Non-Hermitian pseudo mobility edge in a coupled chain system

This study reveals that coupling a non-Hermitian chain exhibiting skin localization with a delocalized chain induces a pseudo mobility edge in the complex energy plane that separates localized and extended states while enabling unidirectional transport, with the system's transition to trivial extended phases characterized by a quantized winding number under specific boundary conditions.

The Gist

Imagine two parallel train tracks running side-by-side.

Track A is a strange, one-way street. If a train (representing a particle or signal) tries to move left, it gets pushed back hard. If it tries to move right, it zooms forward easily. In physics terms, this is a "non-Hermitian" chain with a "skin effect." Because of this one-way bias, if you put a train on this track, it doesn't spread out; it gets squashed and piled up at one specific end of the track.

Track B is a normal, fair street. Trains can move left and right equally well. If you put a train here, it spreads out evenly across the whole track. It never gets stuck at the ends.

Now, imagine these two tracks are connected by short bridges (called "rung couplings") that allow trains to hop from one track to the other.

The Main Discovery: The "Pseudo" Border

The researchers in this paper asked: What happens when you connect these two very different tracks?

They found that if the bridges between the tracks are weak, something fascinating happens. The "squashing" behavior from the strange Track A starts to leak over to Track B.

However, it doesn't squash everything. Instead, it creates a weird, invisible border in the middle of the energy spectrum (think of this as a border between "fast" and "slow" trains):

1. On one side of the border: The trains on both tracks get squashed and piled up at the ends. They are "localized."
2. On the other side of the border: The trains on both tracks remain spread out and free to roam. They are "extended."

This border is what the authors call a "Pseudo Mobility Edge."

Why "Pseudo"?

In normal physics, if you have a "Mobility Edge" (a border between stuck and free), the stuck stuff usually stops moving entirely. It's like a traffic jam where cars can't go anywhere.

But in this strange system, the "stuck" trains aren't actually stopped. Because of the one-way nature of Track A, even the trains that are piled up at the end are being amplified and pushed in one direction. It's like a traffic jam where the cars are all crammed into one spot, but they are also being supercharged and shot forward like a rocket. They are "localized" in position but "active" in movement. That's why it's a pseudo (fake) mobility edge—it looks like a stop, but it's actually a high-speed launchpad.

The Role of the Bridges

The behavior changes depending on how strong the bridges are:

• Weak Bridges: You get the "Pseudo Mobility Edge." Some trains are free, some are squashed and amplified.
• Strong Bridges: If you make the bridges very strong, the two tracks merge into one big system.
  • If both tracks have open ends (no loops), the whole system eventually gets squashed at the ends.
  • If one track is a loop (connected back to itself), the "squashing" effect is broken. The whole system becomes free and spread out again.

The "Bulk-Defect" Secret

The paper also discovered a deep mathematical secret connecting the two tracks.

Imagine you look at the system where both tracks are loops (Periodic Boundary Conditions). You can count how many times the energy of the system "winds" around a circle. This is called a Winding Number.

The researchers found that this winding number acts like a crystal ball.

• If the winding number is 2 (or -2), it predicts that if you cut the loops open (making them straight tracks), you will get the "Pseudo Mobility Edge" phase.
• If the winding number is 0, it predicts that cutting the loops open will result in a system where everything is free and spread out.

They call this a "Bulk-Defect Correspondence." It means the hidden topological properties of the "perfect" loop system (the bulk) perfectly predict what happens when you introduce a "defect" (cutting the loop open).

Summary in Plain English

The paper shows that when you connect a "one-way, sticky" system to a "normal, free" system:

1. The sticky behavior can spread to the normal system, but only for certain types of energy.
2. This creates a split where some states are stuck at the edge (but still moving fast in one direction) and others are free.
3. This split is controlled by a hidden mathematical number (the winding number) that you can calculate just by looking at the system as a loop, even before you cut it open.

This helps scientists understand how strange, one-way quantum effects behave when they interact with normal matter, revealing that these effects are surprisingly fragile and sensitive to how the system is connected at the edges.

Technical Summary

1. Problem Statement

The paper addresses a fundamental question in non-Hermitian physics: Can a mobility edge (ME) exist in a clean (disorder-free) non-Hermitian system?

• Context: Traditionally, mobility edges (energies separating localized and extended states) arise in disordered systems or quasiperiodic potentials. In non-Hermitian systems, the Non-Hermitian Skin Effect (NHSE) causes all eigenstates to localize at boundaries when translational symmetry is broken (Open Boundary Conditions, OBC).
• The Gap: It is unclear whether a system exhibiting NHSE can simultaneously host a phase where some states are localized (skin modes) while others remain extended, effectively creating a "mobility edge" without disorder. Furthermore, the interplay between NHSE and different boundary conditions in coupled systems remains underexplored.

2. Methodology

The authors propose and analyze a minimal hybridized Hatano-Nelson (HN) ladder model consisting of two coupled 1D chains of length $N$:

• Chain A (Non-Hermitian): A clean HN chain with non-reciprocal nearest-neighbor hoppings ($J e^{\pm \gamma}$), which exhibits NHSE under OBC.
• Chain B (Hermitian): A clean chain with reciprocal hoppings ($J$), which supports delocalized (extended) states.
• Coupling: The chains are coupled via site-to-site reciprocal hopping ($V$) and an on-site potential bias ($\mu$) between the sublattices.

