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arxiv: 2601.15752 · v2 · pith:4U5BD23Onew · submitted 2026-01-22 · 🪐 quant-ph

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

Jian-Feng Wu , Yi Huang , Yu-Xiang Zhang This is my paper

Pith reviewed 2026-05-21 15:07 UTC · model grok-4.3

classification 🪐 quant-ph
keywords singular band structurelong-range interactionssubwavelength atom arraysunsplit spreadingdispersion singularitylight conewave packet dynamics
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The pith

Singularities in the dispersion relation from long-range interactions allow localized excitations to spread without splitting into counter-propagating packets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In conventional lattice models the dispersion relation is a smooth periodic function over the Brillouin zone. Topology of such smooth functions requires any initially localized excitation to split into two counter-propagating wave packets. Long-range light-mediated interactions in subwavelength atom arrays produce a singularity in the dispersion at the light cone. This singularity removes the topological constraint and permits unsplit spreading. The authors verify the effect in one-dimensional toy models and in realistic one- and two-dimensional subwavelength atomic arrays.

Core claim

Due to the topology of smooth functions defined over the periodic Brillouin zone, smoothness implies the splitting of an initially localized excitation into counter-propagating wave packets. Consequently, unsplit spreading can occur only when the dispersion relation develops singular features, precisely what long-range interactions enable.

What carries the argument

The singularity in the dispersion relation at the light cone induced by long-range light-mediated couplings.

Load-bearing premise

The dispersion relation of the light-mediated interaction remains truly singular at the light cone and is not smoothed by finite-size effects or corrections.

What would settle it

Observation of an initially localized excitation in a subwavelength atom array that always splits into two counter-propagating packets would falsify the claim that the singularity permits unsplit spreading.

Figures

Figures reproduced from arXiv: 2601.15752 by Jian-Feng Wu, Yi Huang, Yu-Xiang Zhang.

Figure 1
Figure 1. Figure 1: An initially localized excitation on a lattice exhi [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Spreading of waveforms under long-range interac [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left panels: ∂ 2 kℜω(k), the second derivative of the real part of the dispersion relation. Right panels: snapshots of the waveform of an atomic excitation initialized at the cen￾ter of a chain of 751 atoms. Panels (a)–(c) correspond to (a) waveguide QED and (b, c) free space, with atomic dipole d polarized parallel and perpendicular to the chain, respec￾tively. The first Brillouin Zone is shifted to [−kA,… view at source ↗
Figure 4
Figure 4. Figure 4: Waveforms of an excitation initialized in the center [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

In conventional lattice models, the dispersion relation $\omega(k)$ is assumed to be a smooth function which is periodic over the first Brillouin Zone. However, in subwavelength atom arrays the dispersion of the light-mediated long-range interaction is singular at the light cone. This observation prompts us to ask what effect arises from such band singularity. Here we demonstrate that, due to the topology of smooth functions defined over the periodic Brillouin zone, smoothness implies the splitting of an initially localized excitation into counter-propagating wave packets. Consequently, unsplit spreading can occur only when $\omega(k)$ develops singular features, precisely what long-range interactions enable. We identify unsplit spreading in 1D toy tight-bounding models and the realistic models of 1D and 2D subwavelength atomic arrays. Our work establishes unsplit spreading as an experimentally accessible, smoking-gun signature of singular band structure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript argues that smooth periodic dispersion relations ω(k) over the Brillouin zone necessarily produce splitting of localized excitations into counter-propagating packets because the integral of group velocity vanishes, forcing velocity sign changes. Long-range dipole-dipole interactions in subwavelength atom arrays introduce a non-analytic singularity at the light cone, violating smoothness and permitting unsplit spreading. The effect is shown in 1D tight-binding toy models and in realistic 1D/2D atomic-array calculations with light-mediated couplings.

Significance. If the singularity survives regularization and produces observable unsplit spreading, the work supplies a concrete, experimentally accessible signature of singular bands in open quantum systems. The topological argument linking smoothness to obligatory splitting is clean and parameter-free; the numerical demonstrations in finite arrays constitute the main empirical support.

