Fibonacci number systems and the localization criterion in the many-body Aubry-Andr'e model

arXiv:2608.03458 · cond-mat.stat-mech, cond-mat.dis-nn, cond-mat.mes-hall, math-ph, math.MP · Submitted 2026-08-04 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Fibonacci number systems and the localization criterion in the many-body Aubry-Andr'e model".

Mira: The paper introduces an extension of number systems, termed the fractional Fibonacci number system, to interpret the localization phase diagram of the many-body non-interacting Aubry-Andr´e model.

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So we’ve looked at what this paper is actually doing, and now we need to talk about who put this work together. The paper is titled "Fibonacci number systems and the localization criterion in the many-body Aubry-Andr´e model," and it’s authored by Balazs Hetényi, along with collaborators from Budapest University of Technology and Economics, MTA-BME Lendület ”Momentum” Open Quantum Systems Research Group, and the Institute for Solid State Physics and Optics at HUN-REN Wigner Research Centre for Physics.

Mira: It's interesting to see the affiliations; it shows this work is coming from a solid theoretical physics background, which makes sense since they are dealing with number systems and complex many-body models like the Aubry-Andr´e model. The authors are clearly experts in their respective fields of condensed matter theory.

Lev: From a researcher perspective, seeing affiliations like that suggests this work is built on a foundation of rigorous theoretical physics, which is necessary when you’re trying to propose new mathematical tools for physical phenomena. I’d be interested in knowing if they have worked on similar mathematical formalisms before or if this approach is entirely novel to this field.

Kai: They are definitely pushing the envelope by introducing the fractional Fibonacci number system specifically for interpreting localization diagrams, which suggests they're connecting discrete mathematics directly to continuous physical properties of these quantum systems. It’s a creative way to bridge that gap.

Mira: That connection is what makes it so compelling; they aren't just applying a known concept; they are building a new number system designed for this specific purpose, which suggests the mathematical structure itself is key to unlocking the physical insight into why localization happens in these models.

Lev: If their mathematical tool proves successful, it opens up avenues for applying similar approaches to other complex systems where standard phase diagrams are hard to interpret because they involve quasiperiodic or highly structured potentials. It’s about developing a universal way of looking at these kinds of problems mathematically rather than just solving them case by case.

Kai: Exactly, and that universality is what makes this paper relevant beyond just the Aubry-Andr´e model. If this method can be generalized, it could become a foundational tool for analyzing localization in many other quantum systems where the potential landscape is highly structured but not perfectly periodic or perfectly random.

The paper's summary: Mira: Moving on to the actual summary of "Fibonacci number systems and the localization criterion in the many-body Aubry-Andr´e model," it boils down to them proposing this fractional Fibonacci number system to classify particle densities. They use this new system to establish a clear rule: if a density for a certain system size is written in this way, then we can predict whether the system will be localized for any finite potential strength.

Kai: So, they’re essentially saying that the whole point of their study is to create a criterion: check the fractional Fibonacci representation of the density; if it stays constant as you increase system size, it's localized everywhere. If it doesn't stay constant, then you get a metal-insulator transition at W=2t <ref:2608.03458#pg0>.

Lev: That’s a very concrete statement about the physics they are investigating. It translates abstract phase diagram analysis into a rule involving number theory and density representation rather than just looking at the raw energy spectra or correlation functions. It’s an interesting way to frame the problem of localization in terms of how particles are distributed.

Mira: They also detail how this system interpolates between the Fibonacci system for natural numbers and base-phi, which is what allows it to handle both integer and real density values in a unified framework, which is a significant mathematical achievement here.

Kai: I think that handling both regimes at once is crucial because physical systems don't always have neatly defined integer properties; they can be continuous, but the fractional Fibonacci system seems to capture that reality better than purely discrete or purely continuous tools.

Lev: From a hardware standpoint, if we can use this system to pre-screen potential parameters, it means we spend less time running full simulations just to find out which parameter set leads us into a localized regime versus a metallic one. It’s about using the mathematical tool to guide the experimental setup efficiently.

Mira: And they show numerical examples of how different densities for different system sizes decompose using this Zeckendorf theorem, illustrating how the number of terms in that decomposition changes with size, which is quite illustrative about the scaling behavior before we get into the deeper limits.

Kai: So it’s like they are providing a set of precise mathematical fingerprints for particle arrangements that tell us exactly what kind of phase we are looking at in this model. It’s turning a complex physical question into a classification problem based on number theory.

The paper's improvements: Mira: Now, regarding the improvements the authors suggest, they are essentially proposing the refinement of their criterion for localization by defining how to handle irrational densities that fall outside their primary category. They introduce three distinct categories for irrational densities to give a more detailed picture than just two options.

Kai: So they are refining the classification beyond just rational versus irrational, by carving out a third group—one that arises from taking the infinite limit of Fibonacci ratios or finite sums thereof, which has its own specific transition behavior at W=zero.

Lev: That extra category is crucial because it addresses those tricky irrational densities that don't fit neatly into the two main groups, meaning they need a specialized rule for them, and this suggests the original classification might be too simplistic for real physical systems.

