Golden Chain: Fibonacci Anyons & Beyond

• Golden Chain is a polysemous term primarily denoting the one-dimensional Fibonacci anyon chain where state growth follows the golden ratio.
• The model employs nearest-neighbor Hamiltonians with AFM and FM couplings resulting in distinct infinite-randomness phases and quantifiable effective central charges.
• It also extends to forbidden-word Hamiltonians and geometric constructions, connecting Fibonacci recursion to golden-angle and golden-ratio universality in physics.

Golden Chain is a polysemous technical expression whose principal meaning in contemporary mathematical physics is the one-dimensional chain of interacting Fibonacci anyons, so called because its state-count growth is controlled by the golden ratio. In newer Hamiltonian-code work, the same name is used for the $K=3$ base rung of a Fibonacci forbidden-word hierarchy that forbids only the local word $SS$. Geometric literature also supports a looser chain interpretation through polygonal constructions whose segment ratios converge to golden-ratio geometry, while adjacent work on the golden angle provides a useful contrast between chain-like golden constructions and a nonconstructible golden division of the circle (0807.1123, Amaral, 9 Nov 2025, Jacak, 2012, Freitas, 2021).

1. Fibonacci-anyon meaning

In the anyonic literature, the golden chain is the one-dimensional chain of interacting Fibonacci anyons $\tau$. Its defining fusion rule is

$\tau\otimes\tau=\mathbf 1\oplus\tau,$

so two neighboring anyons may fuse either to the trivial topological charge $\mathbf 1$ or back to $\tau$. The chain is called “golden” because the Hilbert-space dimension obeys the Fibonacci recursion

$D_N=D_{N-1}+D_{N-2},$

hence $D_N\sim \tau^N$ with $\tau=(1+\sqrt5)/2$. This Hilbert space is not a tensor product of local on-site Hilbert spaces; a convenient basis labels the links between neighboring anyons by $\mathbf 1$ or $SS$0, subject to the rule that two consecutive $SS$1’s are forbidden (0807.1123).

The nearest-neighbor Hamiltonian is written in projector form,

$SS$2

with $SS$3 random in the disordered setting and $SS$4 specifying whether the bond projects onto the AFM or FM fusion channel. In the sign convention used for the four-site problem,

$SS$5

where $SS$6 is AFM-like and favors fusion to $SS$7, while $SS$8 is FM-like and favors fusion to $SS$9. The graphical calculus is controlled by the Fibonacci $\tau$0-matrix

$\tau$1

together with the no-tadpole rule and the evaluation of a disconnected loop to $\tau$2 (0807.1123).

2. Disorder, strong-disorder RG, and infinite-randomness phases

For the disordered golden chain, the strong-disorder Ma-Dasgupta RG exploits the closure of the Fibonacci fusion algebra. If the strongest bond is AFM, two neighboring anyons are forced into the trivial channel $\tau$3, and second-order perturbation theory generates an effective bond

$\tau$4

In logarithmic variables $\tau$5, this becomes asymptotically additive. If the strongest bond is FM, the pair fuses into an effective $\tau$6 cluster and the neighboring couplings are renormalized as

$\tau$7

so FM decimation both removes one site and flips adjacent bond signs (0807.1123).

These decimation rules yield two infinite-randomness phases. The pure AFM case flows to an anyonic random-singlet phase with

$\tau$8

and effective central charge

$\tau$9

Whenever a finite density of FM bonds is present, the AFM random-singlet fixed point is unstable and the system flows to a mixed FM/AFM infinite-randomness phase with

$\tau\otimes\tau=\mathbf 1\oplus\tau,$0

In that mixed phase, the density of undecimated sites scales as $\tau\otimes\tau=\mathbf 1\oplus\tau,$1, hence $\tau\otimes\tau=\mathbf 1\oplus\tau,$2 and $\tau\otimes\tau=\mathbf 1\oplus\tau,$3 (0807.1123).

The mixed phase is not a random-singlet phase. FM decimations generate effective $\tau\otimes\tau=\mathbf 1\oplus\tau,$4-clusters and tree-like trivalent structures, while AFM decimations intermittently terminate branches. Linearized RG gives

$\tau\otimes\tau=\mathbf 1\oplus\tau,$5

near the AFM random-singlet point and

$\tau\otimes\tau=\mathbf 1\oplus\tau,$6

near the mixed point, so the former is unstable and the latter stable. The entanglement entropy of the mixed phase scales as

$\tau\otimes\tau=\mathbf 1\oplus\tau,$7

corresponding to

$\tau\otimes\tau=\mathbf 1\oplus\tau,$8

Because $\tau\otimes\tau=\mathbf 1\oplus\tau,$9, the disorder-driven flow raises the effective central charge, which rules out a $\mathbf 1$0-theorem for this effective central charge in the disordered anyonic setting (0807.1123).

