Competing skin effect and quasiperiodic localization in the non-Hermitian Su-Schrieffer-Heeger chain: Reentrant delocalization, spectral topology destruction, and entanglement suppression

This study reveals that in a non-Hermitian Su-Schrieffer-Heeger chain, the competition between the skin effect and Aubry-André-Harper quasiperiodic disorder generates a unique reentrant delocalization regime and distinct five-phase landscape, while simultaneously destroying point-gap topology and suppressing entanglement entropy.

The Gist

Imagine a long, narrow hallway filled with people (these are the electrons in our quantum system). In a normal hallway, people can walk freely from one end to the other. But in this paper, we are looking at a very strange, "non-Hermitian" hallway where the rules of physics are slightly broken, and two powerful forces are fighting to control where the people stand.

Here is the story of that fight, explained simply.

The Setting: A Hallway with Two Rules

The hallway is a Su–Schrieffer–Heeger (SSH) chain. Think of this as a hallway with alternating wide and narrow doors between rooms. This specific pattern gives the hallway a "topological" personality—it has a special "edge" where people naturally like to hang out, even if they aren't supposed to.

Now, we introduce two opposing forces:

1. The "Wind" (The Non-Hermitian Skin Effect):
   Imagine a strong, one-way wind blowing down the hallway. No matter where you start, the wind pushes everyone toward the right wall. In physics, this is called the Skin Effect. It forces almost all the people to pile up in a massive, dense crowd at one end of the hallway, leaving the rest of the hallway empty.

2. The "Obstacle Course" (Quasiperiodic Disorder):
   Now, imagine someone places a series of strange, irregular obstacles (like boulders or tripwires) along the floor. These aren't random; they follow a specific, repeating-but-never-exactly-the-same pattern (like the rhythm of a song that never quite resolves). This is the Aubry–André–Harper (AAH) disorder.

   • If the obstacles are weak, people can still walk through.
   • If the obstacles are strong, they trap people in small pockets, preventing them from moving anywhere. This is localization.

The Big Discovery: A Five-Act Play

The researchers asked: What happens when the Wind and the Obstacle Course fight against each other?

They mapped out a "battlefield" (a phase diagram) and found five distinct zones where the outcome changes:

• Zone 1: The Free Walkers.
  The wind is weak, and the obstacles are weak. People walk freely down the hallway. The hallway keeps its special "topological" edge states.
• Zone 2: The Trapped.
  The wind is weak, but the obstacles are strong. Everyone gets stuck in small pockets along the floor. They can't move, but they aren't piled up at the wall.
• Zone 3: The Pile-Up.
  The wind is strong, but the obstacles are weak. Everyone is blown into a massive crowd at the right wall. The hallway is empty everywhere else.
• Zone 4: The Total Lockdown.
  Both the wind and the obstacles are very strong. The obstacles are so thick that even the wind can't push people to the wall. Everyone is trapped in tiny, isolated spots.
• Zone 5: The "Re-Entrant" Surprise (The New Discovery!).
  This is the most exciting part. Imagine the wind is strong (trying to pile people at the wall), but you start adding a moderate amount of obstacles.
  • What happens? The obstacles actually help the people escape the pile-up!
  • Why? The obstacles act like a "speed bump" for the wind. They break the wind's ability to push everyone to the wall, scattering the crowd back into the middle of the hallway. The people become more free than they were before.
  • The Catch: If you add too many obstacles, they trap everyone again.
  • The Result: As you increase the obstacles, the people go from "Piled Up" $\rightarrow$ "Free" $\rightarrow$ "Trapped." This "coming back to life" in the middle is called Re-entrant Delocalization. It's like a traffic jam that suddenly clears up because of a construction zone, only to jam again later.

Other Cool Findings

1. The Map is Broken in Two Places
Usually, when a system changes, it happens all at once. Here, the researchers found that the "Wind" destroys the system's shape (its complex loops) before the "Obstacles" stop the people from moving. It's like a building losing its blueprints before the walls actually collapse. These two events happen at different times.

