Fibonacci Quasicrystal Overview

• Fibonacci quasicrystal is a deterministic aperiodic structure generated by recursive inflation and cut-and-project methods, embodying self-similarity and hyperuniform order.
• It exhibits singular-continuous spectra, multifractal eigenstates, and localization transitions that illuminate the interplay between fractality and quantum transport.
• Robust topological invariants yield edge states and higher-order modes, enabling advanced applications in photonics, superconductivity, and quantum information.

A Fibonacci quasicrystal is a deterministic aperiodic structure characterized by strict self-similarity and hyperuniform order, whose spatial arrangement and spectral properties arise from the recursive inflation rule and the cut-and-project formalism. With distinctive singular-continuous spectra, multifractal eigenstates, and topologically quantized invariants inherited from higher-dimensional crystalline models, Fibonacci quasicrystals constitute canonical models for the interplay of quasiperiodic order, fractality, localization, and topology in quantum and classical systems. They underpin rigorous mathematical, physical, and algebraic frameworks across condensed matter, photonics, and quantum information. Their critical states and Cantor-type energy bands, together with the robustness of topological invariants in non-crystalline settings, have generated exhaustive studies of their transport, thermodynamics, and topological applications.

1. Construction Methods and Structural Properties

Fibonacci quasicrystals are generated by repeated application of the inflation (substitution) rules on a binary alphabet (A, B), typically: $A \to AB,\quad B \to A$ The resulting infinite word encodes the one-dimensional sequence of "long" (L) and "short" (S) tiles, whose density ratio converges to the golden mean $\varphi = (1+\sqrt{5})/2$ (Corradetti et al., 2023, Jagannathan, 2020). The sequence is fundamentally nonperiodic yet strictly ordered; the length of the $n$th word is the $n$th Fibonacci number $F_n$, giving robust recurrence properties.

The cut-and-project method gives an equivalent geometric construction:

• Embed $\mathbb{Z}^2$ in $\mathbb{R}^2$.
• Project lattice points onto a line at slope $1/\varphi$, accepting those whose transverse projection lies in an interval ("acceptance window").
• The resulting set $\{x_j\}$ forms the Fibonacci tiling, with inter-site spacings $\{1,\varphi\}$.

Site positions can also be written using a floor-function formula: $$x_n = n + \left\lfloor \frac{n}{\varphi} \right\rfloor, \qquad n \in \mathbb{Z}$$ or, more generally,

$$x_n^{(\alpha,\beta)} = \left\lfloor \frac{n}{\varphi} + \alpha \right\rfloor + n\varphi + \beta$$

for parameter choices $(\alpha,\beta)$ matching the acceptance domain (Corradetti et al., 2023).

Hyperuniformity manifests as bounded fluctuations in tile counts, producing pure-point Bragg diffraction; reciprocal lattice vectors are $q_{h,k}=2\pi(h+\omega k)$ with $\omega=1/\varphi$ (Jagannathan, 2020). Discrete scale invariance arises from the inflation symmetry.

In higher dimensions, models such as the icosahedral Fibonacci-modified icosagrid employ the Fibonacci chain to space planar families with E8-lattice connections, yielding robust icosahedral symmetry via golden-ratio-based rotations (Fang et al., 2015).

2. Spectral Structure, Multifractality, and Localization

The tight-binding Hamiltonian (site or bond modulation depending on the context) represents the prototypical electronic model: $$H = -\sum_n t_n\, (c_n^\dag c_{n+1} + \text{h.c.}) + \sum_n \epsilon_n\, c_n^\dag c_n$$ with $t_n \in \{ t_A, t_B \}$ or $\epsilon_n \in \{ \epsilon_A, \epsilon_B \}$ in Fibonacci order. For any non-zero quasiperiodic modulation, the energy spectrum is a singular-continuous Cantor set of zero Lebesgue measure (Jagannathan, 2020, Varma et al., 2016). Eigenstates display multifractal scaling with generalized dimensions $D_q$, nontrivial participation ratios, and absence of truly extended or localized states.

