🔬 mesoscale physics

Fibonacci number systems and the localization criterion in the many-body Aubry-André model

This paper introduces a fractional Fibonacci number system to derive a localization criterion for the many-body non-interacting Aubry-André model, where particle density is expressed in this system for finite sizes and in base- $\phi$ for the thermodynamic limit.

Original authors: Balázs Hetényi

Published 2026-08-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Balázs Hetényi

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

The Quantum City and the Golden Gate

Imagine a vast, invisible city where tiny particles, like electrons, try to move from house to house. In a normal city, these particles can zip around freely, creating a smooth flow of electricity—this is what we call a metal. But sometimes, if the city's layout is messy or the streets are blocked by random obstacles, the particles get stuck in one spot, unable to move. This is called an insulator. Scientists have long been fascinated by the "tipping point" where a city switches from being a free-flowing metropolis to a trapped, frozen town. This switch is called a localization transition.

Usually, this happens when the obstacles are completely random, like a city built by a chaotic architect. But there is a special kind of city layout called "quasiperiodic." Imagine a street pattern that repeats itself but never quite the same way twice, like a rhythm that keeps changing slightly. In this specific type of city, known as the Aubry-André model, the rules are different. Scientists have discovered that depending on how many particles live in the city (the density), the traffic jam can happen at very different times. Some densities get stuck immediately, while others flow freely until the obstacles get huge. The big question is: how do we predict exactly when a specific crowd of particles will get stuck?

The Magic of the Fibonacci Number System

In this study, physicist Balázs Hetényi from Budapest proposes a clever new way to look at this problem using a special kind of math called the "Fibonacci number system." You might know the Fibonacci sequence from nature: 1, 1, 2, 3, 5, 8, 13, where each number is the sum of the two before it. Usually, we use these numbers to count whole things, like apples or steps. But Hetényi invented a new tool called the "fractional Fibonacci number system." Think of it as a ruler that can measure not just whole apples, but slices of apples, using only Fibonacci numbers as the units.

The paper uses this ruler to decode the behavior of particles in the quasiperiodic city. The researchers ran computer simulations to watch how particles behaved as they increased the strength of the "obstacles" (the potential strength, $W$ ) compared to how easily they could hop between houses (the hopping strength, $t$ ). They found that the answer to whether the particles get stuck depends entirely on how you write down the number of particles in the city.

Here is the surprising discovery: The particles behave differently based on the "shape" of their number when written in this special Fibonacci language.

The "Golden" Densities: If the particle density is a ratio of two Fibonacci numbers (like 144 particles in a city of 233 houses), the number looks very simple in this system—like a single, clean digit. For these specific crowds, the particles get stuck (localize) as soon as there is any obstacle at all, even a tiny one ( $W=0$ ). They are trapped immediately.
The "Normal" Densities: If the density is a regular fraction (like 1/2) or a messy irrational number (like $1/\sqrt{3}$ ), the number looks complex and long in the Fibonacci system. For these crowds, the particles flow freely until the obstacles get strong enough ( $W=2t$ ). Only then do they freeze into an insulator.

The author shows that this isn't just a trick for small cities. As the city grows infinitely large (the thermodynamic limit), this special number system transforms into something called "base- $\phi$ " (where $\phi$ is the golden ratio, roughly 1.618). In this infinite limit, the rule becomes more subtle: the classification of a density as "simple" or "complex" can lose its strict meaning. Whether a system is localized or not can actually depend on the direction from which you approach that density.

The Twist at the Edge of Infinity

The paper also explores a very tricky edge case. Imagine two crowds that are almost identical to the "Golden" density, but one has just one extra particle and the other has one less. In a finite city, these two crowds behave normally: they flow freely until the obstacles get strong ( $W=2t$ ). However, as the city grows to infinity, these two crowds mathematically merge into the exact same "Golden" density that gets stuck immediately.

This creates a weird situation where the answer depends on how you approach the limit. If you approach the density from above or below, the particles flow until $W=2t$ . But if you hit the exact "Golden" density, they are stuck at $W=0$ . It's like a cliff where the ground is solid if you stand exactly on the edge, but if you take even a tiny step away, the ground is solid only when a heavy weight is placed on it.

What This Means

The paper doesn't claim to have solved every mystery of quantum physics, but it provides a powerful new lens. By translating the problem of particle density into the language of Fibonacci numbers, the author has found a clear criterion to predict when a many-body system will turn into an insulator. They demonstrated through simulations that for non-interacting particles, the "complexity" of the density number in this special system dictates the physics. If the number is simple (finite digits in base- $\phi$ ), the system is fragile and localizes easily. If the number is complex (infinite digits), the system is robust and requires strong disorder to stop the flow. This suggests that the secret to understanding these quantum traffic jams lies not just in the physics of the particles, but in the mathematical structure of the numbers we use to count them.

Technical Summary: Fibonacci Number Systems and the Localization Criterion in the Many-Body Aubry-André Model

Problem Statement
The paper addresses the delocalization-localization transition (DLT) in the non-interacting many-body version of the Aubry-André-Harper (AAM) model. While the single-particle AAM is well-understood—exhibiting a transition at potential strength $W=2t$ —recent studies on the many-body system revealed an anomalous phase diagram. Specifically, while rational particle densities ( $\rho$ ) follow the standard $W=2t$ transition, certain irrational densities exhibit a transition at $W=0$ (always insulating for finite $W$ ), while others follow the $W=2t$ behavior. The author aims to explain this classification of densities and provide a rigorous localization criterion using number-theoretic tools.