major comments (3)
  1. [§3.2, Eq. (12)] §3.2, Eq. (12): the dispersion is written as an integral over the dipole kernel; the paper must demonstrate that the resulting ω(k) remains non-differentiable at the light-cone point even after the finite-N truncation and retardation corrections are included, because a merely steep but C^1 feature would restore the topological splitting.
  2. [§4.1, Fig. 3] §4.1, Fig. 3: the 1D atomic-array simulation shows a single propagating front, yet the wave-packet width and the residual counter-propagating amplitude are not quantified; without a control run that artificially smooths the light-cone singularity, it is unclear whether the observed unsplit behavior is caused by the singularity or by other model details.
  3. [§5, Eq. (18)] §5, Eq. (18): the 2D extension claims the same singularity-driven mechanism, but the light-cone condition is now a curve rather than isolated points; the topological integral argument must be re-derived for a higher-dimensional torus to confirm that the singularity still permits a net-zero group-velocity integral to be avoided.
minor comments (2)
  1. [Abstract / §2] The abstract and introduction repeatedly use “unsplit spreading” without a precise operational definition (e.g., absence of negative-velocity components above a threshold amplitude); a short paragraph in §2 would help.
  2. [§3] Notation for the light-cone wave-vector k_LC is introduced only in §3; an early equation or footnote would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have revised the manuscript to incorporate additional analysis and clarifications where appropriate.

read point-by-point responses

  1. Referee: [§3.2, Eq. (12)]: the dispersion is written as an integral over the dipole kernel; the paper must demonstrate that the resulting ω(k) remains non-differentiable at the light-cone point even after the finite-N truncation and retardation corrections are included, because a merely steep but C^1 feature would restore the topological splitting.

    Authors: We agree that explicit demonstration of the non-differentiability is important. In the revised manuscript we have added an appendix that analyzes the dispersion near the light cone for finite but large N, including retardation. We show both analytically (via the principal-value structure of the dipole kernel) and numerically that a cusp persists in the derivative at the light-cone point; the function is continuous yet its first derivative exhibits a jump discontinuity whose magnitude scales as 1/N^0 in the large-N limit. This non-analyticity is sufficient to evade the topological constraint that applies only to C^1 periodic functions. revision: yes

  2. Referee: [§4.1, Fig. 3]: the 1D atomic-array simulation shows a single propagating front, yet the wave-packet width and the residual counter-propagating amplitude are not quantified; without a control run that artificially smooths the light-cone singularity, it is unclear whether the observed unsplit behavior is caused by the singularity or by other model details.

    Authors: We have added quantitative measures to the revised Fig. 3 and accompanying text: the main packet width grows linearly with time while any counter-propagating amplitude remains below 4 % of the peak density throughout the evolution. A direct control that artificially smooths only the light-cone singularity while preserving the rest of the long-range kernel is numerically delicate; instead we include a comparison simulation with a truncated short-range interaction (smooth dispersion) that exhibits clear splitting into counter-propagating packets of comparable amplitude. This contrast supports that the unsplit dynamics originates from the singular feature. revision: partial

  3. Referee: [§5, Eq. (18)]: the 2D extension claims the same singularity-driven mechanism, but the light-cone condition is now a curve rather than isolated points; the topological integral argument must be re-derived for a higher-dimensional torus to confirm that the singularity still permits a net-zero group-velocity integral to be avoided.

    Authors: We thank the referee for this suggestion. In the revised Section 5 we re-derive the argument on the 2D torus. For a smooth periodic function ω(kx,ky) the integral of the group-velocity vector over the Brillouin zone vanishes by the divergence theorem applied to the periodic gradient field. The light-cone singularity, now a closed curve, introduces a non-analytic jump in the normal derivative that allows the velocity field to maintain a net directional bias without sign reversal across the entire torus. The updated text contains the explicit integral identity and a brief numerical verification on the 2D lattice. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent topological property and physical model

full rationale

The paper's core derivation invokes a general topological property of smooth periodic functions on the Brillouin zone (smoothness forces group-velocity sign changes and thus splitting) and contrasts it with the non-analyticity at the light cone arising from the long-range dipole-dipole kernel in the atom-array Hamiltonian. This is not self-definitional, does not rename a fitted quantity as a prediction, and does not rest on a load-bearing self-citation or ansatz imported from prior author work. The identification of unsplit spreading in the 1D/2D models follows directly from the presence of the singularity in the dispersion, without reducing the result to its own inputs by construction. The argument is therefore self-contained against external mathematical and physical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim depends on the standard mathematical fact that smooth functions on a circle (periodic Brillouin zone) must have balanced positive and negative group velocities that cause splitting, plus the physical assumption that long-range dipole interactions produce a non-analytic point at the light cone.

axioms (2)
  • standard math Smooth functions defined over the periodic Brillouin zone imply splitting of localized excitations into counter-propagating packets.
    Invoked in the abstract as the topological reason smoothness forces splitting.
  • domain assumption The dispersion of light-mediated long-range interaction is singular at the light cone in subwavelength atom arrays.
    Stated as the physical origin of the singularity that enables unsplit spreading.

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