Observation of flat-band skin effect

This paper theoretically predicts and experimentally demonstrates a unique flat-band skin effect in non-Hermitian systems, where the phenomenon arises from the spectral topology of surrounding dispersive bands rather than the flat band itself, exhibiting a counterintuitive disappearance at high non-Hermiticity and singular gap-closing behavior at exceptional points.

The Gist

Imagine a crowded dance floor where everyone is trying to move to the music. Usually, in a physics system, waves (like sound or light) travel across this floor, moving from one spot to another at different speeds depending on their energy. This is like a dispersive band: the waves spread out, and their speed depends on how "energetic" they are.

But sometimes, the dance floor is designed in a special way (due to specific symmetries) that creates a flat band. Here, the "dance moves" (waves) have zero speed. They get stuck in tiny, compact clusters called Compact Localized States (CLS). Imagine a group of dancers who are perfectly synchronized in a small circle but never leave that circle, no matter how long the music plays. In normal physics, these stuck dancers stay exactly where they are.

The Twist: The "Skin Effect"

Now, imagine we introduce a "non-Hermitian" element to the dance floor. In physics terms, this means the system is open to the outside world, exchanging energy or particles, making it slightly "unbalanced" (like having some dancers who gain energy and others who lose it).

Usually, in these unbalanced systems, a phenomenon called the Non-Hermitian Skin Effect (NHSE) happens. Think of it like a strong wind blowing across the dance floor. Even if the dancers were supposed to stay in the middle, the wind pushes all the waves to one edge of the floor, piling them up against the wall. This is the "skin effect."

The Big Discovery: Flat Bands Can Get "Skinny" Too

The authors of this paper discovered something surprising: Even the stuck, zero-speed dancers (the flat band) can be pushed to the edge by this wind. They call this the Flat-Band Skin Effect (FBSE).

However, this doesn't happen all the time. It's a bit like a magic trick that only works under very specific conditions:

1. The "Encirclement" Rule: For the stuck dancers to get pushed to the edge, the "wind" (the non-Hermitian parameters) must be strong enough to make the other moving dancers (the dispersive bands) form a loop around the stuck ones on a complex map.
2. The Re-entrant Surprise: If you make the wind too strong, the magic stops. The loop breaks, the stuck dancers are no longer "encircled," and they suddenly stop moving to the edge, returning to their original spots. This is counterintuitive: usually, more wind means more pushing, but here, too much wind actually stops the effect.

The Experiment: A Mechanical Lattice

To prove this, the researchers built a physical model using 36 mechanical rotors (like spinning arms) connected by springs and motors.

• They tuned the motors to create the "unbalanced" conditions (the wind).
• They shook one specific arm (the source) and watched how the vibration spread.
• Result: When the conditions were just right, the vibration didn't stay near the source; it traveled all the way to the far end of the chain and piled up there, even though the system was designed to have "stuck" waves. When they cranked the motors too hard, the vibration stayed put again.

The "Ghost" Connection

The paper also explains why this happens using a concept called biorthogonality. Imagine the dancers have two sides: a "Right Side" (where they physically are) and a "Left Side" (a ghostly partner that influences them).

• In normal situations, both sides are in the same place.
• In this Flat-Band Skin Effect, the "Right Side" of the dancers stays spread out across the floor, but their "Left Side" partners get sucked to the opposite edge.
• Because the system's response depends on both sides, the "Left Side" being at the edge pulls the whole system's reaction to that edge. It's as if the dancers are being pulled by a ghostly rope tied to the wall.

The "Zipper" and the "Singularity"

The researchers also found that at the exact moment the effect starts or stops, the system hits a special point called an Exceptional Point (EP3).

• Think of the energy levels of the dancers as zippers. Usually, the zippers for the "stuck" group and the "moving" group are separate.
• At this special point, the zippers merge. Three different types of waves (two from the moving group and one from the stuck group) fuse together into a single, singular state.
• This fusion creates a "kink" in the geometry of the system. If you try to walk smoothly across this point, the system's behavior jumps discontinuously, like stepping off a cliff.

Summary

In simple terms, the paper shows that even waves that are supposed to be completely stuck in place can be forced to the edge of a system, but only if the surrounding waves form a specific loop around them. If you push the system too hard, the loop breaks, and the waves stop moving. This reveals a new, weird way that energy can be controlled and localized in systems that exchange energy with their environment.

