Fibonacci Chain: Quasiperiodicity & Applications

• Fibonacci chain is a one-dimensional quasiperiodic structure generated by substitution rules, featuring golden-ratio scaling and long-range order.
• It exhibits multifractal spectra and critical eigenstates, with detailed insights from tight-binding models and wavefunction experiments.
• Applications span superconductivity, photonics, and nonharmonic Fourier analysis, leveraging its self-similarity and cut-and-project geometry.

The Fibonacci chain is a one-dimensional quasiperiodic structure generated by the Fibonacci substitution, most commonly written as $A \to AB,\ B \to A$ or equivalently $L \to LS,\ S \to L$. In geometric form it is a tiling or Delone set built from two segment lengths in the golden-ratio proportion, and in operator-theoretic form it is a tight-binding chain whose hoppings or on-site terms follow the same aperiodic word. Across these realizations, the Fibonacci chain serves as a canonical model of long-range order without translational invariance, combining substitution self-similarity, cut-and-project geometry, singular-continuous spectra, critical eigenstates, and nontrivial Fourier structure (Lai et al., 14 Mar 2025, Fang et al., 2016, Wang et al., 2024).

1. Construction and equivalent formulations

The standard symbolic construction begins from a binary alphabet and the inflation rule $A \to AB,\ B \to A$, or, in long/short notation, $L \to LS,\ S \to L$. Iteration produces finite approximants of Fibonacci length $F_n$, and in the infinite limit yields the Fibonacci word. In the $m$-bonacci formalism, the Fibonacci chain is the $m=2$ specialization of the Rauzy substitution $\sigma_2: 1 \mapsto 12,\ 2 \mapsto 1$, with incidence matrix

$M_{\sigma_2}=\begin{pmatrix}1&1\1&0\end{pmatrix},$

dominant eigenvalue $\phi=(1+\sqrt{5})/2$, and left eigenvector $L \to LS,\ S \to L$0. The infinite Fibonacci word $L \to LS,\ S \to L$1 has digit frequencies $L \to LS,\ S \to L$2 and $L \to LS,\ S \to L$3 (Lai et al., 14 Mar 2025).

A geometric realization identifies the letters with two segment types. One conventional choice sets $L \to LS,\ S \to L$4 and $L \to LS,\ S \to L$5, so that $L \to LS,\ S \to L$6. Segment endpoints or tile origins are cumulative sums of these lengths. An equivalent Beatty–Sturmian representation writes the length sequence as

$L \to LS,\ S \to L$7

which takes values in $L \to LS,\ S \to L$8 (Fang et al., 2016).

The same chain also arises by cut-and-project from a higher-dimensional lattice. In this formulation, one projects lattice points from a strip in $L \to LS,\ S \to L$9 to a line of slope $A \to AB,\ B \to A$0, obtaining the same quasiperiodic order. This viewpoint is standard in discussions of Fibonacci quasicrystals and underlies the perpendicular-space or “conumber” description of local environments (Wang et al., 2024).

A further specialization, important for harmonic analysis, encodes the Fibonacci chain as a Delone set $A \to AB,\ B \to A$1 whose consecutive gaps are exactly $A \to AB,\ B \to A$2 and $A \to AB,\ B \to A$3. In the notation of $A \to AB,\ B \to A$4-bonacci chains, these gaps come from the bi-infinite Fibonacci word through the rule $A \to AB,\ B \to A$5, followed by cumulative summation (Lai et al., 14 Mar 2025).

2. Geometric organization and self-similarity

The Fibonacci chain is long-range ordered but not periodic. In bond-modulated tight-binding realizations, the hopping amplitudes are binary, $A \to AB,\ B \to A$6, ordered by the Fibonacci word; in on-site models, $A \to AB,\ B \to A$7 or $A \to AB,\ B \to A$8 take two values arranged in the same sequence. The absence of translational symmetry is compensated by inflation symmetry and by a well-defined local-environment hierarchy (Wang et al., 2024, 2207.13755).