Mira: Furthermore, they define a second category of irrational numbers that can be represented in this system as fractions of the form M/F n, where F n is any Fibonacci number, which allows them to describe a whole range of physical densities precisely.

Kai: That fractional representation is what lets them describe all those specific densities mentioned in their examples, like rho = one/two and this shows the system’s flexibility <ref:2608.03458#pg0>. It's not just about simple integers anymore; it’s about being able to represent these specific physical scenarios accurately.

Lev: If the authors can successfully prove that this structure holds across a wider range of irrational numbers, it validates their method as a robust tool for describing more complex physics than just the initial examples they tested. It moves beyond illustrative cases to a general physical principle.

Mira: The implication is that the improvement isn't just adding complexity; it’s establishing a more comprehensive mathematical map for all relevant density configurations in this model, which should allow us to predict phase behavior across a much broader range of conditions than before.

Kai: So they are giving us a better tool to look at the paper, which means we can test predictions on a wider array of physical parameters and see where the transition points actually lie. It’s an upgrade from their initial framework.

Conclusion: Mira: To wrap up this discussion on "Fibonacci number systems and the localization criterion in the many-body Aubry-Andr´e model," we see that the main conclusion is that the paper successfully introduces a fractional Fibonacci number system to create a localization criterion for these models. The system allows us to classify particle densities, and based on this classification, we can predict whether a system will be localized for finite potential strength or if it will exhibit a metal-insulator transition at W=2t <ref:2608.03458#pg0>.

Kai: That’s the main message: the paper provides a way to translate complex many-body physics into a testable mathematical rule based on density representation, which is really useful for connecting theory to what we might actually measure in our labs.

Lev: I think the most important part is that they established this criterion for systems where we can reliably use the fractional Fibonacci system, and if that holds, it gives us a solid foundation for applying this logic to characterize many-body localization in other physical contexts.

Mira: And the authors' own limitations are that they have to state plainly: their method is primarily focused on the Aubry-Andr´e model, and they haven't fully explored how these number systems behave outside of it, which means we don’t have a complete picture for all possible quantum systems.

Kai: So while the paper gives us a very specific answer for this model, it sets up a clear path for future work to extend this criterion to other Hamiltonians and different types of models where these number-theoretic tools might apply.

Lev: I think the long-term utility lies in developing these number-theoretic tools as general analytical frameworks that can help us tackle localization problems in disordered or quasiperiodic systems more systematically than current methods allow.

Mira: So, this paper is a significant step forward by offering a new mathematical language to interpret phase diagrams, and it gives us a framework to predict where metal-insulator transitions are likely to happen based on the density's representation in the fractional Fibonacci number system.

Department of Theoretical Physics, Budapest University of Technology and Economics · MTA-BME Lendület ”Momentum” Open Quantum Systems Research Group, Institute of Physics · Institute for Solid State Physics and Optics, HUN-REN Wigner Research Centre for Physics

cond-mat.stat-mech, cond-mat.dis-nn, cond-mat.mes-hall, math-ph, math.MP

Submitted: 2026-08-04

Updated: 2026-08-04

Journal ref: J. Phys. A: Math. Theor. 59 32LT01 (2026)

DOI: 10.1088/1751-8121/ae91a5

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 78/100

The gist: The paper introduces an extension of number systems, termed the fractional Fibonacci number system, to interpret the localization phase diagram of the many-body non-interacting Aubry-Andr´e model.

Key concepts

Fractional Fibonacci Number System: This is an extended number system that interpolates between standard Fibonacci numbers and base-phi (golden ratio) representation. It allows for the precise representation of fractions like M/Fn, where Fn is a Fibonacci number, offering a finer way to categorize particle densities.
Aubry-Andr´e Model (AAM): This is a many-body model describing interacting particles with hopping and potential energy terms. The study focuses on the non-interacting version of this model to understand how particle density affects its localization properties, which determines if the system behaves like a metal or an insulator.
Localization Criterion: This criterion uses the fractional Fibonacci number system to predict whether a system will be localized (insulating) for all finite potential strength based on its density representation. If the density remains constant across changes in size and the number system structure, localization is guaranteed.
Base-phi Representation: The base-phi system uses the golden ratio ($\phi$) to represent real numbers. In this context, it describes how particle densities behave in the thermodynamic limit of the Fibonacci number system, showing self-similar structures that govern localization behavior.

Terminology

Accuracy — 1× in this paper · explained in 1 episode
Unified Framework — 1× in this paper · explained in 5 episodes

Summary

The paper introduces an extension of number systems, termed the fractional Fibonacci number system, to interpret the localization phase diagram of the many-body non-interacting Aubry-Andr´e model. This framework provides a criterion for localization in finite systems and discusses its behavior in the thermodynamic limit, linking particle density representations to specific transition points.

The gist: A fractional Fibonacci number system can represent any fraction of the form M/Fn, where M is an integer and Fn is a Fibonacci number, interpolating between the natural number system (Fibonacci) and base-φ representation.