Key Methodological Features:

• Boundary Conditions (BCs): The study systematically compares three BC configurations:
  1. Periodic BC (PBC): Both chains closed.
  2. Open BC (OBC): Both chains open (ends disconnected).
  3. Mixed BC (MBC): Chain A (HN) is Open, while Chain B (Hermitian) is Periodic.
• Diagnostic Tools:
  • Inverse Participation Ratio (IPR): To distinguish between localized ($IPR \sim \text{const}$) and extended ($IPR \sim 1/N$) states.
  • Fractal Dimension (FD): Calculated from the scaling of average IPR to characterize the nature of the states ($0 < FD < 1$ for pseudo-ME, $FD \approx 1$ for extended).
  • Spectral Winding Number ($w$): A topological invariant defined under PBC to characterize point-gap vs. line-gap phases.

3. Key Contributions

A. Discovery of the "Pseudo Mobility Edge"

The authors demonstrate that under Mixed Boundary Conditions (MBC) and weak interchain coupling ($V$), the system exhibits a Non-Hermitian Pseudo Mobility Edge.

• Phenomenon: In the complex energy plane, the spectrum separates into two distinct regions:
  1. Localized Region: States with skin-localized profiles (high IPR).
  2. Extended Region: States with delocalized profiles (low IPR).
• Distinction from Anderson Localization: Unlike Anderson localization where transport is suppressed, the localized states in this system allow for unidirectional signal amplification. Hence, the term "pseudo" mobility edge is used to distinguish it from the transport-blocking ME of disordered systems.

B. Boundary Condition Sensitivity and NHSE Propagation

• OBC Case: As coupling $V$ increases, the NHSE from Chain A propagates to Chain B. Eventually, for large $V$, all states in the hybridized ladder become skin-localized.
• MBC Case: The periodic boundary on Chain B prevents the full takeover of NHSE. Instead, a stable phase emerges where localized and extended states coexist. The range of this "pseudo-ME" phase is strictly determined by the point-gap condition of the PBC spectrum.

C. Establishment of "Bulk-Defect Correspondence"

This is a major theoretical contribution. The authors establish a direct link between the topological properties of the system under PBC and the localization properties under MBC.

• The Correspondence:
  • The Pseudo Mobility Edge phase under MBC corresponds exactly to the Point-Gap phase (with non-zero spectral winding number $w = \pm 2$) under PBC.
  • The Extended phase under MBC corresponds to the Line-Gap phase (with $w = 0$) under PBC.
• Interpretation: The open boundary of the HN chain in the MBC setup acts as a "defect." The topological winding number of the bulk (defined under PBC) predicts whether states near this defect (the open end) will be localized or extended. This validates a "bulk-defect correspondence" in quasi-1D non-Hermitian systems.

4. Key Results

1. Spectral Transitions:

   • Under PBC, the system transitions from a point-gap (loop spectrum) to a line-gap (separated bands) as $V$ increases. The critical coupling is $V_c = \pm \frac{2}{\cosh \gamma + 1}\sqrt{\cosh \gamma [(\cosh \gamma + 1)^2 - \mu^2]}$.
   • Under MBC, this spectral transition maps directly to the transition between the pseudo-ME phase and the fully extended phase.

2. Quantized Winding Number Jump:

   • The transition from the pseudo-ME phase to the extended phase is marked by a quantized jump in the spectral winding number $w$ from $\pm 2$ to $0$.
   • This winding number serves as a topological order parameter for the localization transition.

3. Scaling Analysis:

   • IPR Scaling: In the pseudo-ME phase, the maximum IPR scales as $N^0$ (localized), while the minimum IPR scales as $N^{-1}$ (extended).
   • Fractal Dimension: The FD is strictly between 0 and 1 in the pseudo-ME phase, confirming the coexistence of different localization types.

4. Robustness: The pseudo-ME phase is robust against perturbations in system parameters ($V, \mu$) as long as the point-gap condition is maintained.

5. Significance

• New Localization Mechanism: The work reveals a novel mechanism for generating mobility edges in clean non-Hermitian systems, driven purely by the interplay of non-reciprocity and boundary conditions, rather than disorder.
• Topological Insight: It deepens the understanding of the Non-Hermitian Skin Effect, showing that it is not an all-or-nothing phenomenon but can be tuned to create hybrid phases.
• Bulk-Defect Correspondence: The finding that a bulk topological invariant (winding number) can characterize the localization nature of states around a boundary "defect" extends the concept of Bulk-Boundary Correspondence (BBC) to non-Hermitian systems with mixed boundaries.
• Applications: The "pseudo mobility edge" implies that specific energy modes can be selectively amplified and transported directionally while others remain localized. This has potential applications in topological lasing, high-performance sensing, and unidirectional signal amplification in photonic and acoustic systems.

In summary, the paper uncovers a rich phase diagram in a simple coupled chain system, demonstrating that non-Hermitian topology can induce a "pseudo mobility edge" where localization and delocalization coexist, governed by a robust bulk-defect correspondence.