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Anderson transition in disordered Hatano-Nelson systems

Silvio Barandun

Abstract

We illuminate the fundamental mechanism responsible for the transition between the non-Hermitian skin effect and defect-induced Anderson localization in the bulk via the study of Lyapunov exponents. We obtain a proof that the change of the topological invariant associated with an eigenvalue coincides with the eigenvector crossover from non-Hermitian skin effect to Anderson localization, establishing a universal criterion for localization behavior.

Anderson transition in disordered Hatano-Nelson systems

Abstract

We illuminate the fundamental mechanism responsible for the transition between the non-Hermitian skin effect and defect-induced Anderson localization in the bulk via the study of Lyapunov exponents. We obtain a proof that the change of the topological invariant associated with an eigenvalue coincides with the eigenvector crossover from non-Hermitian skin effect to Anderson localization, establishing a universal criterion for localization behavior.
Paper Structure (10 sections, 3 theorems, 28 equations, 5 figures)

This paper contains 10 sections, 3 theorems, 28 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. The non-Hermitian Lloyd Model
  4. Numerical simulations
  5. General asymptotic results
  6. Weakly disordered Schrödinger model
  7. Phase transition
  8. Numerical simulations
  9. Conclusion
  10. Acknowledgements

Key Result

Theorem 3.1

Let $H_{\text{HN}}$ be a Hatano-Nelson Hamiltonian as in eq: HN Hamiltonian with i.i.d. Cauchy distributed potentials of scale $s\in\mathop{\mathrm{\mathbb{R}}}\nolimits_{>0}$. Then the Lyapunov exponent associated to a frequency $\lambda \in \mathop{\mathrm{\mathbb{R}}}\nolimits$ satisfies up to an error that behaves linearly in $s$ for $\lambda \in \mathbb{W}$ and quadratically for $\lambda \no

Figures (5)

  • Figure 1: The transition of an eigenvalue from inside to outside the topological winding region $\mathbb{W}$ coincides with the eigenvector crossover from non-Hermitian skin effect to Anderson localization, establishing a universal criterion for localization behavior.
  • Figure 2: Cumulative distribution function for the Cauchy distribution for different scale parameters $s$.
  • Figure 3: Numerical simulation of $1000$ realizations of $H_{\text{HN}}^{\gamma=1}$ with potentials with Cauchy distributions of variable scale. On one hand one can appreciate the excellent agreement between the average computed Lyapunov exponent and the predicted value of \ref{['eq: layp herm to not herm']} and \ref{['eq: L0 Lloyd']}. On the other hand, the prediction power of \ref{['thm: main Lloyd']} remains accurate up to $s=1$. We remark that $\mathbb{P}[\vert X_{s=1}\vert > 1] = 0.5$ and $\mathbb{P}[\vert X_{s=2}\vert > 1] = 0.7$ for $X_{s}\sim$ Cauchy of scale $s$.
  • Figure 4: Grey dots represent pairs $(\lambda, \operatorname{softargmax}(v))$ where $(\lambda, v)$ is an eigenpair of one of 1000 samples of $H_{\text{HN}}^1$ for a Cauchy distribution of scale $s=0.1$. $\mathbb{W}$ represent the topological region associated to the deterministic $H_{\text{HN}}^1$. The transition from condensation to localization in the bulk is observed.
  • Figure 5: We show the convergence of average mean error as in \ref{['eq: mean error']} for a system of $N=10^5$ over $10$ runs. The stagnation of the error is due to the finite size of the system.

Theorems & Definitions (6)

  • Definition 1: Lyapunov exponent
  • Theorem 3.1
  • Definition 2: Homogeneous Markov process
  • Definition 3: Brownian motion model
  • Proposition 1: pastur.figotin1992Spectra
  • Theorem 4.1