🔬 mesoscale physics

Interplay of Quasiperiodic Criticality and the Non-Hermitian Skin Effect

This paper analytically demonstrates that in a non-Hermitian Hatano-Nelson model with quasiperiodically modulated hopping, parameter regimes hosting critical eigenstates under periodic boundary conditions can exhibit the non-Hermitian skin effect under open boundary conditions, a phenomenon confirmed through exact Lyapunov exponent calculations and extended to long-range and multiband systems.

Original authors: Zhangyuan Chen, Xianqi Tong, Xiaosen Yang

Published 2026-07-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Zhangyuan Chen, Xianqi Tong, Xiaosen Yang

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 the music isn't random noise, but a perfectly repeating, yet never-ending pattern—like a rhythm that shifts just slightly every time it loops. In the world of quantum physics, this is called a quasiperiodic lattice. Usually, dancers (particles) on such a floor either spread out evenly across the room or huddle tightly in one corner. But there's a special, tricky zone in between where they do something weird: they stay in a state of "critical" tension, neither fully spread out nor fully stuck.

Now, imagine the dance floor itself is a bit unfair. It's non-Hermitian, meaning the rules of the dance are lopsided. If a dancer tries to step to the right, they might find the floor slippery and slide easily, but stepping left is like wading through mud. This imbalance is called nonreciprocal hopping. In physics, this unfairness usually causes a phenomenon known as the Non-Hermitian Skin Effect (NHSE). Think of it like a crowd of people at a concert who, because the exit on the left is jammed and the right is wide open, all suddenly pile up against the right wall, leaving the rest of the room empty.

The Big Discovery
The researchers in this paper, Zhangyuan Chen, Xianqi Tong, and Xiaosen Yang, asked a fascinating question: What happens if you combine this "unfair floor" with the "critical tension" zone?

They built a mathematical model (a Hatano–Nelson model) to test this. They found that the answer depends entirely on how you look at the dance floor.

The "Closed Loop" View (Periodic Boundary Conditions): If you imagine the dance floor is a giant circle where the right edge connects back to the left edge, the dancers in the critical zone stay right where they are. They don't pile up. They are spread out in a complex, fractal pattern that looks messy but is actually stable.
The "Open Door" View (Open Boundary Conditions): But the moment you cut the loop and make it a straight line with two distinct ends (an open floor), the magic changes. The dancers who were calmly standing in the critical zone suddenly get pushed by the unfair rules. They all rush to one side or the other, piling up against the wall. This is the Skin Effect appearing right inside the critical zone.

How They Knew It Was True
The team didn't just guess; they used a clever mathematical trick called a non-unitary gauge transformation. Imagine taking a photo of the dancers and stretching or shrinking the image locally to make the "slippery" floor look "normal" and fair again. Once they did this, they could use standard math to calculate exactly how the dancers would behave.

They derived a precise formula for something called the Lyapunov exponent (a number that tells you how fast a dancer's movement dies out or grows).

If this number is positive, the dancers pile up on the right.
If it's negative, they pile up on the left.
If it's zero, they stay spread out.

They tested their math with computer simulations on a system size of N = 233 (a specific number of dance spots). They also checked a few other scenarios:

Long-range hopping: What if dancers could jump over their neighbors? They simulated this with nearest-neighbor and next-nearest-neighbor jumps, and the pile-up effect still happened.
Two-band systems: What if there were two types of dancers (like a duet)? They tested a Rice–Mele model with N = 89 unit cells, and again, the critical states turned into skin states when the floor was open.

What They Ruled Out
The paper is very clear about what doesn't happen. They explicitly show that the "critical" states (the ones that are neither fully stuck nor fully free under closed loops) do not remain in that critical, non-piling state when you open the boundaries. Instead of staying put, they transform: the parameter regimes that host critical states under closed loops exhibit the Non-Hermitian Skin Effect under open boundaries. The idea that these critical states would remain immune to the pile-up is ruled out by their analysis.

The Bottom Line
The authors found that the "unfairness" of the floor (nonreciprocity) and the "pattern" of the music (quasiperiodicity) work together to create a surprise. The critical states, which seem stable in a closed loop, are actually sitting on a time bomb. As soon as you open the boundaries, they collapse into a pile-up against the wall.

They didn't just suggest this might happen; they provided an exact analytical expression for the boundary between the left-pile and right-pile regimes. Their simulations and math show that this behavior isn't a fluke of a simple model—it persists even when you add long jumps or extra bands of dancers.

