⚛️ quantum physics

Anomalous Localization and Mobility Edges in Non-Hermitian Quasicrystals with Disordered Imaginary Gauge Fields

This paper investigates a non-Hermitian Aubry-André-Harper chain with disordered imaginary gauge fields, revealing an anomalous transition between erratic skin effect and localized phases distinguished by Lyapunov exponents and spectral winding, as well as a unique mobility edge separating Anderson-localized states from macroscopic-accumulation states.

Original authors: Guolin Nan, Zhijian Li, Feng Mei, Zhihao Xu

Published 2026-04-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Guolin Nan, Zhijian Li, Feng Mei, Zhihao Xu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a crowded dance floor where everyone is trying to move to the music. In a normal world (what physicists call a "Hermitian" system), if the music gets too chaotic or the floor gets too slippery, the dancers eventually stop moving and huddle in small, scattered groups. This is called localization.

But this paper explores a very strange, "non-Hermitian" dance floor. Here, the rules of physics are slightly twisted: the floor isn't just slippery; it has invisible, one-way currents (like a conveyor belt) that push dancers in specific directions. This is the Non-Hermitian Skin Effect.

Usually, on this weird floor, if the currents are strong, everyone gets pushed to one specific wall of the room and piles up there. This is the "Skin Effect."

The Big Twist: The "Erratic" Dance Floor

The researchers in this paper introduced a new variable: disorder. They made the one-way currents random and unpredictable from spot to spot. Instead of a smooth conveyor belt pushing everyone to the left wall, the floor now has random, tiny currents pointing in different directions everywhere.

They discovered something bizarre happens here:

The "Erratic" Skin Effect (ENHSE): Instead of piling up at the wall, the dancers pile up in random, irregular spots in the middle of the room. It looks like a mess, but it's actually a specific type of "localization" caused by the random currents.
The "Anderson" Localization: If you turn up the chaos (the "quasiperiodic modulation"), the dancers stop following the currents entirely and just huddle in random, scattered spots, ignoring the currents.

The Mystery: How to Tell Them Apart?

Here is the tricky part: To a casual observer, both groups look the same. In both cases, the dancers are stuck in small clusters, and the "fractal dimension" (a fancy way of measuring how spread out they are) is zero for both. They both look "localized."

The researchers found a clever way to tell them apart, like using a special pair of glasses:

The Lyapunov Exponent (The "Growth Rate"): In the "Erratic" phase, the dancers' movement patterns don't grow or shrink exponentially; they are stable. In the "Anderson" phase, they grow or shrink rapidly.
The "Center of Mass" Wobble: In the Erratic phase, the dancers in a specific realization of the floor all tend to cluster around the same few random spots. If you look at the whole group, their "center of gravity" wobbles a lot depending on which random floor you picked. In the Anderson phase, the dancers are scattered so evenly across the room that the center of gravity is very stable.

The "Mobility Edge" Surprise

In normal physics, there is a concept called a Mobility Edge. Imagine a line on the dance floor. On one side, dancers can run freely (extended states). On the other side, they are stuck (localized states).

The researchers found that in this weird, non-Hermitian world, this "Mobility Edge" still exists, but it's anomalous (weird).

Normal World: The line separates "Running Dancers" from "Stuck Dancers."
This Paper's World: The line separates two different types of "Stuck Dancers."
- On one side of the line, dancers are stuck in the "Erratic Skin" piles (macroscopic accumulation).
- On the other side, they are stuck in the "Anderson" scattered piles.
- Crucially: There are no "Running Dancers" on this floor! The whole system is stuck, just stuck in two different ways.

The "Drift" and the "Winding Number"

The paper also looked at how a single dancer (a wave packet) moves over time.