Conumber indexing makes that hierarchy explicit. For an $A \to AB,\ B \to A$9-site approximant, the conumber

$L \to LS,\ S \to L$0

orders sites according to similarity of their local surroundings. In the off-diagonal Fibonacci Hamiltonian, this separates sites into “atom” sites, with weak bonds on both sides, and “molecule” sites, with one strong bond. Atom sites number $L \to LS,\ S \to L$1 and occupy a central conumber window, while molecule sites number $L \to LS,\ S \to L$2 and occupy two side windows. In both real space and perpendicular space, patterns recur under inflation or deflation by powers of $L \to LS,\ S \to L$3, with the superconducting study reporting recurrence scaled by $L \to LS,\ S \to L$4 (Wang et al., 2024).

The $L \to LS,\ S \to L$5-bonacci construction gives a complementary geometric description in terms of a unidimensional quasicrystal $L \to LS,\ S \to L$6. For $L \to LS,\ S \to L$7, consecutive gaps in $L \to LS,\ S \to L$8 take exactly the two values $L \to LS,\ S \to L$9 and $F_n$0, and the upper density is determined by the digit frequencies of the bi-infinite Fibonacci word. This Delone-set viewpoint is tailored to nonharmonic Fourier analysis and diffraction (Lai et al., 14 Mar 2025).

Experimental work with dielectric resonator chains has shown that conumber ordering also reorganizes the measured local density of states into a nearly symmetric fractal pattern, with central spectral clusters concentrated on atomic sites and side clusters on molecular sites. That observation directly links cut-and-project geometry, local environments, and experimentally accessible wavefunction intensities (2207.13755).

3. Spectrum, multifractality, and dynamics

The Fibonacci chain is a standard example of a system with singular-continuous spectrum and critical eigenstates. In the off-diagonal model with hoppings $F_n$1 and $F_n$2 arranged by the Fibonacci word, the modulation ratio is $F_n$3, and real-space renormalization separates the chain into molecular and atomic sectors. In the strong-modulation limit $F_n$4, participation moments obey multiplicative renormalization rules, and the generalized dimensions of individual wavefunctions take the form

$F_n$5

with $F_n$6. At $F_n$7, $F_n$8, while multifractality appears at order $F_n$9. The same work identifies a leading-order symmetry under permutation of site and energy indices, $m$0 (Macé et al., 2016).

Microwave-resonator experiments have directly observed this multifractality. For the measured local density of states, the conumber-sorted pattern is fractal, and the averaged generalized dimensions agree well with perturbative renormalization-group predictions in the quasiperiodic limit. The same experiment extracted renormalization factors $m$1 and $m$2 at $m$3, close to the theoretical values $m$4 and $m$5 (2207.13755).

In the on-site Fibonacci model, all single-particle eigenstates are critical for any $m$6, and noninteracting transport is anomalous with a continuously varying exponent. Reported spreading exponents are $m$7 at $m$8, $m$9 at $m=2$0, $m=2$1 at $m=2$2, and $m=2$3 at $m=2$4, interpolating from ballistic at $m=2$5 toward localized behavior as $m=2$6 (Varma et al., 2019). For the isotropic Fibonacci XY chain, this anomalous one-body transport has a rigorous many-body consequence: Lieb–Robinson bounds become sub-ballistic, with the standard $m=2$7 light cone replaced by $m=2$8, where admissible $m=2$9 are precisely those above the upper transport exponent $\sigma_2: 1 \mapsto 12,\ 2 \mapsto 1$0 of the one-body Fibonacci Hamiltonian (Damanik et al., 2014).