So, in the world of these quantum dancers, the lesson is: Don't trust the calm. Even in the most balanced, critical-looking crowd, if the floor is slightly unfair and the doors are open, everyone is going to rush to the exit.

Technical Summary: Interplay of Quasiperiodic Criticality and the Non-Hermitian Skin Effect

Problem Statement
This work investigates the localization phenomena in non-Hermitian systems where quasiperiodic modulation coexists with nonreciprocal hopping. Specifically, the authors address the interplay between quasiperiodic critical eigenstates—states that are neither fully extended nor exponentially localized—and the non-Hermitian skin effect (NHSE), a phenomenon where bulk eigenstates accumulate at boundaries under open boundary conditions (OBC). A central question is whether parameter regimes that host quasiperiodic critical states under periodic boundary conditions (PBC) can exhibit the NHSE when open boundaries are imposed, and how nonreciprocity arising from unequal modulation strengths influences this behavior.

Methodology
The authors analyze an off-diagonal Hatano–Nelson (HN) model with quasiperiodically modulated hopping amplitudes. The nonreciprocity in the system is generated not by constant asymmetric hopping, but by unequal modulation strengths of the right ( $J^R_n$ ) and left ( $J^L_n$ ) hopping amplitudes.

Non-Unitary Gauge Transformation: The core analytical tool employed is a non-unitary gauge transformation ( $\psi_n = \phi_n G_n$ ). This transformation maps the non-Hermitian system with OBC into an equivalent Hermitian quasiperiodic system. This mapping allows for the exact analytical evaluation of the Lyapunov exponent ( $\lambda$ ) in the thermodynamic limit.
Analytical Derivation: By applying the transformation, the authors derive an exact expression for $\lambda$ as a function of the hopping parameters. The phase boundary separating left-skin and right-skin regimes is determined by the condition $\lambda = 0$ .
Numerical Verification and Scaling: The analytical predictions are verified using numerical simulations. Key metrics include:
- Directional Inverse Participation Ratio (dIPR): Used to quantify the accumulation direction of eigenstates under OBC.
- Inverse Participation Ratio (IPR) and Finite-Size Scaling: Used under PBC to distinguish between extended, localized, and critical states. The fractal dimension $\beta$ is extracted via finite-size scaling of the mean IPR (MIPR) to identify critical regimes ( $0 < \beta < 1$ ).
Generalizations: The analysis is extended to two broader classes of models:
- Long-range hopping Hamiltonians (including next-nearest neighbors).
- A two-band non-Hermitian Rice–Mele lattice with quasiperiodic modulations.

Key Results

Exact Analytical Phase Boundary: The study provides an exact analytical expression for the Lyapunov exponent and the phase boundary between left-skin and right-skin regimes. The boundary depends on the ratio of modulation strengths ( $\mu_R/\mu_L$ ) and the uniform hopping components ( $t$ ).
Boundary-Dependent Localization: The comparison between OBC and PBC reveals a distinct phenomenon: parameter regimes that host quasiperiodic critical states under PBC (identified by a fractal dimension $0 < \beta < 1$ ) can exhibit the NHSE under OBC. In these regimes, the eigenstates accumulate at the boundaries, with the direction of accumulation (left or right) determined by the sign of the Lyapunov exponent.
Reversal of Skin Direction: As modulation parameters are varied, the Lyapunov exponent changes sign, indicating a reversal in the direction of skin localization. The point where $\lambda = 0$ corresponds to the phase boundary where the NHSE vanishes.
Robustness of the Phenomenon: The findings are not restricted to the nearest-neighbor single-band HN chain.
- In long-range hopping models, the non-unitary gauge transformation still successfully removes nonreciprocity, and the condition $\lambda = 0$ remains governed by the accumulated logarithmic imbalance of hoppings. The NHSE persists in critical regimes.
- In the two-band Rice–Mele model, parameter regimes hosting critical states under PBC similarly exhibit the NHSE under OBC.

Significance and Claims
The paper claims to provide an analytically controlled perspective on how quasiperiodicity, modulated nonreciprocity, and boundary conditions jointly shape the non-Hermitian skin effect in critical regimes. The primary contribution is the demonstration that the NHSE is not exclusive to localized or extended phases but can emerge specifically in quasiperiodic critical regimes when boundary conditions are switched from PBC to OBC. The work highlights the boundary-sensitive nature of critical localization in the presence of quasiperiodically modulated nonreciprocity. The authors suggest these findings may offer guidance for future experimental realizations in non-Hermitian systems.

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