The multifractality of both spectrum and wavefunctions is accessed via thermodynamic formalisms, RG analysis, and trace map recursions: $x_{n+1} = 2x_nx_{n-1} - x_{n-2}$ with invariants $$I = x_{n-1}^2 + x_n^2 + x_{n+1}^2 - 2x_{n-1}x_nx_{n+1} - 1$$ derived from transfer matrices (Jagannathan, 2020, Bruin et al., 2013).

Quantum metrics provide sensitive probes for localization hierarchy in the quasicrystal, outperforming the inverse participation ratio (IPR) in resolving spatial delocalization structure. The position-space metric $g^{(x)}$ quantifies symmetry-center separation, while the phason component $g^{(\varphi)}$ encodes sensitivity to quasi-periodic shifts. The sum of both defines a total quantum metric lower-bounded by the gap-label integer $\nu$: $$\Omega_x + \Omega_\varphi \geq |\Delta| / \pi \quad [2506.15575]$$ Scale invariance is preserved in the hierarchy of metrics and mixed Chern numbers.

3. Topological Invariants, Edge States, and Higher-Order Topology

Fibonacci quasicrystals possess dense gaps in their spectrum, each labeled by: $N_{\mathrm{gap}} = p + q\,\varphi^{-1} \mod 1$ where $q$ is the Chern number, a topological invariant inherited from the 2D ancestor (Harper or SSH model) (Levy et al., 2015, Kraus et al., 2012).

Topological boundary phenomena manifest as:

• Edge states traversing gaps under variation of the phason; the number and direction of crossings correspond to $|q|$ and $\mathrm{sign}(q)$ (Baboux et al., 2016).
• Generalized Fabry-Perot cavity modes, with resonance condition $\theta_{\mathrm{cav}}(E,\varphi) = 2\pi m$; winding of $\theta_{\mathrm{cav}}$ in $\varphi$ provides direct measurement of Chern numbers (Levy et al., 2015).
• Higher-order topology: Projections of higher-dimensional SSH models onto Fibonacci chains and squares yield inherited edge and corner states. In two dimensions, the Cartesian product of Fibonacci chains supports protected, zero-dimensional corner modes, even in the absence of well-defined Wannier centers (Ouyang et al., 26 Jan 2024).

Topological equivalence with the Harper model is proven by continuous deformation between the two, with unchanging bulk gap structure and invariants (Kraus et al., 2012). Photonic and polaritonic experiments have demonstrated topological pumping and direct measurement of winding numbers (Verbin et al., 2014, Singh et al., 2015, Baboux et al., 2016).

4. Transport Regimes, Phase Transitions, and Thermodynamic Applications

Fibonacci quasicrystals exhibit metallic conduction at generic fillings, despite the absence of extended states. Kohn's localization tensor $\lambda^2$ diverges at most electron densities, but saturates at special fillings $\rho = 1/\varphi^n$ where the Fermi energy lies in a mini-gap—yielding true band insulating behavior (Varma et al., 2016). The spectrum thus contains a dense hierarchy of metallic and insulating regimes.

Quantum transport and entanglement diagnostics differentiate extended, critical, and localized states:

• Extended: Absence of Lyapunov exponent ($\lambda \to 0$), uniform Wigner distributions in $x$ and peak in $p$.
• Critical: Power-law scaling, sub-extended phase-space distributions, reduced entanglement entropy at the critical transition.
• Metallic-insulator-like transitions are observed as conductance drops sharply; entropy acting as a thermodynamic order parameter (Xu et al., 23 Oct 2024).

Fibonacci quasicrystals function as quantum working mediums in heat cycle engines (Otto-type): phase regime determines operation as engine, heater, accelerator, or refrigerator.

5. Algebraic and Quantum Information Structures: Aperiodic Algebras and Codes

The quasiperiodic point set $F(\Omega)$, indexed by elements in $\mathbb{Z}[\varphi]$, underpins infinite-dimensional aperiodic Lie, Witt, and Jordan algebras. Explicitly:

• Lie algebra: Generators $e_x$ at $x \in F(\Omega)$, with windowed commutator

$[e_x, e_y] = (y-x)\,\chi_\Omega(x^*+y^*)\,e_{x+y}$

• Witt algebra: Integer-indexed modes $L_n$, with aperiodic extension

$$[L_n, L_m] = (n-m)\chi_\Omega(F(n)^*+F(m)^*)L_{n+m}$$

and a Virasoro central extension.

