Spectral Topology and Delocalization in Disordered Hatano-Nelson Chains

Abstract: The unidirectional Hatano-Nelson chain serves as the fundamental non-Hermitian building block of the Su-Schrieffer-Heeger (SSH) model. We investigate its Anderson localization properties under diagonal binary disorder. For weak disorder, the complex eigenvalue spectrum forms a single closed loop, which bifurcates into two distinct loops at a critical disorder threshold. Correspondingly, the spectral winding number ν undergoes a transition from ν = 1 in the weak-disorder regime, through ν = 1/2 at the critical point, to ν = 0 in the strong-disorder limit. We show that the eigenstates are exponentially localized, with a localization length that varies analytically as a function of the loop parameter. Notably, at weak and critical disorder, the spectrum hosts two completely delocalized states with diverging localization lengths. This divergence is directly correlated with the non-trivial spectral winding number. These findings remain robust under various boundary conditions, with the exception of strictly open boundaries.