The Winding Number: This is like a compass that tells you which way the "invisible currents" are swirling.
The Drift: If the compass says "Left," the dancer drifts left. If it says "Right," they drift right.
The Magic: If you look at one specific dance floor, the dancer drifts in a specific direction. But if you average out thousands of different random floors, the left-drifters and right-drifters cancel each other out. The result? The average dancer looks like they are just standing still, behaving exactly like a dancer on a normal, boring floor.

The Takeaway

This paper is a detective story about a strange quantum world.

The Crime: Dancers are getting stuck in two different ways that look identical to the naked eye.
The Clue: By measuring how their "center of gravity" wobbles and how their energy levels behave, the researchers found a way to distinguish the "Erratic Pile-Up" from the "Scattered Huddle."
The Twist: The boundary between these two states (the Mobility Edge) doesn't separate "moving" from "stuck." It separates "stuck in a pile" from "stuck in a scatter."
The Lesson: Even in a chaotic, non-Hermitian world, there are hidden patterns (topology) that dictate how things move and where they get stuck. If you know how to look (using spectral winding and center-of-mass fluctuations), you can see the invisible currents driving the chaos.

In short: Just because everyone is stuck, doesn't mean they are stuck for the same reason. And sometimes, the "stuck" behavior is actually a sign of a hidden, swirling topological order.

1. Problem Statement

The paper investigates the interplay between non-Hermiticity, quasiperiodicity, and spatially disordered non-reciprocity.

Context: While the Non-Hermitian Skin Effect (NHSE) typically causes eigenstates to accumulate at boundaries, and Anderson localization causes states to be distributed throughout the lattice, recent work identified an "Erratic Non-Hermitian Skin Effect" (ENHSE) where states accumulate at irregular bulk positions determined by disorder.
Gap: It remains unclear how to distinguish ENHSE states from conventional Anderson-localized states, as both exhibit vanishing fractal dimensions in the thermodynamic limit. Furthermore, the existence and nature of mobility edges (boundaries separating localized and extended states) in non-Hermitian systems with disordered gauge fields are not well understood. Specifically, can a mobility edge separate two distinct types of localized states (e.g., Anderson vs. ENHSE) rather than extended vs. localized states?

2. Methodology

The authors study a one-dimensional generalized Aubry-André-Harper (AAH) model with the following features:

Hamiltonian: A tight-binding model with nearest-neighbor (NN) and next-nearest-neighbor (NNN) hopping.
Disordered Imaginary Gauge Field: Implemented via asymmetric hopping amplitudes $J_L^j = J_1 e^{-h_j}$ and $J_R^j = J_1 e^{h_j}$ , where $h_j$ is a Bernoulli random variable ( $\pm \Delta h$ ). This introduces spatially varying non-reciprocity.
Quasiperiodic Potential: An onsite potential $\lambda_j = \lambda \cos(2\pi\alpha j)$ with irrational $\alpha$ .
Cases Studied:
1. Standard AAH limit ( $J_2 = 0$ ): To investigate the transition between fully ENHSE and fully localized phases.
2. Generalized AAH limit ( $J_2 \neq 0$ ): To investigate the emergence of mobility edges.
Analytical Tools:
- Gauge Transformation: A non-unitary transformation maps the non-Hermitian Hamiltonian to a Hermitian generalized AAH model, allowing the use of known Hermitian results for Lyapunov exponents.
- Diagnostics: Fractal dimension ( $D$ ), Lyapunov exponent ( $\gamma$ ), center-of-mass fluctuation ( $S$ ), spectral winding numbers, and wave-packet dynamics.
- Boundary Conditions: Comparisons between Periodic Boundary Conditions (PBC) and Open Boundary Conditions (OBC) to analyze spectral topology and skin effects.

3. Key Contributions and Results

A. Anomalous Localization Transition ( $J_2 = 0$ )

The authors identify a transition from a fully ENHSE phase to a fully Anderson-localized phase at a critical modulation strength $\lambda_c = 2$ .

