Non-Abelian gauge effect for the non-Hermitian Hatano-Nelson model in cylindrical geometry

Non-Abelian gauge offers a powerful route to engineer novel topological phenomena. Here, we systematically investigate a two-dimensional non-Hermitian Hatano-Nelson model incorporating an SU(2) non-Abelian gauge, demonstrating the emergence of Hopf-link bulk braiding topology in the complex energy spectrum solely with $x$-direction nearest-neighbor couplings. Given that the limitations of exceptional point topology in fully capturing the rich non-Hermitian skin effect (NHSE) under non-Abelian influence, we introduce a polarization parameter derived from the generalized Brillouin zone. This parameter quantitatively discerns left-, right-, and notably, both-side skin modes, with its accuracy corroborated by directly encoding real-space eigenstate. Our findings reveal that non-Abelian gauge provides unprecedented influence over NHSE, compared with Abelian gauge and without gauge cases. Furthermore, we uncover the pronounced clustering and size-effect-independent bipolar localization behavior of the eigenstates at the topological boundaries, as characterized by dynamics evolution and inverse participation ratio. This work establishes a paradigm for synthesizing and manipulating non-Hermitian topological phases driven by non-Abelian structures, opening avenues for topological engineering and holding promise for experimental realization in synthetic platforms.