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Topological pumping of light governed by Fibonacci numbers

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Pith's one-line read Quasi-periodic topological pumping is the Fibonacci limit of periodic pumps, with Chern numbers F_n and velocity set by the golden ratio.

desk verdict First experiment on bichromatic quasi-periodic Thouless pumping, with a clean Fibonacci-Chern story, but the exact recurrence is conjectured for the small-n regime the experiment actually tests. read the letter →

arxiv 2509.04910 v1 pith:JCKHSDJB submitted 2025-09-05 physics.optics

Ruihan Peng , Kai Yang , Qidong Fu , Yanli Chen , Peng Wang , Yaroslav V. Kartashov , Vladimir V. Konotop , Fangwei Ye This is my paper · ORCID
classification physics.optics
keywords topologicalpumpingquantizedtransportquasi-periodicdrivingFibonaccinumbersgoldenratiobestrationalapproximationsChernnumberphotorefractiveopticallattices
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to show that topological pumping does not require a strictly periodic drive: a bi-chromatic pump whose two frequencies are in an irrational ratio can be understood as the limit of periodic pumps whose frequency ratios are the best rational approximations to that irrational number. For the golden-ratio case, the approximant periods are successive Fibonacci numbers $F_n$, and the paper argues that the Chern number of the pumped band in the $n$-th approximant obeys the Fibonacci recurrence $C^\nu_{n+1}=C^\nu_n+C^\nu_{n-1}$, giving $C^1_n=F_n$ for the excited band. A one-cycle pump then moves a light beam by $F_n$ lattice periods, while the average pumping velocity converges to $\alpha/\phi$, where $\alpha$ is the small sliding angle of the sublattices. The paper reports experimental observation in photorefractive crystals for the first three approximants, with measured center-of-mass displacements consistent with the Fibonacci rule and with the velocity approaching the golden-ratio value.

What carries the argument

The central object is the periodic approximant: replace the irrational frequency ratio by the best rational approximation $F_n/F_{n+1}$, so that the quasi-periodic Hamiltonian $H_\phi$ is replaced by a periodic $H_n$ with period $Z_n=2\pi F_{n+1}/\alpha$. The argument then treats $H_{n+1}$ as $H_n+W_n$ with the small correction $W_n=(-1)^n \alpha z/(F_{n+1}F_{n+2})\,\partial V_n/\partial\zeta_n+O(V_n/F_{n+2}^2)$; a perturbation calculation for the Chern number per cycle yields $C^\nu_{n+1}=C^\nu_n+C^\nu_{n-1}+O(1/F_{n+2})$. Because Chern numbers are integers and the $O(1/F_{n+2})$ term is non-integer, the correction must vanish, giving the exact Fibonacci recurrence, conjectured rather than proven for all $n$. The seeds $C^1_0=0$ and $C^1_1=1$ then force $C^1_n=F_n$. A parity-time symmetry relation that halves the Chern number over half a period is what makes the displacement measurable in a 20 mm crystal.

What would settle it

Evaluate the Berry curvature for approximants $n=4$ and $n=5$ in the same optical potential and check numerically whether $C_4=5$ and $C_5=8$ exactly; if the non-integer $O(1/F_6)$ correction has magnitude at least $1/2$, the Fibonacci rule fails there. Alternatively, a crystal long enough to observe half a period of the fourth approximant should show a half-cycle center-of-mass shift of about $4$ transverse lattice periods; a clear deviation from that would falsify the recurrence.

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Extended reading notes

Core claim

In the paper's own terms, the central discovery is a Fibonacci law for quantized transport in quasi-periodic pumps. When the golden-ratio conjugate $\phi^{-1}=(\sqrt{5}-1)/2$ is approximated by ratios $F_n/F_{n+1}$ of consecutive Fibonacci numbers, the Hamiltonian becomes $Z_n$-periodic with $Z_n=2\pi F_{n+1}/\alpha$, and the excited-band Chern number is $C^1_n=F_n$. Equivalently, after one pumping cycle the beam center moves by $F_n$ transverse lattice periods, and the cycle-averaged velocity $v_n=\alpha F_n/F_{n+1}$ tends to $\alpha/\phi$. The same asymptotic perturbation argument gives the general recurrence for any band, and the paper verifies the rule numerically for approximants $n=1,\dots,6$ and experimentally for $n=1,2,3$.

