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Arbitrarily Fast Quantum Dispersion in Long-Range Crystals

This paper constructs the first long-range crystals exhibiting arbitrarily fast polynomial quantum dispersion by utilizing highly oscillatory Weierstrass functions and developing a new Fourier decay theory that simultaneously resolves a 1980 question regarding the local time of such functions.

Original authors: Gaétan Leclerc, Mostafa Sabri, Tuomas Sahlsten

Published 2026-08-28

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CC BY 4.0

Original authors: Gaétan Leclerc, Mostafa Sabri, Tuomas Sahlsten

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

In the quantum world, particles do not simply sit still or travel in straight lines like tiny bullets. Instead, they exist as waves, spreading out and evolving over time according to the rules of the Schrödinger equation. Two distinct behaviors describe how these waves move through a material. The first is transport, which measures how fast the center of the wave packet travels from one place to another. The second is dispersion, which describes how the wave flattens and thins out as it spreads, losing its concentration. In many familiar materials, such as perfect crystals made of atoms arranged in a neat, repeating grid, these behaviors are well understood. The waves tend to move at a steady speed and spread out at a predictable, moderate rate. Scientists have long believed that there is a natural limit to how quickly a wave can flatten out in such ordered systems, a limit dictated by the smoothness of the material's internal structure.

However, a new study challenges this assumption by exploring a different kind of crystal: one that allows particles to jump not just to their immediate neighbors, but to distant points across the lattice. These are called long-range crystals. While the atoms still form a repeating pattern, the connections between them stretch far and wide, creating a complex web of interactions. Researchers Gaétan Leclerc, Mostafa Sabri, and Tuomas Sahlsten have constructed the first examples of such crystals that exhibit a startling phenomenon: the quantum waves inside them can flatten out with arbitrarily fast polynomial speed. They proved that by carefully arranging these long-distance jumps, they could make the wave disperse at polynomial rates that are higher than any previously known. This discovery reveals that the roughness of the internal structure, rather than hindering the wave, actually acts as a powerful engine for rapid spreading.

The team achieved this by designing a crystal where the probability of a particle jumping to a distant site follows a specific, self-similar pattern. Imagine a structure where the connections repeat at scales of one, ten, one hundred, one thousand, and so on, with the strength of the connection dropping off at each step. This arrangement creates a mathematical object known as a Weierstrass function, which is famous for being continuous everywhere but smooth nowhere. It is a jagged, fractal-like curve that looks rough at every level of magnification. In previous studies, such roughness was thought to be an obstacle to rapid dispersion, as smooth waves usually spread more predictably. The researchers found the opposite to be true. The extreme, self-similar roughness of their crystal's structure causes the quantum wave to cancel itself out in a highly efficient way, leading to a decay in concentration that can be made as fast as desired by simply increasing the gap between the jump distances.

To understand how this works, one must look at the wave's behavior in a mathematical space where its movement is described by a single function. The researchers showed that the speed of dispersion depends on how quickly the oscillations of this function cancel each other out. In their long-range crystals, the self-similar nature of the jumps creates a rigid, repeating pattern of oscillations. By using a sophisticated method involving transfer operators—a tool borrowed from the study of chaotic dynamical systems—they demonstrated that these oscillations interfere with one another so effectively that the wave flattens out incredibly fast. The more irregular the structure becomes, the stronger this cancellation effect, allowing the wave to disperse at a polynomial rate that can be tuned to be arbitrarily high. This finding overturns the intuition that disorder slows things down, showing instead that a specific type of structured disorder can accelerate the spreading of quantum information.

The study also places firm boundaries on what is possible. While the researchers showed that the speed of dispersion can be made faster than any previously known polynomial rate, they proved that it can never be exponential. In other words, the wave cannot flatten out with a speed that doubles every second forever; there is a hard ceiling to how fast this can happen. This limit arises from the fundamental spectral properties of the crystal, which prevent the wave from decaying too quickly. Furthermore, the team showed that if the long-range jumps are arranged in a simple, monotonic way—where the jump strength simply decreases as the distance increases—then the dispersion cannot be faster than a standard, slow rate. This highlights that the rapid dispersion is not a result of the long-range jumps alone, but specifically requires the intricate, self-similar arrangement of those jumps. The question of whether an intermediate regime of super-polynomial but sub-exponential dispersion can occur remains open.