2. The "Silence" and the "Noise"
In physics, "entanglement" is a measure of how connected the people are to each other.

• When the Wind dominates, everyone is crammed into a corner. They are so close they stop interacting in a complex way. The "connection" drops to near zero. It's a silent, dead zone.
• When the Obstacles are added (in that sweet spot of Zone 5), they scatter the crowd. Suddenly, the people start interacting again, and the "connection" (entanglement) comes back to life. The disorder actually restores the system's complexity.

3. Why the Hallway Design Matters
The researchers compared this "alternating door" hallway (SSH) to a "uniform door" hallway. They found that the alternating design is crucial. Without those alternating doors, the "Re-entrant" surprise (Zone 5) doesn't happen. The specific structure of the hallway creates the stage for this complex dance between the wind and the obstacles.

The Bottom Line

This paper shows that when you mix a one-way wind with a patterned obstacle course, you don't just get a mess. You get a rich landscape with five different behaviors. Most importantly, you can find a "sweet spot" where adding more obstacles actually frees up the system, allowing it to escape the grip of the wind before getting trapped again.

It's a reminder that in the quantum world, sometimes a little bit of chaos (disorder) is exactly what you need to break a bad habit (the skin effect) and let things flow again.

Technical Summary

1. Problem Statement

The paper investigates the complex interplay between two distinct physical phenomena in one-dimensional lattice systems:

1. Non-Hermitian Skin Effect (NHSE): The anomalous accumulation of an extensive number of eigenstates at a single boundary under open boundary conditions (OBC), driven by nonreciprocal hopping.
2. Aubry–André–Harper (AAH) Quasiperiodic Disorder: A deterministic disorder potential that drives a metal-insulator transition (localization) distinct from random Anderson localization.

While the individual effects of NHSE and AAH disorder have been studied in isolation (e.g., in non-Hermitian AAH chains or Hermitian SSH models), their simultaneous presence in a dimerized (Su–Schrieffer–Heeger or SSH) chain remains unexplored. The authors aim to determine how the SSH sublattice structure modifies the competition between directional skin accumulation and isotropic quasiperiodic localization, specifically looking for new phases, topological transitions, and entanglement behaviors.

2. Methodology

The authors employ a comprehensive approach combining analytical derivations and extensive numerical simulations:

• Model Definition: They define a non-Hermitian SSH-AAH Hamiltonian on a chain of $N$ unit cells (2N sites) with open boundary conditions.
  • Parameters: $v$ (intracell hopping), $w$ (intercell hopping), $\delta$ (nonreciprocity parameter breaking Hermiticity), and $\lambda$ (AAH modulation strength).
  • Potential: $V_n = \lambda \cos(2\pi\alpha n + \phi)$, where $\alpha$ is the inverse golden ratio.
• Analytical Approach:
  • Similarity Transformation: They apply an imaginary gauge transformation ($\hat{S} = \text{diag}(r, 1, r, 1, \dots)$) to map the non-Hermitian intracell hopping to an effective Hermitian hopping $v_{\text{eff}} = \sqrt{v^2 - \delta^2}$.
  • Localization Boundary: Using the self-duality of the AAH model, they derive an analytical expression for the critical localization boundary: $\lambda_c(\delta) = 2\sqrt{v_{\text{eff}}w}$.
• Numerical Techniques:
  • Exact Diagonalization: Computation of right and left eigenvectors for non-Hermitian matrices.
  • Phase Averaging: Averaging diagnostics over multiple phase offsets ($\phi$) of the AAH potential to eliminate sample-specific artifacts.
  • Finite-Size Scaling: Analyzing system sizes up to $N=100$ to distinguish true phase transitions from finite-size effects.
• Diagnostics:
  • Inverse Participation Ratio (IPR) & Fractal Dimension ($D_2$): To distinguish extended, localized, and multifractal states.
  • Skin Asymmetry ($A$): To quantify directional accumulation vs. isotropic localization.
  • Topological Invariants: Biorthogonal polarization (for band topology) and spectral winding number (for point-gap topology).
  • Biorthogonal Entanglement Entropy: To probe quantum correlations.

