Anderson transition in disordered Hatano-Nelson systems

This paper establishes a universal criterion for the transition between the non-Hermitian skin effect and Anderson localization in disordered Hatano-Nelson systems by proving that the change in topological invariants coincides with the crossover in eigenvector behavior, as demonstrated through Lyapunov exponent analysis.

The Gist

Imagine you have a long, crowded hallway filled with people (these are the "particles" or "waves" in a quantum system). In a perfectly orderly hallway, people might walk freely back and forth. But in this paper, we are looking at two specific ways people get stuck in this hallway when things get messy or "disordered."

The author, Silvio Barandun, is investigating a tug-of-war between two different ways people get trapped: The "Skin Effect" and "Anderson Localization."

Here is the story of the paper, broken down into simple concepts:

1. The Two Types of Traps

The Non-Hermitian Skin Effect (The "Crowded Door" Effect)
Imagine the hallway has a slight wind blowing from left to right. Even if the hallway is perfectly empty and clean, this wind pushes everyone toward the right door. Everyone piles up at the exit, leaving the middle of the hallway empty.

• In the paper: This happens because the system is "non-Hermitian" (a fancy math term meaning the rules aren't perfectly symmetrical). The "wind" (represented by a parameter called $\gamma$) forces all the waves to condense at one edge of the system.

Anderson Localization (The "Obstacle Course" Effect)
Now, imagine you scatter random obstacles (like chairs or trash cans) all over the hallway. In a normal world, if you throw a ball, it might bounce around and eventually stop. But in a quantum world, if the obstacles are random enough, the waves get confused. They bounce off the obstacles so chaotically that they get stuck right where they are, unable to move forward or backward. They get "localized" in the middle of the hallway, not at the door.

• In the paper: This is the classic "Anderson Localization" caused by disorder (randomness).

2. The Big Question: When Does the Switch Happen?

For a long time, scientists knew about these two traps. But they didn't have a clear rule for when a system switches from being a "Skin Effect" (everyone at the door) to an "Anderson Localization" (everyone stuck in the middle).

The author asks: Is there a magic switch? Can we predict exactly when the wind stops being strong enough to push everyone to the door, and the random obstacles take over?

3. The "Magic Map" (The Topological Invariant)

The paper's big breakthrough is finding a "Magic Map" (mathematically called a topological invariant or a winding region).

• The Analogy: Imagine the hallway has a special "Safe Zone" drawn on the floor.
  • If a person (an eigenvalue) is standing inside this Safe Zone, the "wind" is strong enough to push them to the door (Skin Effect).
  • If a person steps outside this Safe Zone, the wind isn't strong enough to fight the random obstacles. They get stuck in the middle (Anderson Localization).

The author proves that the moment a person crosses the line of this Safe Zone, their behavior instantly changes from "piling up at the door" to "getting stuck in the middle."

4. The "Noise" Factor

Here is the most surprising part of the discovery, which is different from normal physics:

• In normal (Hermitian) physics: If you add even the tiniest speck of dust (noise) to a hallway, people immediately get stuck in the middle. The "wind" is instantly defeated.
• In this non-Hermitian physics: The "wind" is surprisingly tough! You can add a little bit of dust, and the wind still pushes everyone to the door. You need to add a minimum amount of dust (a specific threshold of disorder) before the wind finally loses and people get stuck in the middle.

The paper calculates exactly how much "dust" (disorder) is needed to break the "Skin Effect."

5. How They Proved It

The author used a mathematical tool called Lyapunov Exponents.

• The Analogy: Think of this as a "stability meter."
  • If the meter reads negative, it means the system is unstable in a way that pushes things to the edge (Skin Effect).
  • If the meter reads positive, it means the system is chaotic and traps things in the middle (Anderson Localization).

The author showed that this meter perfectly aligns with the "Magic Map" (the Safe Zone). When the math says you are inside the Safe Zone, the meter is negative. When you step outside, the meter flips to positive.

Summary: Why This Matters

This paper is like finding the instruction manual for a very strange, chaotic hallway.

1. It tells us that there is a clear, universal rule (the "Magic Map") that predicts whether waves will pile up at the edge or get stuck in the middle.
2. It proves that non-Hermitian systems (the "windy" ones) are more robust against disorder than we thought; they need a specific "kick" of chaos to break the skin effect.
3. It bridges the gap between two major fields of physics: the study of perfect crystals (topology) and the study of messy, random materials (disorder).

In short: The paper gives us a crystal-clear rule to predict when a quantum system will behave like a crowd rushing to an exit, and when it will behave like people getting lost in a maze.

Technical Summary

1. Problem Statement

The paper investigates the fundamental transition between two distinct localization regimes in disordered non-Hermitian quantum systems:

1. Non-Hermitian Skin Effect (NHSE): A phenomenon where all eigenmodes of a system condense and localize at one edge of the system, driven by non-reciprocal hopping (non-Hermiticity).
2. Anderson Localization: A phenomenon where disorder causes eigenstates to localize in the bulk of the system due to destructive interference.

The Core Question: In a Hatano-Nelson (HN) system with disorder, how does the system transition from NHSE (edge localization) to Anderson localization (bulk localization)? Specifically, the author seeks to establish a universal, a priori criterion (based on the unperturbed system's topology) to predict this transition without needing to solve the specific disordered instance.