Interacting dynamics depend strongly on regime. Exact-diagonalization work at $\sigma_2: 1 \mapsto 12,\ 2 \mapsto 1$1 found an ergodic-to-many-body-localized transition in the range $\sigma_2: 1 \mapsto 12,\ 2 \mapsto 1$2, together with distinctive “magic-angle” peaks in the density distribution of the MBL phase (Macé et al., 2018). By contrast, a high-temperature transport study at weak interaction reported diffusion for any studied potential strength and no evidence of many-body localization near $\sigma_2: 1 \mapsto 12,\ 2 \mapsto 1$3, with diffusion constants from boundary-driven Lindblad dynamics and unitary dynamics in agreement (Varma et al., 2019). A dynamical-typicality study at $\sigma_2: 1 \mapsto 12,\ 2 \mapsto 1$4 found superdiffusion for $\sigma_2: 1 \mapsto 12,\ 2 \mapsto 1$5, subdiffusion for $\sigma_2: 1 \mapsto 12,\ 2 \mapsto 1$6, and MBL-like saturation at large $\sigma_2: 1 \mapsto 12,\ 2 \mapsto 1$7 (Chiaracane et al., 2021). This suggests a parameter-dependent dynamical landscape rather than a single universal transport scenario.

The optical response of the off-diagonal Fibonacci chain inherits the same hierarchy. Exact diagonalization of the Kubo conductivity shows a multitude of sharp peaks grouped into intra-$\sigma_2: 1 \mapsto 12,\ 2 \mapsto 1$8, $\sigma_2: 1 \mapsto 12,\ 2 \mapsto 1$9, and $M_{\sigma_2}=\begin{pmatrix}1&1\1&0\end{pmatrix},$0 transitions, with a self-similar collapse under a frequency rescaling $M_{\sigma_2}=\begin{pmatrix}1&1\1&0\end{pmatrix},$1 at large hopping ratio $M_{\sigma_2}=\begin{pmatrix}1&1\1&0\end{pmatrix},$2. In the corresponding periodic approximant, Bloch momentum conservation suppresses most of this structure (Iijima et al., 2022).

4. Fourier structure, diffraction, and harmonic analysis

Two distinct Fourier-theoretic uses of the Fibonacci chain are prominent. One concerns nonharmonic Fourier series whose frequencies are the points of the Fibonacci Delone set $M_{\sigma_2}=\begin{pmatrix}1&1\1&0\end{pmatrix},$3; the other concerns direct Fourier transforms of the symbolic or geometric sequence itself.

For the first problem, the Fibonacci chain enters as a frequency set in sums

$M_{\sigma_2}=\begin{pmatrix}1&1\1&0\end{pmatrix},$4

with $M_{\sigma_2}=\begin{pmatrix}1&1\1&0\end{pmatrix},$5. The relevant quantity is the upper density

$M_{\sigma_2}=\begin{pmatrix}1&1\1&0\end{pmatrix},$6

Beurling’s generalization of Ingham’s inequality then implies that for every bounded interval $M_{\sigma_2}=\begin{pmatrix}1&1\1&0\end{pmatrix},$7 with $M_{\sigma_2}=\begin{pmatrix}1&1\1&0\end{pmatrix},$8, there exist $M_{\sigma_2}=\begin{pmatrix}1&1\1&0\end{pmatrix},$9 such that

$\phi=(1+\sqrt{5})/2$0

In the Fibonacci normalization where $\phi=(1+\sqrt{5})/2$1, the critical length is $\phi=(1+\sqrt{5})/2$2. The same analysis gives explicit gap estimates, including $\phi=(1+\sqrt{5})/2$3, $\phi=(1+\sqrt{5})/2$4, and the general bound

$\phi=(1+\sqrt{5})/2$5

with refinements at Fibonacci lengths $\phi=(1+\sqrt{5})/2$6 (Lai et al., 14 Mar 2025).

The second Fourier problem studies the discrete Fourier transform of the length sequence $\phi=(1+\sqrt{5})/2$7,

$\phi=(1+\sqrt{5})/2$8

and plots the cloud $\phi=(1+\sqrt{5})/2$9 in the complex plane. Successive approximants display a cardioid-like “cycloidal fractal signature,” with an observed scale factor of approximately $L \to LS,\ S \to L$00 per inflation step and an odd/even mirror flip of orientation. The reported pointwise dimension is approximately $L \to LS,\ S \to L$01. Sensitivity tests show that truncating or scrambling the Fibonacci order destroys the cardioid closure, whereas changing the ratio $L \to LS,\ S \to L$02 away from $L \to LS,\ S \to L$03 mainly rescales the pattern. The visualization therefore emphasizes substitution order rather than the metric choice of segment lengths (Fang et al., 2016).

