Singular band Induced by Long-Range Interaction Enables Unsplit Spreading of Localized Excitations

This paper demonstrates that long-range light-mediated interactions in subwavelength atom arrays induce singularities in the band structure, which uniquely enable the unsplit spreading of localized excitations—a phenomenon forbidden in conventional smooth-band lattice models where excitations inevitably split into counter-propagating wave packets.

The Gist

Imagine you drop a single drop of ink into a glass of water. In a normal, calm glass, the ink doesn't just stay in one spot; it spreads out. Usually, it spreads in a circle, getting wider and wider. But in the quantum world of atoms arranged in a grid (a lattice), things get a bit stranger.

This paper explores a specific question: When you excite a single atom in a grid, does that energy spread out as one big, smooth blob, or does it split into two separate waves moving in opposite directions?

Here is the simple breakdown of their discovery:

1. The "Smooth Road" Rule (Short-Range Interactions)

Think of a standard grid of atoms like a smooth, perfectly paved road. In physics, this is called a "smooth band."

• The Rule: If the road is perfectly smooth and continuous, the laws of physics (specifically, a bit of math involving shapes and loops) dictate that a drop of energy must split.
• The Analogy: Imagine you are standing in the middle of a long, smooth hallway and you clap your hands. The sound waves travel to your left and to your right. They separate. You can't have the sound just get louder in one direction without a matching wave going the other way.
• The Result: In normal, short-range systems, an excited atom always sends out two wave packets moving in opposite directions. It's like a fork in the road; the energy splits.

2. The "Broken Road" Exception (Long-Range Interactions)

Now, imagine the road isn't smooth. Imagine it has a sudden, sharp cliff or a jagged spike in the middle. This paper studies systems where atoms talk to each other over long distances (like light bouncing between them), which creates these "jagged" or singular spots in the physics rules.

• The Discovery: When the "road" has a sharp singularity (a break in smoothness), the rules change. The energy does not split.
• The Analogy: Imagine you are on a road that suddenly ends in a sheer cliff at a specific point. If you try to walk, you can't smoothly turn left or right around a corner because the corner doesn't exist in the usual way. Instead, the energy spreads out like a single, widening puddle. It gets bigger, but it stays as one single piece.
• The Result: The authors found that in these specific "long-range" systems, the excitation spreads out without ever splitting into two separate groups. It remains a single, unified wave.

3. The "Magic Mirror" Effect

The paper also looked at real-world setups, like tiny arrays of atoms in a vacuum or inside a waveguide (a pipe for light).

• In these setups, there is a special zone where atoms are "subradiant." Think of this as a magic mirror that traps the energy.
• In this trapped zone, the "road" becomes jagged (singular). Because of this, the energy spreads out as a single, unbroken wave, even though the atoms are arranged in a grid.
• The authors showed that if you change the spacing of the atoms, you can switch between the "splitting" mode (smooth road) and the "unsplit" mode (jagged road).

4. Why This Matters (The "Smoking Gun")

The authors call this "unsplit spreading" a smoking-gun signature.

• The Analogy: If you see a car driving down a road and it suddenly splits into two cars going opposite ways, you know the road is smooth. But if you see a car that just gets wider and wider without ever splitting, you know for a fact that the road has a hidden, jagged cliff in it.
• The Claim: By watching how an excited atom spreads, scientists can now tell if the system has these special, singular long-range interactions. It's a way to "see" the invisible jaggedness in the physics of the system.

Summary

• Normal Systems (Smooth): Energy splits into two waves (Left and Right).
• Long-Range Systems (Jagged/Singular): Energy spreads as one single, widening wave.
• The Takeaway: The paper proves that this "unsplit spreading" is a direct result of the "jagged" nature of the energy bands caused by long-range forces. It's a new way to identify these special quantum systems in the lab.

Technical Summary

Technical Summary: Singular Band Induced by Long-Range Interaction Enables Unsplit Spreading of Localized Excitations

Problem Statement
In conventional lattice models with short-range interactions, the dispersion relation $\omega(k)$ is assumed to be a smooth, periodic function over the first Brillouin Zone. A fundamental dynamical consequence of this smoothness is that an initially localized excitation inevitably splits into two counter-propagating wave packets. While long-range interactions (decaying as $1/r^\alpha$) are known to induce anomalous phenomena such as point spectra (for $\alpha < d$) and subradiance, the specific dynamical signature of band singularities in the context of wave packet spreading has not been systematically characterized. The authors address the question: What observable consequences arise from the singularities in the dispersion relation induced by long-range interactions, specifically regarding the splitting or non-splitting of localized excitations?

Methodology
The authors employ a combination of topological analysis, stationary-phase approximation, and numerical simulations to investigate the dynamics of localized excitations.