• Jordan algebra: Quasiaddition $x \triangleright y = \varphi^2 x - \varphi y$, Jordan product $$e_x \circ e_y = \frac12(e_{x\triangleright y} + e_{y\triangleright x})$$, flexible yet non-unital, with rich ideal structure (Corradetti et al., 2023).

Finite-dimensional quotients yield coefficients entering integrable models (e.g., KdV-type flows) governed by the algebraic identities.

In quantum information, the Fibonacci Quasicrystal Inflation Code (QIC) encodes the full combinatorial structure via local Hamiltonians forbidding "00" substrings. The dimension of the code subspace matches the fusion space of $N$ Fibonacci anyons, facilitating simulation of anyon braiding using local gates and preserving topological relations (Amaral, 25 Jun 2025). Geometric and combinatorial statistics (e.g., gap distributions, Bragg diffraction at $k = 2\pi(m+n\varphi)$) are matched exactly by the codeword ensemble.

6. Response to Disorder and Perturbations

The Fibonacci chain responds to disorder with pronounced structural sensitivity fixed by the renormalization path of positions:

• Placement of on-site impurities perturbs localization selectively, with the amplitude map symmetry between site and energy-level renormalization paths.
• The transition regime between quasiperiodic order and localization is characterized by path-grouped families in inverse participation ratio, with superposition holding for weak, separated impurities; beyond a disorder threshold, full localization and symmetry breakdown occurs (Moustaj et al., 2020).
• The fractal Fourier signature (complex cardioid) is highly sensitive to sequence order but not to tile-length ratio—a stringent test for quasicrystalline order (Fang et al., 2016).

7. Proximity, Superconductivity, and Experimental Realizations

Superconducting proximity effect in Fibonacci chains yields long-range (power-law) spatial penetration of induced order parameter, with the induced amplitude modulated by topological winding numbers of edge states. Critical wavefunctions enhance the proximity compared to normal metals. The spectral flow under the phason reveals fingerprints of the underlying Chern labels, providing a route for experimental detection via, e.g., STM or tunneling spectroscopy (Rai et al., 2019, Kobiałka et al., 20 May 2024).

Competition between critical QC subgap states and topological superconducting gaps (Majorana bound states) is mutually exclusive: MBS occurs only outside QC gaps; vice versa, critical subgap states vanish inside the topological gap. Short Fibonacci approximants optimize topological gap enhancement and broaden the parameter region for MBS realizability. These effects are experimentally accessible with STM fabrication of atomic chains (Kobiałka et al., 20 May 2024).

Experimental platforms include:

• Photonic waveguide arrays demonstrating topological pumps and edge states (Verbin et al., 2014, Singh et al., 2015).
• Polaritonic microcavities with direct observation and winding-number measurement (Baboux et al., 2016).
• Ultracold atomic Fibonacci optical lattices, offering control of phason dynamics and multifractal spectra (Singh et al., 2015).

Summary Table: Core Structural Features

Dimension	Construction Rule	Spectral Property
1D chain	$A \to AB$, $B \to A$	Cantor set, critical states, multifractality
Cut-and-project	Slope $1/\varphi$, window $\Omega$	Pure-point diffraction, Bragg peaks at $2\pi(m+n\varphi)$
Algebraic	$\mathbb{Z}[\varphi]$, star-map	Aperiodic Lie/Witt/Jordan algebras

Fibonacci quasicrystals thus provide a mathematically rigorous, physically rich paradigm for understanding the emergence, classification, and application of quasiperiodicity, fractal spectra, and topological quantization in low-dimensional systems (Corradetti et al., 2023, Levy et al., 2015, Jagannathan, 2020, Marsal et al., 18 Jun 2025, Ouyang et al., 26 Jan 2024).