Technical Summary

Technical Summary: Observation of Flat-Band Skin Effect

Problem Statement
Symmetry-protected ideal flat bands in one-dimensional (1D) Hermitian lattices are characterized by compact localized states (CLS) and vanishing group velocity. While the non-Hermitian skin effect (NHSE) is a well-established phenomenon where eigenmodes of dispersive bands accumulate at open boundaries due to nontrivial point-gap topology, the interplay between NHSE and flat bands populated by CLS remains largely unexplored. Specifically, it is unclear whether flat bands, which typically possess trivial spectral topology (appearing as a single point in the complex energy plane), can exhibit skin effects, and how such effects relate to the surrounding dispersive bands.

Methodology
The authors investigate 1D non-Hermitian lattices supporting an ideal flat band using a combination of theoretical modeling, numerical simulation, and experimental realization.

• Theoretical Model: A three-band non-Hermitian Bloch Hamiltonian is constructed with a specific rank-deficient structure ($M=2$), ensuring a zero-energy flat band regardless of non-Hermiticity. The model includes reciprocal ($t_1, t_2$) and non-reciprocal ($\gamma_1, \gamma_2$) hopping parameters.
• Numerical Analysis: The authors analyze the system under both periodic boundary conditions (PBC) and open boundary conditions (OBC). To diagnose the skin effect in the flat band, they utilize the Green's function $G(E_\eta) = (E_\eta - H_{OBC})^{-1}$ with $E_\eta \to 0$. They define a spatially averaged response $\chi$ (analogous to a center of mass) to quantify the localization of the flat-band modes. They also employ the Generalized Brillouin Zone (GBZ) formalism, replacing the Bloch wavevector $k$ with a complex non-Bloch wavevector $\beta$, to analyze exceptional points (EPs).
• Experimental Realization: An active mechanical lattice consisting of 36 spring-coupled rotational harmonic oscillators (12 unit cells) is fabricated. Non-reciprocal coupling is achieved via programmable brushless DC motors driven by active feedback loops. The system is tuned to a resonant frequency of 10.9 Hz. Steady-state responses to local harmonic excitations are measured to map the spatial distribution of the flat-band modes.

Key Contributions and Results

1. Discovery of Flat-Band Skin Effect (FBSE): The study demonstrates that flat-band modes can exhibit a skin effect (FBSE) under OBC, where the response localizes at the boundary. Crucially, unlike conventional NHSE in dispersive bands which arises from the band's own nontrivial topology, the FBSE emerges only when the flat band is enclosed within the point gap formed by the dispersive bands under PBC.
2. Re-entrant Behavior: The FBSE exhibits a counterintuitive re-entrant behavior. It appears within a finite range of non-Hermitian parameters (where the dispersive bands enclose the flat band) but disappears when non-Hermitian parameters become large, causing the line gaps between dispersive bands to reopen and the flat band to fall outside the spectral loops.
3. Role of Left Eigenvectors (LEVs): Theoretical analysis reveals that while the right eigenvectors (REVs) of the flat band remain compact localized states (CLS) distributed across the bulk even in the FBSE regime, the corresponding biorthogonal left eigenvectors (LEVs) become exponentially localized at the opposite boundary with divergent amplitudes. The FBSE is driven by this biorthogonal response effect.
4. Third-Order Exceptional Points (EP3s): The authors identify that the gap between the flat band and dispersive bands closes at higher-order exceptional points (EP3s) under both PBC and OBC. These EP3s occur on the GBZ plane where the non-Bloch wavevectors of the flat band coalesce with those of two dispersive band modes.
5. Singular Quantum Geometry: The gap-closing at EP3s leads to a discontinuity in the quantum distance across the EP. Specifically, the quantum distance between biorthogonal LEVs and REVs ($d_{LR}$) remains finite and anisotropic near the EP, a signature of singular flat bands not typically found in 1D Hermitian systems.
6. Experimental Verification: The mechanical lattice experiments successfully observe the FBSE. In the parameter region where the flat band is enclosed by the dispersive bands, excitations localize at the right edge. In regions where the flat band is outside the gap (large non-Hermiticity), the response remains localized near the source. The experiments also confirm the standard NHSE in the dispersive bands across all parameters.

Significance
The paper claims to reveal a new class of non-Hermitian phenomena unique to flat-band systems. The primary significance lies in demonstrating that the skin effect in flat bands is not an intrinsic property of the flat band itself (which has trivial topology) but is induced by the nontrivial point-gap topology of the surrounding dispersive bands. This challenges the conventional understanding that flat bands in 1D systems are non-singular and highlights the role of higher-order exceptional points (EP3s) in non-Hermitian quantum geometry. The work suggests new possibilities for controlling localization and quantum geometry in metamaterials, photonics, and condensed matter physics, particularly through the manipulation of spectral loops and exceptional points.