Beyond the speed of the waves, the research sheds light on the nature of the crystal's spectrum, which describes the possible energy levels the particles can occupy. In these rapidly dispersing crystals, the energy spectrum is purely continuous, meaning the particles can occupy a smooth range of energies rather than being locked into discrete levels. This is a significant departure from some other quantum systems where the spectrum can be fragmented or singular. The ability to control the dispersion rate also has implications for transport. The researchers found that in certain regimes of their crystals, the particles can move faster than the standard ballistic speed, a phenomenon known as super-ballistic transport. This suggests that by engineering the long-range connections in a material, one could potentially control how quickly energy or information spreads through it.

The mathematical tools developed to prove these results have broader applications. The team established a new lemma, a mathematical rule for estimating the behavior of oscillatory integrals, which applies to functions that are rough and self-similar. This tool helps explain why the local time—a measure of how much time a function spends at a specific value—exists and is smooth for these fractal-like functions. This answers a question that had remained open since 1980 regarding the behavior of deterministic, nowhere-differentiable functions. By showing that these functions can have smooth local times under certain conditions, the researchers connected the physics of quantum dispersion to deep questions in analysis and probability.

Ultimately, this work demonstrates that the rules governing quantum transport are more flexible than previously thought. By moving beyond the constraints of local interactions and embracing a specific type of long-range, self-similar structure, it is possible to create materials where quantum waves spread with unprecedented polynomial speed. The findings do not just describe a theoretical curiosity; they provide a concrete blueprint for how the geometry of a system dictates its dynamic behavior. The researchers have shown that the key to ultra-fast dispersion lies not in smoothing out the material, but in embracing a precise, fractal-like roughness that turns the wave's own oscillations into a mechanism for rapid spreading. This opens a new chapter in the understanding of quantum dynamics, suggesting that the future of material design may lie in the careful engineering of long-range connections to harness these powerful dispersive effects.

Technical Summary: Arbitrarily Fast Quantum Dispersion in Long-Range Crystals

Problem Statement
The paper addresses the phenomenon of quantum dispersion in Schrödinger operators on the integer lattice $\mathbb{Z}$ . While transport (the movement of wave packet mass) and dispersion (the flattening of the wave packet) are well-understood in locally finite periodic systems, the behavior of "long-range" crystals—where hopping amplitudes $w_k$ decay but are non-zero for infinitely many distances—remains less explored. Previous work established that long-range hopping can lead to exotic spectral types and transport regimes, but the speed of dispersion was thought to be bounded by universal polynomial rates (e.g., $|t|^{-1/3}$ ) derived from classical oscillatory integral estimates (stationary phase, van der Corput lemmas). These estimates rely on the smoothness of the Floquet function (the symbol of the operator). The central question is whether removing the local finiteness constraint allows for dispersion rates faster than any fixed polynomial, and if so, what mechanisms drive such behavior.

Methodology
The authors construct a specific class of long-range Hamiltonians $H_w$ defined by hopping weights supported on a lacunary sequence $\lambda^j$ with geometrically decaying amplitudes $\mu^j$ . The Floquet function $h(\theta)$ for these operators is identified as a Weierstrass function:
$h(\theta) = \sum_{j=0}^\infty \mu^j \phi(\lambda^j \theta)$
where $\phi$ is a non-constant real-analytic 1-periodic function.

The core methodological innovation is the development of a new Fourier decay theory for the push-forward of Lebesgue measure under these rough, autosimilar maps. Since the Floquet functions are typically only Hölder continuous ( $C^\alpha$ ) rather than differentiable, classical stationary phase methods fail. Instead, the authors employ a transfer operator approach inspired by Dolgopyat's method (originally used for mixing rates in hyperbolic dynamical systems).