1. Topological No-Go Theorem: The paper establishes a theoretical constraint for smooth bands. By assuming the dispersion $\omega(k)$ is twice continuously differentiable ($C^2$) and periodic, the authors utilize the properties of smooth functions on a circle ($S^1$) and the Gauss–Bonnet theorem for 2D tori ($T^2$). They demonstrate that for a smooth dispersion, the second derivative $\partial^2_k \omega(k)$ (or the determinant of the Hessian in 2D) must cross zero an even number of times. These zero-crossings correspond to extrema in group velocity, necessitating the formation of both right-moving and left-moving wave packets, thereby forbidding "unsplit spreading."
2. Stationary-Phase Approximation: To analyze the waveform evolution $\psi(x,t)$, the authors use the stationary-phase approximation, linking the amplitude peaks of the wavefunction to the zeros of the second derivative of the dispersion relation. They benchmark this approximation against exact numerical simulations, noting that while it may not be quantitatively precise for open systems, it correctly captures the qualitative profile (splitting vs. unsplitting).
3. Model Simulations: The authors simulate three distinct classes of models:
   • 1D Power-Law Tight-Binding Models: Investigating hopping amplitudes $1/r^\alpha$ for $\alpha = 1, 2, 3$ to identify thresholds where smoothness is violated.
   • 1D Light-Mediated Interactions: Modeling atoms coupled to 1D waveguides and free space, where the interaction is mediated by photon exchange.
   • 2D Subwavelength Atom Arrays: Simulating realistic 2D arrays in free space where the lattice constant $a$ is smaller than the resonant wavelength ($\pi/a > k_0$).

Key Results

• The No-Go Theorem for Smooth Bands: The authors prove that unsplit spreading is topologically forbidden for any lattice with a smooth dispersion relation. The continuity and periodicity of $\omega(k)$ and its derivatives force the existence of at least two group velocity extrema with opposite signs, leading to inevitable splitting.
• Unsplit Spreading via Singularities: Unsplit spreading becomes possible only when the dispersion relation develops singular features (non-smoothness).
  • In 1D power-law models, unsplit spreading is observed for $\alpha=1$ (where $\omega(k)$ is discontinuous) and $\alpha=2$ (where the first derivative is discontinuous, creating a cusp). For $\alpha=3$, where the second derivative is singular but the function remains $C^1$, the system reverts to splitting because $\partial^2_k \omega(k)$ still possesses zeros.
  • In light-mediated interactions, the dispersion relation becomes singular at the light cone ($|k| = k_0$).
• Subradiant States as a Mechanism: In subwavelength atom arrays, the light cone divides the Brillouin zone into a radiative sector and a subradiant (coherent) sector. The subradiant states effectively act as a post-selection mechanism. Within this sector, the dispersion can exhibit singular features that prevent the formation of counter-propagating packets.
  • 1D Results: In waveguide QED and free-space chains, specific parameters (e.g., $k_A = 0.6\pi$) lead to a unique minimum in the second derivative of the real part of the dispersion within the subradiant branch, resulting in unsplit spreading. Conversely, parameters leading to multiple minima (e.g., $k_A = 0.15\pi$) result in splitting.
  • 2D Results: In 2D subwavelength arrays, the authors observe that for certain lattice parameters (e.g., $k_A = 1.2\pi$), the excitation diffuses without splitting, whereas other parameters (e.g., $k_A = 0.3\pi$) lead to splitting. This behavior correlates with the topology of the zero-set of the Hessian determinant in the subradiant zone.

Significance and Claims
The paper claims to establish "unsplit spreading" as a direct, experimentally accessible, "smoking-gun" signature of singular band structures induced by long-range interactions.

• Fundamental Distinction: The work delineates a fundamental gap between short-range and long-range interacting systems. While short-range models always yield smooth dispersions and thus splitting dynamics, long-range interactions can induce singularities that enable unsplit propagation.
• Topological Interpretation: The authors reframe the problem of wave packet spreading as a topological constraint on the Brillouin zone, extending the analysis from 1D circles to 2D tori using the Gauss–Bonnet theorem.
• Reinterpretation of Existing Data: The authors identify that unsplit spreading was likely present but overlooked in prior quantum simulation experiments (specifically Ref. [39], a trapped-ion simulation of a 1D long-range XY model with $\alpha \approx 0.75$).
• Open-System Dynamics: The paper highlights that in realistic light-mediated systems, the singular band structure is often realized within the subradiant sector of an open quantum system, offering a natural pathway to circumvent the no-go theorem through dissipation-induced post-selection.

The authors conclude that while their work focuses on non-interacting (free) Hamiltonians, the presence of singular dispersions may have traceable effects on interacting many-body phenomena, such as thermalization and entanglement growth, in long-range systems. They do not propose new experimental setups but rather provide a theoretical framework to interpret existing and future observations of localized excitation spreading.