Critical Golden Chain Models

• Critical Golden Chain is a family of one-dimensional models defined by Fibonacci fusion constraints, forbidden-word Hamiltonians, and Temperley–Lieb algebra structures.
• It utilizes a forbidden-word hierarchy to enforce specific pattern avoidance, yielding a frustration-free Hamiltonian with a computable entropy staircase.
• Applications include modeling conformal critical points, anyonic fusion path constraints, and disordered systems exhibiting infinite-randomness phases.

Searching arXiv for recent and foundational papers on the critical golden chain and related Hamiltonian constructions. arXiv search query: "critical golden chain Fibonacci anyons Temperley-Lieb" The critical golden chain denotes a family of closely related one-dimensional models built from Fibonacci fusion constraints, Temperley–Lieb (TL) algebra, and, in a recent reformulation, forbidden-word Hamiltonians derived from the Fibonacci word. In the forbidden-word hierarchy, the base rung $K=3$ forbids only the pattern $\texttt{SS}$ and defines the “golden chain” as a commuting, frustration-free projector Hamiltonian whose ground states are exactly the binary strings with no consecutive $S$’s. In the conventional anyonic formulation, the golden chain is the canonical Fibonacci anyon chain built from the fusion rule $\tau\times\tau=1+\tau$, with integrable points governed by TL generators at loop parameter $\phi=(1+\sqrt5)/2$. The adjective “critical” therefore has model-dependent meaning: in the forbidden-word hierarchy it refers to an entropic plateau and a unique algebraic TL compatibility, whereas in the non-commuting Fibonacci anyon chain it refers to conformal critical points, and in disordered variants it refers to infinite-randomness phases (Amaral, 9 Nov 2025, Corcoran et al., 2024, 0807.1123).

1. Forbidden-word definition and Hamiltonian construction

In the hierarchy introduced by “A Hierarchy of Fibonacci Forbidden-Word Hamiltonians: From the Golden Chain to the Plastic Chain and Aperiodic Order” (Amaral, 9 Nov 2025), the underlying alphabet is $\Sigma=\{S,L\}$ with the identification $S\equiv1$ and $L\equiv0$. The Fibonacci (Sturmian) word $w_\infty$ is the fixed point of the substitution $0\to01$, $\texttt{SS}$0. A minimal forbidden factor (MFF) is a finite word $\texttt{SS}$1 that does not occur in $\texttt{SS}$2, while every proper factor of $\texttt{SS}$3 does occur.

There is exactly one Fibonacci MFF of length $\texttt{SS}$4 for each $\texttt{SS}$5, and none at other lengths. Under $\texttt{SS}$6, $\texttt{SS}$7, the first examples are

$\texttt{SS}$8

A constructive “boundary-flip” recursion generates $\texttt{SS}$9 from $S$0 by flipping three boundary letters.

For a fixed rung $S$1, let $S$2. On an $S$3-site chain with Pauli $S$4, the literal-pattern projector onto $S$5 at sites $S$6 is

$S$7

The rung-$S$8 Hamiltonian is

$S$9

All terms commute and the model is frustration-free. Its zero-energy space is exactly the set of $\tau\times\tau=1+\tau$0-strings of length $\tau\times\tau=1+\tau$1 avoiding every $\tau\times\tau=1+\tau$2.

If $\tau\times\tau=1+\tau$3 denotes the number of valid words of length $\tau\times\tau=1+\tau$4, then

$\tau\times\tau=1+\tau$5

where $\tau\times\tau=1+\tau$6 is the Perron eigenvalue of the Aho–Corasick avoidance automaton for $\tau\times\tau=1+\tau$7. The sequence decreases monotonically: $\tau\times\tau=1+\tau$8 and tends to $\tau\times\tau=1+\tau$9, corresponding to convergence toward the Fibonacci subshift.

2. Base rung $\phi=(1+\sqrt5)/2$0: the golden chain in the hierarchy

At the base rung, $\phi=(1+\sqrt5)/2$1, or equivalently the forbidden bit pattern $\phi=(1+\sqrt5)/2$2. The Hamiltonian is

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

with

$\phi=(1+\sqrt5)/2$4

Thus $\phi=(1+\sqrt5)/2$5 penalizes every occurrence of adjacent $\phi=(1+\sqrt5)/2$6’s (Amaral, 9 Nov 2025).

Its kernel consists of all binary strings of length $\phi=(1+\sqrt5)/2$7 with no consecutive $\phi=(1+\sqrt5)/2$8’s. If $\phi=(1+\sqrt5)/2$9 is the number of admissible length-$\Sigma=\{S,L\}$0 words, then

$\Sigma=\{S,L\}$1

Hence

$\Sigma=\{S,L\}$2

so the ground-space growth constant is $\Sigma=\{S,L\}$3.