3. Key Contributions

The paper makes five primary contributions to the field of non-Hermitian topological matter:

1. Discovery of a Five-Phase Landscape: Identification of five distinct regimes in the $(\lambda, \delta)$ phase diagram, including a previously unreported competition regime exhibiting reentrant delocalization.
2. Analytical Localization Boundary: Derivation of the modified critical condition $\lambda_c(\delta) = 2\sqrt{v^2-\delta^2}\sqrt{w}$, demonstrating how nonreciprocity reduces the effective bandwidth and enhances susceptibility to localization.
3. Decoupled Topological Transitions: Demonstration that the destruction of point-gap topology (spectral winding) occurs at a different parameter strength than the band-topological transition (Zak phase/biorthogonal polarization).
4. Entanglement Suppression and Recovery: Observation that while the skin effect suppresses entanglement entropy to near-zero, sufficiently strong quasiperiodic disorder can partially restore entanglement by disrupting directional accumulation.
5. Role of Sublattice Structure: Proof that the SSH dimerization is essential for the five-phase structure; the non-dimerized limit (uniform chain) lacks the reentrant competition regime.

4. Key Results

A. The Five Distinct Regimes

The phase diagram reveals:

• Region I (Topological/Extended): Small $\lambda$, small $\delta$. Extended bulk states with topological edge states.
• Region II (AAH-Localized): Large $\lambda$, small $\delta$. Bulk states localized by quasiperiodic disorder; edge states destroyed.
• Region III (Skin-Localized): Small $\lambda$, large $\delta$. Directional accumulation of states at the boundary (NHSE).
• Region IV (Fully Localized): Large $\lambda$, large $\delta$. Both mechanisms contribute to localization; skin effect is overwhelmed by disorder.
• Region V (Competition/Reentrant): Intermediate $\lambda$ and moderate $\delta$. A novel regime where intermediate disorder disrupts the skin accumulation, leading to a reentrant partial delocalization. States become less localized than in the pure skin regime before eventually localizing again at high disorder.

B. Reentrant Delocalization

In Region V, the IPR shows a non-monotonic dip as $\lambda$ increases.

• Mechanism: At moderate $\lambda$, the quasiperiodic potential introduces competing localization centers in the bulk, redistributing wavefunction weight away from the boundary and breaking the directional coherence required for the skin effect.
• Validation: Finite-size scaling confirms this dip sharpens with increasing system size, ruling out finite-size artifacts.

C. Spectral Topology Destruction

• Increasing AAH strength ($\lambda$) scatters eigenvalues in the complex energy plane.
• The spectral winding number (point-gap topology) is destroyed (loops collapse to real lines) at a critical $\lambda$ before the band gap closes. This confirms that point-gap topology is more fragile to quasiperiodic disorder than band topology.

D. Entanglement Dynamics

• Skin Effect: Suppresses entanglement entropy ($S$) to near-zero values ($S \sim 10^{-9}$), consistent with previous predictions for NHSE.
• Disorder Recovery: Adding strong AAH disorder to the skin regime partially restores $S$. The quasiperiodic potential traps some states in the bulk, preventing complete boundary accumulation and re-establishing quantum correlations.

E. Importance of SSH Dimerization

Comparing the SSH model ($v \neq w$) with the non-dimerized limit ($v = w$):

• The non-dimerized model (standard non-Hermitian AAH) exhibits only a single localization-skin transition.
• The SSH dimerization introduces a topological gap and a two-band structure, which creates the intermediate Region V and the reentrant behavior.

5. Significance and Implications

• Theoretical Insight: The work challenges the assumption that localization mechanisms simply add up; instead, they can compete to produce non-monotonic, reentrant phases. It highlights that sublattice structure is a critical design parameter for engineering non-Hermitian phases.
• Experimental Realization: The predicted phenomena (reentrant delocalization, spectral loop collapse, entanglement recovery) are directly measurable in photonic waveguide arrays, topolectrical circuits, and cold-atom systems. Specifically, the non-monotonic transmission coefficient as a function of modulation strength serves as a clear experimental signature.
• Control of Quantum States: The finding that deterministic (quasiperiodic) disorder can restore entanglement suppressed by non-Hermiticity offers a new pathway for controlling quantum information in open systems without relying on random noise.

In conclusion, this paper establishes a rich phase diagram for non-Hermitian topological systems, revealing that the competition between skin effects and quasiperiodic disorder in dimerized chains leads to unique phenomena like reentrant delocalization and disorder-induced entanglement recovery, fundamentally distinct from single-band models.