Load-bearing premise

The argument depends on the leftover approximation error being smaller than half a unit, so that the whole-number Chern indices are forced to follow the exact Fibonacci pattern; the proof only controls this for large approximants, while the experiments use the first three, where that guarantee is not yet established.

Editorial extensions

If this is right

  • In each successive approximant the transported displacement is quantized as $F_n$ lattice periods, so the displacement sequence itself obeys the Fibonacci recursion $Y_{n+1}=Y_n+Y_{n-1}$.
  • The average pumping velocity converges to $\alpha/\phi$, giving quasi-periodic pumping a well-defined, topology-determined speed even though no global period exists.
  • Because the Chern number is insensitive to lattice depth, the pumping rate does not change with applied voltage and tolerates local disorder in the crystal.
  • The same approximant construction generalizes to any irrational frequency ratio through its continued-fraction convergents, so the Fibonacci rule is the special case for the golden ratio.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An implication the paper leaves implicit is that for a generic irrational ratio, the approximant Chern numbers should follow a recurrence whose coefficients come from the continued fraction, making the golden-ratio Fibonacci rule one case of a broader transport law.
  • A testable extension would be to repeat the pump in a discrete waveguide array, where diffraction is suppressed and approximants n=4 and n=5 become reachable, and check the predicted center-of-mass shifts before the asymptotic regime sets in.
  • A conceptual consequence, if the velocity limit is robust, is that quantized transport can be defined for entirely aperiodic drives without a well-defined cycle, broadening topological pumping beyond periodic modulation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes that quasi-periodic topological pumping with an irrational frequency ratio can be understood through periodic approximants given by Fibonacci ratios. The authors derive, via perturbation theory, that the Chern numbers of successive approximants satisfy the recurrence C^ν_{n+1}=C^ν_n+C^ν_{n-1} asymptotically for large n, conjecture its validity for all n, and report an optical experiment in photorefractive crystals in which the center-of-mass displacement of a light beam for the first three nonzero approximants is consistent with the Fibonacci sequence and the average pumping velocity approaches α/φ. The experiment uses the best rational approximations 1/2, 2/3, and 3/5 of the golden-ratio conjugate and relies on parity-time symmetry to extract half-period results from a finite sample.

Significance. If the Fibonacci Chern-number recurrence and the convergence statement are established, this is a significant contribution: it provides the first experimental demonstration of quasi-periodic topological pumping via periodic approximants, offers a practical protocol for observing quantized transport in finite samples, and gives a falsifiable prediction for the limiting pumping velocity. The paper is commendably explicit about the parameter-free nature of the perturbation argument, and the experimental robustness to voltage variations is a strength. The numerical verification for n=1..6 and the convergence of the measured velocities toward Eq. (6) are also positive features. However, the central theoretical result is not fully proven for the experimentally accessed regime; it is a conjecture supported by numerical checks, and the paper does not currently supply a bound that would upgrade the asymptotic relation to an exact integer recurrence.