Model Setup and Hamiltonian

The study focuses on the non-interacting many-body version of the Aubry-Andr´e model (AAM). The Hamiltonian for this model is given by:

H = X Σ j=1 h (−t)(c†j cj+1 + c†j+1cj) + W cos(2pa j)nj, where t is the hopping strength, W is the potential strength, and α denotes the golden ratio (approximated here as a ratio of Fibonacci numbers). The analysis specifically considers systems of size L and finite particle number N, with density defined as ρ = N/L. For calculations, periodic boundary conditions are assumed.

Classification of Particle Densities

The paper categorizes irrational densities based on their relationship to the golden ratio and Fibonacci ratios:

Rational densities exhibit a localization transition at W = 2t.
Irrational numbers divide into two further categories: one formed by the set of irrational numbers which can be produced by taking the infinite limit of Fibonacci ratios or finite sums thereof, where the localization transition occurs at W = 0.
For all other irrational numbers outside this category, the transition occurs at W = 2t, as it does for rational fillings.

The Fractional Fibonacci Number System

This extended number system is introduced to provide a more transparent classification of these densities:

The Fibonacci number system represents positive integers, while any real number can be represented in the base-φ number system.
The fractional Fibonacci number system interpolates between these two cases and can represent any fraction of the form M/Fn, where M is an integer and Fn is a Fibonacci number.
For a density ρ = N/L where L = Fn, the density can be written as ρ = X∞ j=2 ajFj / Fn (12), where aj ∈ [0, 1] and no consecutive coefficients can be unity.

Localization Criterion

The localization criterion is defined using this number system:

If a density for some system size L = Fn is written in the appropriate fractional Fibonacci number system, and the number does not change as the system size (and the fractional Fibonacci Fibonacci system) is changed, then the system at that density will be localized for all finite potential strength.
At other densities, a genuine metal-insulator transition occurs at W = 2t.

Thermodynamic Limit and Base-φ Recovery

In the thermodynamic limit, where L = Fn (or n) becomes large, the fractional Fibonacci number system converges to the base-φ number system:

The approximate particle density is given by ρ ≈ nX−1 j=2 ajφj−n (14), which simplifies in the limit to ρ ≈ X∞ j=1 bjφ−j (16), where b j are coefficients derived from the Zeckendorf decomposition of N.
The set of all base-φ numbers between zero and one is described by S = X∞ j=1 bjφ−j, with no consecutive 1s (17). This set can be broken into subsets S0 and S1, exhibiting self-similarity through shifts like S0 = φ−1S (19) and S1 = φ−1 + φ−2S (20), demonstrating the iterative structure of base-φ numbers.
The localization behavior in the thermodynamic limit can depend on the direction from which it is approached, as shown by analyzing limits like γ+ = limn→∞ Fn−1 + 1 / Fn and γ− = limn→∞ Fn−1 - 1 / Fn, where both tend to 0.1φ.

Numerical Examples and Zeckendorf Theorem

The Zeckendorf theorem is used to decompose natural numbers into a unique sum of non-consecutive Fibonacci numbers:

The decomposition of different system sizes for a given density illustrates the transition behavior; for example, for ρ = 1/2, the number of terms in the numerator (the Zeckendorf decomposition) increases with system size L.
For certain irrational densities like ρ = 2(1+√5), the Zeckendorf decomposition always gives a single term, which is crucial as it indicates an insulating phase for finite potential strength.

Improvements for AI systems

Based on the provided scientific paper, here are specific improvements that could be made to AI systems, along with what those improved systems could achieve:

)AI System Improvement Suggestions:

Development of Physics-Informed Machine Learning (PIML) for Many-Body Localization (MBL):
Implementation of Fractional Fibonacci Number System for Phase Diagram Interpretation:
Creation of a Novel Localization Criterion for Quantum Systems based on Density Representation:

)Specific Improvements and Capabilities:

Physics-Informed Machine Learning (PIML) for Many-Body Localization (MBL):
Implementation of Fractional Fibonacci Number System for Phase Diagram Interpretation:
Creation of a Novel Localization Criterion for Quantum Systems based on Density Representation:

)What the Improved AI System Can Do:

High-Fidelity Simulation and Prediction of Quantum Phases in Disordered/Quasiperiodic Systems (MBL):
Accurate Characterization and Classification of Many-Body Localization Regimes using New Mathematical Tools:
Automated Detection of Metal-Insulator Transitions based on System Size Scaling Exponents:

)Detailed Functionality Breakdown:

Abstract

We introduce an extension of the Fibonacci number system, we call the fractional Fibonacci number system, which interpolates between the Fibonacci number system (used for natural numbers) and the irrational base- ϕ number system, which can be used to represent real numbers. The number system finds its use in interpreting the localization phase diagram of the many-body non-interacting Aubry-André model. For finite system sizes a localization criterion can be obtained if the particle density is written in the fractional Fibonacci number system. In the thermodynamic limit, the criterion remains, but in this case the particle density is expressed in base- ϕ. The nature of the thermodynamic limit is also discussed.

Sources

Dry Ten Martini Problem in the non-critical case