The "Anomaly": Unlike the Hermitian AAH model (which transitions from extended to localized), this transition connects two confined regimes.
Diagnostic Failure of Fractal Dimension: Both phases exhibit a vanishing fractal dimension ( $D \to 0$ ) in the thermodynamic limit, making $D$ insufficient to distinguish them.
Successful Diagnostics:
- Lyapunov Exponent ( $\gamma$ ): $\gamma = 0$ in the ENHSE phase (indicating no exponential decay/growth relative to the gauge) but $\gamma > 0$ in the Anderson-localized phase.
- Center-of-Mass Fluctuation ( $S$ ): In the ENHSE phase, eigenstate centers cluster around a few disorder-selected bulk positions ( $S \propto N$ ). In the Anderson phase, centers are uniformly distributed ( $S \propto N^2$ ).
Spectral Signatures:
- Complex-to-Real Transition: Under PBCs, the spectrum is complex in the ENHSE phase and becomes purely real in the localized phase.
- Winding Number: The spectral winding number changes from non-trivial ( $w = \pm 1$ ) in the ENHSE phase to trivial ( $w = 0$ ) in the localized phase. The sign of the winding correlates with the direction of the macroscopic accumulation (left or right).

B. Anomalous Mobility Edges ( $J_2 \neq 0$ )

With weak NNN hopping, the system exhibits an anomalous mobility edge at the same energy location $E_c(\lambda)$ as the Hermitian generalized AAH model.

Physical Meaning: In the non-Hermitian frame, this edge does not separate extended and localized states. Instead, it separates:
- Low-energy states: Anderson-localized states (uniformly distributed centers, real spectrum, trivial winding).
- High-energy states: ENHSE-type macroscopic accumulation states (clustered centers, complex spectrum, non-trivial winding).
Phase Diagram: The system exhibits three regimes:
1. Fully ENHSE: All states accumulate at irregular bulk positions.
2. Coexistence (Mobility Edge): A mix of Anderson-localized and ENHSE states.
3. Fully Localized: All states are Anderson-localized.
Spectral Correlation: The mobility edge coincides with the boundary where the PBC spectrum transitions from complex to real. States below $E_c$ have real energies; states above $E_c$ have complex energies.

C. Dynamical Detection

The authors analyze wave-packet dynamics to provide experimental signatures:

Winding-Dependent Drift: For a single disorder realization, the wave packet exhibits a directional drift (left or right) determined by the sign of the spectral winding number ( $w = +1$ or $-1$).
Group Averaging: If realizations are grouped by the sign of their winding number, the average dynamics preserves the directional drift (opposite directions for opposite signs).
Full Disorder Averaging: When averaging over all disorder realizations (without grouping), the opposite drifts cancel out, restoring Hermitian-like transport (ballistic spreading or localization without net drift). This explains why the non-Hermitian nature might be hidden in standard ensemble-averaged experiments.

4. Significance

New Classification of Localization: The work establishes that localization in non-Hermitian quasicrystals is not binary (extended vs. localized) but can involve distinct types of confined states (ENHSE vs. Anderson) separated by mobility edges.
Robust Diagnostics: It provides practical tools (Lyapunov exponent, center-of-mass fluctuation, spectral winding) to distinguish between these phases, overcoming the limitations of fractal dimension analysis.
Topological-Dynamical Link: It reveals a direct link between spectral topology (winding numbers) and real-space dynamics (directional drift), showing that the "skin effect" in disordered systems is not just a boundary phenomenon but a bulk phenomenon sensitive to disorder realizations.
Experimental Relevance: The findings suggest that while full disorder averaging may mask non-Hermitian effects, analyzing specific disorder realizations or winding-resolved dynamics could reveal these anomalous phases in photonic, acoustic, or electrical circuit platforms.

In summary, this paper redefines the understanding of localization transitions in non-Hermitian quasicrystals, demonstrating that disordered imaginary gauge fields induce a unique "erratic" skin effect and that mobility edges can separate different topological classes of localized states.