major comments (3)
  1. [Section 2.2 and Methods 4.3] The derivation establishes only the asymptotic statement C^ν_{n+1}=C^ν_n+C^ν_{n-1}+O(1/F_{n+2}) for large n. Since Chern numbers are integers, the exact recurrence follows only if the O(1/F_{n+2}) remainder has magnitude below 1/2, but no such bound is provided or justified. For the experimentally studied approximants (n=2,3,4 in the F_n indexing used in Section 2.2), the nominally small parameter takes values 1/3, 1/5, and 1/8, so the condition F_{n+2}≫1 is not satisfied. Because the headline experimental result in Section 2.4 interprets the measured Fibonacci displacements as confirming the Chern-number recurrence in precisely this small-n regime, the central claim currently rests on a conjecture where it is tested. Please either provide a rigorous bound on the remainder (or an exact evaluation) for the experimental parameter range, or explicitly state that the experiment tests a conjecture and provide a quantitative account of the extrapolation.
  2. [Methods 4.3, Eqs. (8)-(10)] The passage from Eq. (8) to Eq. (10) and then to C^ν_{n+1}=C^ν_n+C^ν_{n-1}+O(1/F_{n+2}) is too terse, and the index shift is not spelled out. More importantly, the replacement Ω^ν_n = Ω^ν_{n-1} + O(ε) is treated as uniform in z and k without any control on the remainder in terms of the potential gradients, the band gap, or the adiabatic parameter α. The O(ε) term is therefore an uncontrolled asymptotic estimate rather than a proven bound, and it does not justify the exact integer recurrence. Please present the full derivation and state explicitly which norm or supremum bound is used, and supply evidence that the remainder is below 1/2 for the n values used in the experiment.
  3. [Section 2.3] The statement that the Chern numbers for n=1..6 are computed to be 1,2,3,5,8,13 is reported without any details of the numerical method. Because the exact recurrence is only conjectured for small n, these numerics are the principal evidence for the Fibonacci sequence in the regime tested experimentally. The paper should describe the numerical procedure (grid resolution, number of k-points, number of z-steps, handling of Berry-curvature gauge) and demonstrate convergence of the extracted integer values with respect to these parameters. Without this information, the reader cannot assess whether the computed Chern numbers are reliable.
minor comments (5)
  1. [Section 2.2 and Section 2.3] There is an index inconsistency: Section 2.2 sets C^1_0=0=F_0 and C^1_1=1=F_1, which implies C^1_2=F_2=1, but Section 2.3 lists the computed values for n=1..6 as 1,2,3,5,8,13 (i.e., F_2,...,F_7). Please define the indexing convention for n explicitly and use it consistently throughout.
  2. [Fig. 1 caption] The caption states Z_1:Z_2:Z_3 = 1:1.5:2.5, but from the definition Z_n=2πF_{n+1}/α the ratios for the first three nontrivial approximants are 1:2:3 (or 2:3:5 depending on the offset used in the figure). Please reconcile the figure, the text, and the definition of Z_n.
  3. [Section 2.4] The quoted half-periods of approximately 8, 12, and 19 mm have ratios that do not exactly match the Fibonacci ratios of the Z_n periods; please explain whether this is due to experimental calibration, the use of half versus full periods, or the finite sample geometry.
  4. [Introduction] There is a typo: 'light bean' should be 'light beam' in the sentence 'paraxial light bean propagating in our structure'.
  5. [Eq. (5)] The notation v_n = Y C^1_n / Z_n, together with the earlier definition Y^ν_n = Y C^ν_n, is slightly redundant and could confuse the reader; please define Y^ν_n and Y^{ν=1}_n distinctly and consistently.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Chern-number recurrence is derived, not fitted; the only self-citation (half-period PT relation from Ref. [24]) is an external, parameter-free symmetry result.

full rationale

The central derivation is not circular. The Fibonacci numbers enter as an input through the choice of periodic approximants (Z_n = 2π F_{n+1}/α, with α the sliding angle), but the Chern-number recurrence C^ν_{n+1} = C^ν_n + C^ν_{n-1} is derived in Methods (Eqs. 8–10) from a perturbation expansion in 1/F_{n+2} together with the integer-valuedness of Chern numbers; it is not assumed. The numerical Chern values 1, 2, 3, 5, 8, 13 for the lowest approximants are computed from the band structure (Section 2.3), and the experimental COM displacements in Fig. 3 are compared with the parameter-free prediction of Eq. (6). No fitted parameter is renamed as a prediction, and no displacement datum is used to construct the theoretical curve. The paper explicitly gates the all-n recurrence behind a conjecture: 'we conjecture (and check numerically, see below) its validity for all n≥0'; thus the small-n experimental regime (n=1–3) rests on a conjectural extension rather than on the proven asymptotic statement. That is a rigor gap, not a reduction to inputs. The only self-citation with functional weight is Ref. [24], used to justify that, by parity-time symmetry of H_n(z), the Chern number over half a period is half of the full-period Chern number; this is a published, parameter-free symmetry statement that does not assume the Fibonacci result, so under the stated rules it counts as external support rather than circularity. Overall the derivation is self-contained against the experiment, with at most a minor self-citation that is not circular.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central theoretical result rests on standard Bloch and adiabatic theory plus a perturbation expansion; no parameters are fitted to the reported displacements. The experimental inputs alpha, p1^2, p2^2, and V0 are hand-set controls. The main unproved premise is the smallness of the perturbation correction that turns an asymptotic relation into an exact integer recurrence.