These two Fourier viewpoints are complementary rather than interchangeable. The Beurling–Ingham framework concerns stability of nonharmonic expansions on intervals, while the cardioid-like signature arises from plotting Fourier coefficients of a finite symbolic sequence in the complex plane. A plausible implication is that the same substitution hierarchy can manifest either as sampling stability in harmonic analysis or as nested interference geometry in coefficient space.

5. Superconducting, coupled, and topological extensions

The Fibonacci chain has become a laboratory for superconductivity in quasiperiodic media. In the attractive Hubbard model on the off-diagonal Fibonacci chain, Bogoliubov–de Gennes calculations yield a bulk superconducting transition at a critical temperature obeying a power law in $L \to LS,\ S \to L$04,

$L \to LS,\ S \to L$05

with $L \to LS,\ S \to L$06 controlled by the modulation strength $L \to LS,\ S \to L$07. At half filling, $L \to LS,\ S \to L$08 for weak modulation, while in the strong-modulation limit $L \to LS,\ S \to L$09. The local order parameter is self-similar in both real and perpendicular space, the local densities of states vary from site to site, but the width of the superconducting gap is identical on all sites. At $L \to LS,\ S \to L$10, the $L \to LS,\ S \to L$11–$L \to LS,\ S \to L$12 phase diagram contains insulating domes centered on intrinsic normal-state gaps and surrounded by superconducting regions (Wang et al., 2024).

A distinct proximitized off-diagonal Fibonacci chain exhibits “universal end superconductivity.” Self-consistent BdG calculations find three critical temperatures: a bulk $L \to LS,\ S \to L$13, a left-end $L \to LS,\ S \to L$14, and a right-end $L \to LS,\ S \to L$15. For $L \to LS,\ S \to L$16, the left termination stabilizes as $L \to LS,\ S \to L$17, making $L \to LS,\ S \to L$18 and $L \to LS,\ S \to L$19 essentially independent of the Fibonacci approximant index, while the right termination alternates with parity, producing an $L \to LS,\ S \to L$20-dependent $L \to LS,\ S \to L$21. With the chosen parameters, the maximal enhancement of $L \to LS,\ S \to L$22 reaches up to $L \to LS,\ S \to L$23 relative to $L \to LS,\ S \to L$24 for even $L \to LS,\ S \to L$25, while $L \to LS,\ S \to L$26 increases by up to $L \to LS,\ S \to L$27. The mechanism is attributed to competition between topological bound quasiparticles and critical states (Zhu et al., 2024).

For on-site Fibonacci disorder with attraction, the transition temperature is also enhanced relative to self-averaged analytical theory. The enhancement traces to the full overlap matrix

$L \to LS,\ S \to L$28

and to power-law two-eigenfunction correlations $L \to LS,\ S \to L$29. In the very weak coupling regime, self-averaging breaks down: the distribution $L \to LS,\ S \to L$30 broadens, sample-to-sample fluctuations become strong, and some realizations have $L \to LS,\ S \to L$31 (Sun et al., 2023).

Topological superconducting variants display a different hierarchy. In the Fibonacci–Kitaev chain with quasiperiodic chemical potentials $L \to LS,\ S \to L$32, transfer-matrix analysis of the Majorana zero-energy mode yields a critical pairing $L \to LS,\ S \to L$33 and a topological phase diagram with self-similar fractal structure. The box-counting dimension of the phase boundary is reported as $L \to LS,\ S \to L$34 (Ghadimi et al., 2017).

Coupled Fibonacci chains add further spectral possibilities. Two identical chains coupled uniformly simply produce two shifted copies of the single-chain spectrum; selective coupling of only $L \to LS,\ S \to L$35 sites can make one parity block periodic and Bloch-like while the other remains critical; coupling through intermediate sites yields a macroscopically degenerate set of four-site compact localized states and, in the infinite limit, a perfectly flat quasi band (Moustaj et al., 2022).