A key distinction highlighted is that unlike Hermitian systems where any infinitesimal disorder induces Anderson localization, non-Hermitian systems require a finite minimum noise amplitude to overcome the stability of the NHSE and induce bulk localization.

2. Methodology

The author employs a combination of rigorous mathematical analysis, topological invariants, and numerical simulations.

• Model: The study focuses on the Hatano-Nelson Hamiltonian:
  $$H_{HN}^\gamma = \sum_i (e^{-\gamma}|i+1\rangle\langle i| + e^{\gamma}|i\rangle\langle i+1| + V_i|i\rangle\langle i|)$$
  where $\gamma$ represents the non-Hermitian asymmetry and $V_i$ are i.i.d. random on-site potentials.
• Lyapunov Exponents: The primary tool for analyzing localization is the Lyapunov exponent $L(z)$.
  • $L(z) < 0$: Indicates exponential growth in one direction and decay in the other, signifying NHSE (boundary localization).
  • $L(z) > 0$: Indicates exponential decay in both directions away from a point, signifying Anderson localization (bulk localization).
• Topological Invariants: The paper utilizes the "winding region" $W$ of the symbol function $f_\gamma(z) = e^\gamma z + V + e^{-\gamma}z^{-1}$ associated with the unperturbed (disorder-free) system.
  • $W$ is an ellipse in the complex plane with vertices at $\pm 2\cosh(\gamma)$.
  • The paper posits that the transition occurs exactly when an eigenvalue moves from inside $W$ to outside $W$.
• Analytical Approaches:
  • Lloyd Model: Exact analytical treatment using Cauchy-distributed disorder, allowing for the derivation of the complexified Lyapunov exponent via the Thouless formula.
  • Weak-Disorder Limit: Asymptotic analysis for general distributions with small variance, utilizing results from Pastur and Figotin regarding the Schrödinger model.

3. Key Contributions

1. Universal Criterion for Localization: The paper establishes that the transition from NHSE to Anderson localization coincides precisely with the transition of an eigenvalue from the interior to the exterior of the topological winding region $W$ of the unperturbed system.
2. Proof of Minimal Disorder Threshold: It proves that for non-Hermitian systems, there exists a non-zero minimum noise amplitude required to induce Anderson localization. This contrasts with Hermitian systems where any non-zero noise suffices.
3. Quantitative Error Bounds: For the Lloyd model (Cauchy disorder), the paper provides explicit error estimates showing that the Lyapunov exponent changes sign at the boundary of $W$ with errors scaling linearly ($O(s)$) inside $W$ and quadratically ($O(s^2)$) outside $W$ as the disorder scale $s \to 0$.
4. Linking Topology to Dynamics: It rigorously connects the change in the topological invariant (winding number) associated with an eigenvalue to the crossover of the corresponding eigenvector from edge-localized to bulk-localized.

4. Key Results

Theoretical Results

• Theorem 3.1 (Lloyd Model): For a Hatano-Nelson Hamiltonian with Cauchy-distributed potentials of scale $s$, the Lyapunov exponent $L_\gamma(\lambda)$ satisfies:
  • If $\lambda \in W$ (interior), $L_\gamma(\lambda) < 0$ (NHSE).
  • If $\lambda \notin W$ (exterior), $L_\gamma(\lambda) > 0$ (Anderson Localization).
  • This holds up to small errors dependent on the disorder scale $s$.
• Theorem 4.4 (General Weak Disorder): In the limit of vanishing disorder strength ($\gamma \to 0$) and small variance, the condition $L_\gamma(\lambda) > 0$ is equivalent to $\lambda \notin W$. This confirms the universality of the topological criterion beyond specific distributions.
• Relation Formula: The Lyapunov exponent of the non-Hermitian system is related to the Hermitian one by a simple shift: $L_\gamma(z) = L_0(z) - \gamma$. This reduction allows the complex non-Hermitian problem to be analyzed via the well-understood Hermitian case.

Numerical Simulations

• Figure 3: Simulations of 1,000 realizations with varying Cauchy scales ($s=0.1$ to $2.0$) demonstrate excellent agreement between the computed average Lyapunov exponent and the theoretical prediction. The phase transition (sign change of $L_\gamma$) occurs exactly at the boundary of $W$.
• Figure 4: Visualization of eigenvector localization positions (using softargmax).
  • Eigenvalues inside $W$: Eigenvectors are localized at the system edges (NHSE).
  • Eigenvalues outside $W$: Eigenvectors are distributed in the bulk (Anderson Localization).

5. Significance and Implications

• Predictive Power: The work provides an a priori method to determine the localization behavior of a disordered non-Hermitian system solely based on the topology of the underlying periodic structure and the eigenvalue's position relative to the winding region.
• Fundamental Physics: It clarifies the stability of the NHSE against disorder, showing that the skin effect is robust against weak disorder and only collapses when the disorder strength exceeds a specific threshold determined by the non-Hermiticity parameter $\gamma$.
• Mathematical Framework: The paper bridges the gap between the study of non-Hermitian crystalline systems (dominated by topological invariants) and disordered systems (dominated by Anderson localization), unifying them under the framework of Lyapunov exponents.

6. Future Directions

The author suggests extending this analysis to:

• Higher-dimensional non-Hermitian systems (where the topology of $W$ becomes more complex).
• The critical behavior at the boundary of $W$ (delocalized or multifractal states).
• Systems with correlated disorder rather than i.i.d. disorder.