free parameters (3)
  • Sliding angle alpha = approximately 0.004 (dimensionless)
    Sets the adiabatic driving rate and appears linearly in the claimed asymptotic velocity v=alpha/phi; it is an experimental input chosen for adiabaticity, not fitted to the displacement data.
  • Sublattice depths p1^2, p2^2 = 0.09 and 0.49 (dimensionless)
    Chosen to match experimental beam intensities and keep the upper band isolated; hand-set inputs that do not affect the topological integer in the reported regime.
  • Lattice amplitude V0 = -2.5 (dimensionless, corresponding to 600 V)
    Controls band gaps and diffraction; chosen for the experiment and not fitted to the Fibonacci prediction.
assumptions (6)
  • domain assumption The paraxial Schrodinger equation with photorefractive saturable potential V(y,z)=V0/(1+I) governs the probe beam.
    The experimental observation is interpreted through this linear model; drift, diffusion, and weak nonlinear effects are treated as small perturbations.
  • standard math Thouless relation: displacement over a pumping cycle equals the Chern number of the populated band times the lattice period.
    Central to interpreting measured center-of-mass displacement as topological; cited to Ref. [1].
  • domain assumption Adiabatic following: the beam populates only band nu=1 for the entire propagation.
    The paper asserts a large band gap guarantees this; no direct measurement of band population is reported.
  • domain assumption PT symmetry of H_n(z) implies the Chern number over half a period is half the full-period Chern number.
    Used to interpret half-period measurements in a 20 mm sample; sourced from the authors' previous work (Ref. [24]).
  • standard math The best rational approximations of the golden ratio conjugate are Fibonacci quotients F_n/F_{n+1}, and the asymptotic limit defines quasi-periodic pumping.
    Standard continued-fraction result; underpins the construction of approximants.
  • ad hoc to paper The O(1/F_{n+2}) correction in the perturbation relation is smaller than 1/2 in magnitude, so the integer Chern numbers satisfy the exact Fibonacci recurrence.
    This assumption is not proved; the paper states the asymptotic relation for large n and then conjectures it for all n.

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Pith. "Pith review of Topological pumping of light governed by Fibonacci numbers." pith.science (2026). https://pith.science/paper/JCKHSDJB

@misc{pith2026250904910,
  author       = {Pith},
  title        = {Pith review of: Topological pumping of light governed by Fibonacci numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JCKHSDJB}},
  note         = {Machine review of arXiv:2509.04910}
}
read the original abstract

Topological pumping refers to transfer of a physical quantity governed by the systemtopology, resulting in quantized amounts of the transferred quantities. It is a ubiqui-tous wave phenomenon typically considered subject to exactly periodic adiabatic vari-ation of the system parameters. Recently, proposals for generalizing quasi-periodictopological pumping and identifying possible physical settings for its implementa-tion have emerged. In a strict sense, pumping with incommensurate frequencies canonly manifest over infinite evolution distances, raising a fundamental question aboutits observability in real-world finite-dimensional systems. Here we demonstrate thatbi-chromatic topological pumping with two frequencies, whose ratio is an irrationalnumber, can be viewed as the convergence limit of pumping with two commensuratefrequencies representing the best rational approximations of that irrational number. In our experiment, this phenomenon is observed as the displacement of a light beamcenter in photorefractive crystals induced by two optical lattices. The longitudinalperiods of the lattices, that in the paraxial approximation emulate two pumping fre-quencies, are related as Fibonacci numbers, successively approaching the golden ratio. We observed that a one-cycle displacement of the beam center at each successiveapproximation is determined by the relation between successive Fibonacci numbers,while the average direction of propagation (emulating average pumping velocity) ofthe beam is determined by the golden ratio.

Figures

Figures reproduced from arXiv: 2509.04910 by the authors.

Figure 1
Figure 1. FIG. 1: Conceptual schematic of periodic approximants illustrating light pump [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Quantized transport of light in the lowest periodic approximants for [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Observation of light pumping in the approximants of a quasi-periodic [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Experimental setup [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]

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Reviewed August 15, 2026 · model on record in the stance chip above.