In photonics, a finite Fibonacci-type chain of $L \to LS,\ S \to L$36 waveguides has been used for end-to-end pumping. Quasiperiodicity is encoded in binary spacings $L \to LS,\ S \to L$37 and $L \to LS,\ S \to L$38, and only two waveguides are bent, modulating four boundary bonds along propagation. With this boundary-only interpolation, a localized pumping state migrates adiabatically from one edge to the other over $L \to LS,\ S \to L$39 mm. Controlled defects with $L \to LS,\ S \to L$40 do not destroy the pump and can enlarge the excitation gap (Ghosh et al., 13 May 2026).

6. Symbolic, algebraic, and alternative meanings

Beyond its role as a physical quasicrystal, the Fibonacci chain supports purely algebraic and symbolic constructions. On the model set $L \to LS,\ S \to L$41, one can define a quasicrystal Lie algebra with generators $L \to LS,\ S \to L$42 and bracket

$L \to LS,\ S \to L$43

together with an aperiodic Witt algebra, an aperiodic Virasoro extension, and an aperiodic Jordan algebra based on the quasiaddition $L \to LS,\ S \to L$44. The Jordan product

$L \to LS,\ S \to L$45

is commutative and satisfies the Jordan identity. These constructions are designed to match the original defect-free Fibonacci quasicrystal exactly, not only the earlier modified chain with a point defect (Corradetti et al., 2023).

A different symbolic-dynamics program forbids the minimal forbidden factors of the Fibonacci word. This yields a hierarchy of frustration-free Hamiltonians $L \to LS,\ S \to L$46 whose ground-state languages avoid the minimal forbidden factors up to length $L \to LS,\ S \to L$47. The base rung $L \to LS,\ S \to L$48, which forbids only $L \to LS,\ S \to L$49, reproduces the “golden chain” with ground-state count $L \to LS,\ S \to L$50 and growth constant $L \to LS,\ S \to L$51. The first nontrivial rung $L \to LS,\ S \to L$52, or “Plastic chain,” forbids $L \to LS,\ S \to L$53 and $L \to LS,\ S \to L$54, satisfies

$L \to LS,\ S \to L$55

and has growth constant $L \to LS,\ S \to L$56, the plastic constant satisfying $L \to LS,\ S \to L$57. More generally, $L \to LS,\ S \to L$58 decreases monotonically toward $L \to LS,\ S \to L$59, producing an entropy staircase that flows to the zero-entropy Fibonacci subshift (Amaral, 9 Nov 2025).

The phrase “Fibonacci chain” also has a separate combinatorial meaning. In graph-theoretic number theory, it denotes a permutation $L \to LS,\ S \to L$60 of $L \to LS,\ S \to L$61 such that every adjacent sum $L \to LS,\ S \to L$62 is a Fibonacci number. In this sense, a Fibonacci chain is a Hamiltonian path in the Fibonacci-sum graph $L \to LS,\ S \to L$63, and a Fibonacci necklace is a Hamiltonian cycle. The existence theorem is sharp: such chains exist precisely for $L \to LS,\ S \to L$64, $L \to LS,\ S \to L$65, or $L \to LS,\ S \to L$66 and $L \to LS,\ S \to L$67 with $L \to LS,\ S \to L$68, and no Fibonacci necklaces exist. A billiards construction on rectangles with corners at halves of consecutive Fibonacci numbers produces the admissible chains geometrically (2002.03705).

Taken together, these uses show that “Fibonacci chain” is not a single object but a family of mathematically linked constructions organized by the Fibonacci word: quasicrystal tilings and Delone sets, tight-binding and spin chains, superconducting and photonic devices, aperiodic algebras, constrained symbolic Hamiltonians, and even Hamiltonian-path problems on Fibonacci-sum graphs. The common invariant is the same inflation order, and the accumulated literature shows how much analytic structure that order can sustain.