3. Forbidden-word Hamiltonians and the $\mathbf 1$1 base rung

A later formulation embeds the golden chain into an infinite hierarchy of one-dimensional, frustration-free Hamiltonians built from minimal forbidden factors of the Fibonacci word. In this construction, the alphabet is mapped as

$\mathbf 1$2

with $\mathbf 1$3 interpreted as the trivial/vacuum label and $\mathbf 1$4 as the Fibonacci anyon label. The base rung is $\mathbf 1$5, for which the only forbidden factor is

$\mathbf 1$6

Thus the golden chain is exactly the $\mathbf 1$7 case forbidding adjacent $\mathbf 1$8’s, equivalently $\mathbf 1$9 in binary notation (Amaral, 9 Nov 2025).

The hierarchy is defined by

$\tau$0

with local projectors

$\tau$1

For the golden chain,

$\tau$2

with open boundaries. Its zero-energy subspace is precisely the language of length-$\tau$3 words over $\tau$4 with no adjacent $\tau$5’s: $\tau$6 The ground-state count is

$\tau$7

with $\tau$8 and $\tau$9, so the asymptotic growth constant is

$D_N=D_{N-1}+D_{N-2},$0

Equivalently, the avoidance automaton has adjacency matrix

$D_N=D_{N-1}+D_{N-2},$1

whose Perron eigenvalue is $D_N=D_{N-1}+D_{N-2},$2 (Amaral, 9 Nov 2025).

This formulation recovers the conventional anyonic golden chain at the base rung and then extends beyond it. The next rung, the Plastic chain at $D_N=D_{N-1}+D_{N-2},$3, adds the constraint $D_N=D_{N-1}+D_{N-2},$4, and the growth constants decrease monotonically: $D_N=D_{N-1}+D_{N-2},$5 flowing toward $D_N=D_{N-1}+D_{N-2},$6. The paper interprets this as an entropy staircase and an RG-like flow from the highest-entropy constrained phase to the zero-entropy Fibonacci subshift. Exact Temperley-Lieb braiding compatibility survives only at $D_N=D_{N-1}+D_{N-2},$7: the local valid triples

$D_N=D_{N-1}+D_{N-2},$8

form a $D_N=D_{N-1}+D_{N-2},$9-dimensional sector at the base rung, whereas adding $D_N\sim \tau^N$0 removes $D_N\sim \tau^N$1 and leaves a $D_N\sim \tau^N$2-dimensional sector, destroying the exact three-site Kauffman-Lomonaco Temperley-Lieb structure. This obstruction is summarized by

$D_N\sim \tau^N$3

For small quantum-annealing instances, the $D_N\sim \tau^N$4 case at $D_N\sim \tau^N$5 was reported as trivial for the annealer: quadratic, $D_N\sim \tau^N$6 success, and recovery of all $D_N\sim \tau^N$7 unique theoretical ground states (Amaral, 9 Nov 2025).

4. Geometric chain constructions and the golden angle

In geometry, the phrase “golden chain” is not standardized, but one explicit chain-like construction is Dorota Jacak’s polygonal chain generated by the recurrence

$D_N\sim \tau^N$8

for $D_N\sim \tau^N$9. The paper proves

$\tau=(1+\sqrt5)/2$0

with $\tau=(1+\sqrt5)/2$1, by expressing the iterates through Fibonacci numbers. Applied to a counterclockwise polygonal chain with consecutive orthogonal segments satisfying

$\tau=(1+\sqrt5)/2$2

this yields

$\tau=(1+\sqrt5)/2$3

The local triangles approach Kepler triangles, and when $\tau=(1+\sqrt5)/2$4 the paper explicitly names the construction a “golden polygonal chain” (Jacak, 2012).