Within this framework, the criticality of $\Sigma=\{S,L\}$4 is explicitly not ordinary quantum criticality. The commuting-projector Hamiltonian $\Sigma=\{S,L\}$5 has a discrete spectrum with a unit gap to first excitations, set by $\Sigma=\{S,L\}$6, and is therefore not quantum critical by itself. The model is instead “critical” in two precise senses. First, it is the first plateau of the entropy staircase

$\Sigma=\{S,L\}$7

so $\Sigma=\{S,L\}$8, and every non-redundant added Fibonacci MFF lowers the entropy. Second, $\Sigma=\{S,L\}$9 is the unique rung whose ground-state language coincides with the Fibonacci anyon fusion-path constraint and admits an exact TL realization. This identifies the forbidden-word golden chain as the algebraic entry point to the conventional anyonic golden chain, even though the commuting projector $S\equiv1$0 is not itself the standard non-commuting critical Hamiltonian.

3. Temperley–Lieb structure and the conventional Fibonacci anyon chain

In the standard anyonic formulation, the golden chain is built from the Fibonacci fusion category with simple objects $S\equiv1$1 and fusion rule

$S\equiv1$2

It is called the “golden chain” because the TL parameter equals the quantum dimension of $S\equiv1$3, namely $S\equiv1$4 (Corcoran et al., 2024).

The TL generators $S\equiv1$5 satisfy

$S\equiv1$6

with loop parameter $S\equiv1$7 in the Fibonacci model. In the qubit “inflation-code” realization of the forbidden-word hierarchy, one can choose three-site, non-diagonal operators $S\equiv1$8 acting on the 5-dimensional local sector of valid triples with no $S\equiv1$9, so that the TL relations hold exactly with $L\equiv0$0. The valid triples are

$L\equiv0$1

This exact local rank is decisive. Adding $L\equiv0$2 at $L\equiv0$3 removes one triple and shrinks the local sector to rank $L\equiv0$4, breaking the Jones–Wenzl/TL constraints. For that reason, exact TL braiding compatibility holds only at $L\equiv0$5.

The conventional Fibonacci anyon golden chain has conformal critical points. The antiferromagnetic coupling flows to the tricritical Ising CFT with central charge $L\equiv0$6, while a different sign flows to the 3-state Potts CFT with $L\equiv0$7 (Amaral, 9 Nov 2025). The anyonic Hilbert space may be represented by fusion paths of $L\equiv0$8-anyons, equivalently the Rydberg blockade-constrained spin-$L\equiv0$9 chain in which neighboring down-spins are disallowed (Corcoran et al., 2024). A common misconception is therefore to identify every “golden chain” Hamiltonian with the conformal anyonic model; the hierarchy paper makes the distinction explicit by separating the gapped commuting projector $w_\infty$0 from the non-commuting TL projector Hamiltonians that realize the known CFTs.

4. Higher rungs, the plastic chain, and the entropy staircase

The first nontrivial extension beyond the golden chain is the $w_\infty$1 rung, called the “plastic chain,” defined by

$w_\infty$2

that is, by forbidding $w_\infty$3 and $w_\infty$4. Its Hamiltonian is

$w_\infty$5

with

$w_\infty$6

If $w_\infty$7 is the number of valid length-$w_\infty$8 words, then for $w_\infty$9,

$0\to01$0

with

$0\to01$1

The characteristic polynomial is

$0\to01$2

whose Perron root is the plastic constant $0\to01$3, so

$0\to01$4

A closed form exists: $0\to01$5 where $0\to01$6 is real, $0\to01$7 are the complex roots of $0\to01$8, and the constants are fixed by the initial conditions (Amaral, 9 Nov 2025).

More generally, the hierarchy defines an entropy staircase

$0\to01$9

which converges to the zero-entropy aperiodic fixed point, the Fibonacci subshift. The energy scale for each newly introduced forbidden word is proposed as

$\texttt{SS}$00

with $\texttt{SS}$01 setting the base scale. With $\texttt{SS}$02, this gives a “unit-gap” normalization between adjacent plateaus. This suggests an explicit renormalization-group flow from the high-entropy phase at $\texttt{SS}$03 toward the zero-entropy aperiodic fixed point.

The same paper reports small-instance D-Wave annealing snapshots. For $\texttt{SS}$04, the problem is a trivial quadratic instance with $\texttt{SS}$05 success and recovery of all $\texttt{SS}$06 ground states. For $\texttt{SS}$07, cubic penalties open a clean unit spectral gap and yield moderate success, for example approximately $\texttt{SS}$08 at $\texttt{SS}$09 over $\texttt{SS}$10 reads. For $\texttt{SS}$11, forward annealing becomes fragile because higher-degree penalties make HOBO$\texttt{SS}$12QUBO reduction and embedding dominate, whereas reverse annealing from near-feasible states robustly recovers $\texttt{SS}$13 ground-state success. The same progression clarifies the local-rank obstruction: $\texttt{SS}$14, while $\texttt{SS}$15 for all $\texttt{SS}$16, so higher rungs define constrained aperiodic Hamiltonian codes rather than TL representations.