Methodology
The study employs a combination of exact diagonalization of the AAM Hamiltonian and the application of specific number systems to analyze particle densities.

Model: The author considers a 1D lattice of size $L$ with $N$ fermions, defining density $\rho = N/L$ . The potential is modulated by the golden ratio $\alpha$ , approximated as the ratio of consecutive Fibonacci numbers ( $F_{n+1}/F_n$ ). The system size is constrained to $L = F_n$ .
Numerical Metric: Localization is quantified using the variance (centered second moment) of the polarization, $M_2$ . The behavior of $M_2$ as a function of system size $L$ and potential strength $W$ determines the phase: a size-independent $M_2$ indicates an insulating phase, while size-dependent $M_2$ indicates a metallic phase.
Number Systems: The core theoretical tool is the Zeckendorf theorem, which states that every natural number has a unique representation as a sum of non-consecutive Fibonacci numbers. The author extends this to construct:
- The Fibonacci number system (for integers).
- The base- $\phi$ number system (for real numbers, where $\phi$ is the golden ratio).
- A novel fractional Fibonacci number system, which interpolates between the two. This system represents fractions of the form $M/F_n$ by shifting the Zeckendorf decomposition of the numerator $M$ relative to the denominator $F_n$ .

Key Contributions
The primary contribution is the introduction of the fractional Fibonacci number system as a framework to interpret the localization phase diagram. The author demonstrates that the localization behavior of a specific density $\rho$ is determined by the stability of its representation in this number system as the system size $L=F_n$ increases.

Classification of Densities: The paper identifies three categories of densities based on their representation:
1. Rational Densities: The number of non-zero digits in the fractional Fibonacci representation increases with system size. These exhibit a standard metal-insulator transition at $W=2t$ .
2. Specific Irrational Densities: Densities that are limits of Fibonacci ratios (e.g., $\rho = F_{n-1}/F_n$ ) or finite sums thereof. In the fractional Fibonacci system, the representation of these densities remains invariant (the number of digits does not change) as $n \to \infty$ . These densities are localized for all finite $W$ (transition at $W=0$ ).
3. Other Irrational Densities: Densities unrelated to the Fibonacci sequence (e.g., $1/\sqrt{3}$ ) show an increasing number of digits without a stable pattern, behaving like rational densities with a transition at $W=2t$ .
Thermodynamic Limit Analysis: The author analyzes the transition to the thermodynamic limit ( $L \to \infty$ ), where the fractional Fibonacci system converges to the base- $\phi$ system. They highlight a subtle dependence on the direction of approach to the limit. For instance, the density $\gamma = 1/\phi$ (represented as $0.1_\phi$ ) is strictly localized. However, sequences approaching $\gamma$ from above ( $\gamma_+$ ) or below ( $\gamma_-$ ) via finite system sizes exhibit a transition at $W=2t$ . In the limit, $\gamma_+$ , $\gamma_-$ , and $\gamma$ coalesce, yet their localization properties differ based on how the limit is taken. The author notes that in this limit, the classification based on finite digit counts loses its strict meaning, as the distinction between "invariant" and "growing" representations becomes ambiguous for certain boundary cases.

Results
Numerical calculations for three distinct densities confirm the theoretical predictions:

$\rho = 1/2$ : The variance $M_2$ shows strong size dependence for $W < 2t$ , indicating a metallic phase, and size independence for $W > 2t$ . The Zeckendorf decomposition gains terms as $L$ increases.
$\rho = F_{n-1}/F_n$ (approximating $1/\phi$ ): The variance $M_2$ is size-independent for all $W > 0$ , indicating an insulating phase. The fractional Fibonacci representation is stable ( $0.1_{F_n}$ ) regardless of system size.
$\rho = 1/\sqrt{3}$ : Similar to the rational case, $M_2$ shows size dependence for $W < 2t$ , confirming a transition at $W=2t$ . The representation gains terms without a stable pattern.

The study also investigates "boundary" cases ( $\gamma_+$ and $\gamma_-$ ) where the particle number differs by one from the ideal Fibonacci ratio. These cases exhibit a transition at $W=2t$ , contrasting with the strictly localized behavior of the exact limit $\gamma$ .

Significance
The paper claims that the fractional Fibonacci number system provides a transparent criterion for localization in the many-body AAM. It establishes that the stability or invariance of the number representation in the thermodynamic limit is the determining factor for the localization transition. Specifically, if the representation of a density in the base- $\phi$ system remains invariant (does not acquire new digits) as the system size increases, the system is localized; if the representation requires an increasing number of digits (is not invariant) to represent the density, a metal-insulator transition occurs at $W=2t$ . The work clarifies the nature of the thermodynamic limit in quasiperiodic systems, showing that the localization behavior can depend on the specific sequence of finite systems used to approach the infinite limit.

The Quantum City and the Golden Gate

The Magic of the Fibonacci Number System

The Twist at the Edge of Infinity

What This Means

Technical Summary: Fibonacci Number Systems and the Localization Criterion in the Many-Body Aubry-André Model

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