprocal Chains

Non-Hermitian systems exhibit unique boundary phenomena absent in their Hermitian counterparts, most notably the non-Hermitian skin effect (NHSE). In this work, we explore a lattice model consisting of two coupled non-reciprocal chains, focusing on the interplay between system size, inter-chain coupling, and spectral topology. Using both analytical and numerical approaches, we systematically examine the evolution of the complex energy spectra and spectral winding numbers under periodic and open boundary conditions. Our results uncover a variety of size-dependent localization transitions, including the emergence and instability of concurrent bipolar skin effects in the $W=0$ region, and their crossover to unipolar and conventional bipolar NHSE as the system size increases. Notably, we demonstrate that these size-dependent behaviors persist even beyond the weak-coupling regime, highlighting their universality in non-Hermitian systems with complex spectral structures. This study provides insights into the mechanisms governing skin effects and offers practical guidelines for engineering non-Hermitian topological phases in synthetic lattices.