5. Disorder and infinite-randomness criticality

A distinct use of the term “critical golden chain” arises in the disordered nearest-neighbor Fibonacci anyon chain studied in “Infinite Randomness Phases and Entanglement Entropy of the Disordered Golden Chain” (0807.1123). The degrees of freedom are Fibonacci anyons $\texttt{SS}$17 with

$\texttt{SS}$18

and bonds may prefer either the trivial channel $\texttt{SS}$19 or the $\texttt{SS}$20 channel. Bonds preferring $\texttt{SS}$21 are called antiferromagnetic (AFM), while bonds preferring $\texttt{SS}$22 are called ferromagnetic (FM).

The disordered projector Hamiltonian is

$\texttt{SS}$23

where $\texttt{SS}$24 are random couplings and $\texttt{SS}$25 specifies whether bond $\texttt{SS}$26 favors the AFM or FM channel. In a sign convention useful for local derivations,

$\texttt{SS}$27

with $\texttt{SS}$28 denoting AFM bonds and $\texttt{SS}$29 denoting FM bonds.

Strong-disorder real-space RG decimates the strongest bond $\texttt{SS}$30. For AFM decimation, the two anyons fuse to the trivial sector and are removed, generating

$\texttt{SS}$31

For FM decimation, the pair fuses into an effective $\texttt{SS}$32 cluster, and neighboring couplings renormalize as

$\texttt{SS}$33

Thus FM decimation flips neighboring bond signs and reduces their magnitudes by a factor $\texttt{SS}$34.

Using

$\texttt{SS}$35

the flow exhibits two infinite-randomness fixed points. The AFM random-singlet fixed point has

$\texttt{SS}$36

The mixed fixed point has equal AFM and FM distributions,

$\texttt{SS}$37

The AFM random-singlet phase occurs only when all microscopic bonds are AFM. Any finite density of FM bonds drives the system to the mixed fixed point. Linear stability makes the distinction explicit: $\texttt{SS}$38 about the AFM random-singlet fixed point, and

$\texttt{SS}$39

about the mixed fixed point.

Entanglement scaling provides the effective central charge. In the AFM random-singlet phase,

$\texttt{SS}$40

In the mixed phase,

$\texttt{SS}$41

Because $\texttt{SS}$42 increases along the RG flow from the random-singlet fixed point to the mixed fixed point, the paper concludes that there is no $\texttt{SS}$43-theorem for the effective central charge at these infinite-randomness fixed points.

6. Haagerup analogue and the scope of “golden-chain” criticality

The paper “Integrable and critical Haagerup spin chains” constructs a Haagerup analogue of the golden chain by replacing Fibonacci data with the Haagerup fusion category $\texttt{SS}$44 (Corcoran et al., 2024). The simple objects are

$\texttt{SS}$45

with nontrivial fusion rules including

$\texttt{SS}$46

The constrained fusion-path Hilbert space $\texttt{SS}$47 is obtained by projecting $\texttt{SS}$48 onto the allowed nearest-neighbor sector; of the $\texttt{SS}$49 possible pairs, $\texttt{SS}$50 are allowed, and the dimension grows as

$\texttt{SS}$51

The first integrable Hamiltonian is a projector onto the identity fusion channel $\texttt{SS}$52 for pairs of $\texttt{SS}$53 anyons. Writing $\texttt{SS}$54, its local generators satisfy

$\texttt{SS}$55

with

$\texttt{SS}$56

This is directly parallel to the Fibonacci golden chain at the level of projector construction and TL algebra, but the continuum behavior is different. Numerical evidence indicates that this Haagerup $\texttt{SS}$57 model is gapless with dynamical critical exponent $\texttt{SS}$58, and its half-chain entanglement does not show the standard $\texttt{SS}$59 scaling. A second integrable Haagerup model breaks the $\texttt{SS}$60 topological symmetry while retaining $\texttt{SS}$61, is gapless with $\texttt{SS}$62, and has

$\texttt{SS}$63

with best fits giving $\texttt{SS}$64 for $\texttt{SS}$65.

These comparisons delimit the scope of the term “critical golden chain.” In the Fibonacci setting, the clean anyonic chain has relativistic CFT criticality with $\texttt{SS}$66, the forbidden-word $\texttt{SS}$67 rung is critical only in entropic and algebraic senses, and the disordered chain realizes infinite-randomness criticality. The Haagerup chain is a genuine golden-chain analogue in construction and TL structure, but not in infrared universality. A plausible implication is that “critical golden chain” is best understood as a family resemblance across constrained Hilbert spaces, projector Hamiltonians, and TL algebra, rather than as a single universal critical theory.