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Spectral Topology and Delocalization in Disordered Hatano-Nelson Chains

Supriyo Ghosh, Sergej Flach

TL;DR

The paper investigates non-Hermitian Anderson localization in a unidirectional Hatano-Nelson chain with diagonal binary disorder, revealing a loop-to-two-loops spectral transition and a winding-number change from ν = 1 to ν = 0, with a critical point at h = t_2. Using analytic expressions for E_± = ±\sqrt{h^2 + t_2^2 e^{iq}} and a winding-number diagnostic, it shows the emergence of two purely imaginary, completely extended states when the topology is nontrivial, accompanied by a diverging localization length as q → π. The analysis demonstrates that, in the nontrivial phase, at least two states remain delocalized while all states localize in the trivial phase, and that generalized boundary conditions preserve these results (except for open boundaries). It also contrasts onsite versus hopping disorder, showing hopping disorder cannot localize due to a unitary equivalence to a uniform chain and clarifying the SSH connection via the irreducible block structure.

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.

Spectral Topology and Delocalization in Disordered Hatano-Nelson Chains

TL;DR

The paper investigates non-Hermitian Anderson localization in a unidirectional Hatano-Nelson chain with diagonal binary disorder, revealing a loop-to-two-loops spectral transition and a winding-number change from ν = 1 to ν = 0, with a critical point at h = t_2. Using analytic expressions for E_± = ±\sqrt{h^2 + t_2^2 e^{iq}} and a winding-number diagnostic, it shows the emergence of two purely imaginary, completely extended states when the topology is nontrivial, accompanied by a diverging localization length as q → π. The analysis demonstrates that, in the nontrivial phase, at least two states remain delocalized while all states localize in the trivial phase, and that generalized boundary conditions preserve these results (except for open boundaries). It also contrasts onsite versus hopping disorder, showing hopping disorder cannot localize due to a unitary equivalence to a uniform chain and clarifying the SSH connection via the irreducible block structure.

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.
Paper Structure (10 sections, 33 equations, 3 figures)

This paper contains 10 sections, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model and results
  3. Binary Onsite Disorder
  4. Symmetries and Winding Number
  5. Localization Length
  6. Participation Number
  7. Generalized Boundary Conditions
  8. Binary Disorder in the Hoppings
  9. Connection to the SSH Model
  10. Conclusion and Discussion

Figures (3)

  • Figure 1: Complex energy spectra of the binary disordered uHN chain. Parameters: $t_2 = 1, N \to \infty$. (a) Weak disorder ($h = 0.8 < t_2$); (b) Critical point ($h = 1.0 = t_2$); (c) Strong disorder ($h = 1.2 > t_2$).
  • Figure 2: (a) Spatial profile of $|\psi|^2$ for a localized eigenstate. (b) Probability density function (PDF) of the participation number for 1000 random disorder realizations. Parameters: $t_2 = 1, h = 0.8, N = 1000, q = \pi/2$.
  • Figure 3: Participation number (blue solid lines) and analytical localization length (red solid lines) as a function of $q$. The participation number is averaged over 100 random disorder configurations. Parameters: $t_2 = 1$, $N = 1000$; (a) weak disorder ($h = 0.8$); (b) critical disorder ($h = 1.0$); (c) strong disorder ($h = 1.2$).