A distinct but closely related golden object is the golden angle. If a full circle of angle $\tau=(1+\sqrt5)/2$5 is divided in the golden ratio, the smaller angle is

$\tau=(1+\sqrt5)/2$6

with $\tau=(1+\sqrt5)/2$7. Pedro J. Freitas proves that this angle is not constructible by straightedge and compass. The proof does not proceed through an algebraic-degree test for $\tau=(1+\sqrt5)/2$8; it shows instead that $\tau=(1+\sqrt5)/2$9 and $\mathbf 1$0 are transcendental. Writing

$\mathbf 1$1

one applies the Gelfond-Schneider theorem: if $\mathbf 1$2 were algebraic, then $\mathbf 1$3 would be transcendental, but

$\mathbf 1$4

Hence $\mathbf 1$5 is transcendental, and so are $\mathbf 1$6, therefore also $\mathbf 1$7. A notable contrast follows: $\mathbf 1$8 itself is constructible, but the golden angle is not. The paper nevertheless notes an approximate pentagram-based construction due to Almada Negreiros giving $\mathbf 1$9, differing from the exact value by about $SS$00 (Freitas, 2021).

5. Golden-ratio universality in one-dimensional driven chains

In nonlinear fluctuating hydrodynamics and mode-coupling theory, “golden” also designates a dynamical universality class for one-dimensional systems with two conservation laws. For a conserved mode with scaling form

$SS$01

the possible exponents in the Fibonacci family are Kepler/Fibonacci ratios. In the two-mode case, the golden class occurs precisely when both self-couplings vanish but each mode is nonlinearly driven by the other: $SS$02 Then

$SS$03

and the scaling functions are $SS$04-stable Lévy laws (Popkov et al., 2023).

A central contribution of this analysis is that the allowed universality classes can be read off in closed form from the stationary currents $SS$05, their Jacobian $SS$06, and Hessians $SS$07. The mode-coupling matrices

$SS$08

depend on these current derivatives and on the normal-mode transformation $SS$09. An Onsager-type current symmetry,

$SS$10

with $SS$11 the compressibility matrix, rules out broad classes of microscopic models before stationarity is even proved. At equal mean densities, the paper obtains a sharp dichotomy. If the currents are antisymmetric under interchange of the conserved densities, golden modes can occur only when the two conserved quantities are correlated. If the currents are symmetric, one mode is always diffusive and the other may be KPZ, modified KPZ, $SS$12-Lévy, or also diffusive, but not golden (Popkov et al., 2023).

The same work analyzes a noisy chain of harmonic oscillators as an exactly solvable benchmark. That chain conserves total energy and total stretch-like field $SS$13, has mode velocities

$SS$14

and mode-coupling matrices

$SS$15

It therefore realizes a diffusive mode together with a maximally asymmetric $SS$16-stable Lévy mode rather than the golden class. The significance of the example is methodological: for this noisy harmonic chain, the predictions of mode-coupling theory are exact (Popkov et al., 2023).

6. Broader and implicit uses

Several neighboring literatures employ “golden” and “chain” in ways that do not define a standard object called Golden Chain but nevertheless support a broader interpretive field. In the musical-icosahedron work, the phrase “Golden Chain” does not appear. The paper instead studies the chromatic scale and the Pythagorean chain as ordered tone sequences embedded on regular icosahedra, while golden triangles and golden gnomons encode major and minor triads. It also introduces “Golden Major Minor Self-Duality.” This suggests a musico-geometric extension in which a “golden chain” would mean a chain of tones or triads organized by golden-ratio figures on the icosahedron rather than a named object of the theory (Imai et al., 2021).

The paper on the Golden quantum oscillator likewise does not present a chain model, but it develops a Fibonacci-deformed algebraic framework that is adjacent to golden-chain thinking. With the Golden bases

$SS$17

the oscillator spectrum is

$SS$18

the level spacings satisfy

$SS$19

and the ratio of successive energies tends to the golden ratio: $SS$20 This suggests an indirect algebraic toolkit for Fibonacci-graded excitations rather than a spatial chain Hamiltonian (Pashaev et al., 2011).

A different adjacent usage appears in Golden Riemannian geometry. There too the literal phrase “Golden Chain” is absent; the paper studies manifolds endowed with a Golden structure

$SS$21

and characterizes slant submanifolds by

$SS$22

A plausible implication is that “golden chain” in this setting would refer not to a named object but to the sequence of induced identities linking the ambient Golden structure, tangent-normal decompositions, and slant-submanifold characterizations (Bahadır et al., 2018).

Across these uses, a common invariant is not a single ontology but a recurrent mathematical signature: Fibonacci recursion, Perron growth $SS$23, golden-ratio scaling, or golden-polynomial structure. The most precise technical meaning remains the Fibonacci-anyon chain and its forbidden-word $SS$24 reformulation, but the expression also participates in a wider family of chain-like constructions whose organizing principle is the golden ratio.