Key technical steps include:

Twisted Transfer Operators: Defining operators $L_{\lambda, i\Phi}$ that incorporate the oscillatory phase of the Schrödinger evolution.
Dolgopyat-type Estimates: Proving uniform contraction bounds for these twisted operators in $L^2$ and $C^1$ norms. This relies on the "autosimilarity" of the Weierstrass function, which allows the authors to treat the phase as a dynamical system under multiplication by $\lambda$ .
Separation and Tree Lemmas: Establishing that for sufficiently large lacunarity $\lambda$ , the derivatives of the phase along different branches of the dynamical system separate significantly. This prevents the concentration of the measure and ensures rapid oscillatory cancellation.
Van der Corput Lemma for $C^\alpha$ Functions: Deriving a new lemma that provides polynomial decay rates for oscillatory integrals with $C^\alpha$ phases, where the decay exponent scales with the roughness of the function.

Key Contributions and Results

Arbitrarily Fast Polynomial Dispersion (Theorem 1.1):
The paper constructs the first examples of crystals exhibiting arbitrarily fast polynomial dispersion. For any $N \in \mathbb{N}$ , there exists a long-range crystal such that the dispersive norm decays as $|t|^{-N}$ as $t \to \infty$ . This is achieved by increasing the lacunarity parameter $\lambda$ , which increases the roughness of the Floquet function. Contrary to intuition, this roughness does not obstruct dispersion; rather, the self-similar structure induces strong oscillatory cancellation.
Impossibility of Exponential Dispersion (Proposition 1.2):
The authors prove that no crystal (regardless of hopping range) can exhibit exponential dispersion. This is a purely spectral obstruction: the diagonal matrix element $\langle \delta_0, e^{-itH}\delta_0 \rangle$ is the Fourier transform of a compactly supported spectral measure. By the Paley-Wiener theorem, exponential decay would imply the spectral density extends analytically to a strip, which is impossible for a non-zero compactly supported function.
Regularity Obstructions (Proposition 3.3 & Corollary 3.6):
A converse result is established: if the Floquet function is locally $C^\alpha$ on any open interval, the dispersion cannot be faster than $|t|^{-1/\alpha}$ . Specifically, if the hopping weights are eventually monotone, the Floquet function is $C^1$ away from integers, limiting dispersion to at best $O(1/t)$ . This highlights that the "lacunary" arrangement of interactions is essential for the fast dispersion mechanism.
Local Time Regularity (Corollary 1.5):
As a byproduct of the Fourier decay theory, the paper resolves a question posed by Geman and Horowitz (1980) regarding the local time (occupation density) of Weierstrass functions. They prove that for sufficiently large lacunarity, the local time of deterministic Weierstrass functions exists and is $C^k$ for any $k$ . This demonstrates that rougher autosimilar phases can produce smoother image densities.
Transport Properties:
The paper connects the roughness of the Floquet function to transport. For the constructed Weierstrass crystals, if the Hölder exponent $\alpha < 1/2$ , the transport is "super-ballistic," meaning the second moment of the position operator diverges instantaneously for $t \neq 0$ .

Significance and Claims
The paper claims to fundamentally alter the understanding of quantum dispersion in long-range systems. It demonstrates that:

There is no universal polynomial upper bound on dispersion speed once local finiteness is removed.
Roughness, typically viewed as a source of spectral singularities or transport anomalies, can be engineered (via lacunarity) to act as a mechanism for rapid dispersion through self-similar oscillatory cancellation.
The mechanism relies on a "pseudorandom" property derived from deterministic autosimilarity, providing a deterministic substitute for the randomness often required to generate strong Fourier decay (as seen in fractional Brownian motion).

The authors explicitly state that while arbitrarily fast polynomial decay is achievable, exponential decay is universally forbidden. The existence of a "super-polynomial but sub-exponential" regime remains an open question, though the paper suggests it might be accessible via non-stationary Weierstrass phases or adaptations of the Beurling-Malliavin multiplier theorem. The work also opens new directions for studying random and higher-dimensional long-range crystals, as well as the Fourier decay of images of singular measures under autosimilar maps.

Technical Summary: Arbitrarily Fast Quantum Dispersion in Long-Range Crystals

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