---
title: Critical Golden Chain Models
url: https://www.emergentmind.com/topics/critical-golden-chain
type: topic
---

# Critical Golden Chain Models

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 [2511.10672] [2410.16356] [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” [2511.10672], 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 \(0\to01\), \(1\to0\). A minimal forbidden factor (MFF) is a finite word \(M\in\Sigma^+\) that does not occur in \(w_\infty\), while every proper factor of \(M\) does occur.

There is exactly one Fibonacci MFF of length \(F_k\) for each \(k\ge3\), and none at other lengths. Under \(0\to L\), \(1\to S\), the first examples are
\[
M_{F_3}=\texttt{SS},\qquad
M_{F_4}=\texttt{LLL},\qquad
M_{F_5}=\texttt{SLSLS},\qquad
M_{F_6}=\texttt{LLSLLSLL}.
\]
A constructive “boundary-flip” recursion generates \(M_{F_k}\) from \(M_{F_{k-1}}M_{F_{k-2}}\) by flipping three boundary letters.

For a fixed rung \(K\ge3\), let \(F_K=\{M_{F_3},M_{F_4},\dots,M_{F_K}\}\). On an \(N\)-site chain with Pauli \(Z_i\), the literal-pattern projector onto \(M=M_0M_1\dots M_{m-1}\) at sites \(i,\dots,i+m-1\) is
\[
\Pi^{(M)}_{i:i+m-1}=\bigotimes_{j=0}^{m-1}\frac12\left(I+(-1)^{M_j}Z_{i+j}\right).
\]
The rung-\(K\) Hamiltonian is
\[
H_K=\sum_{M\in F_K} J_M \sum_{i=1}^{N-|M|+1}\Pi^{(M)}_{i:i+|M|-1},\qquad J_M>0.
\]
All terms commute and the model is frustration-free. Its zero-energy space is exactly the set of \(\Sigma\)-strings of length \(N\) avoiding every \(M\in F_K\).

If \(D_K(N)\) denotes the number of valid words of length \(N\), then
\[
D_K(N)=\Theta(\lambda_K^N),
\]
where \(\lambda_K\) is the Perron eigenvalue of the Aho–Corasick avoidance automaton for \(F_K\). The sequence decreases monotonically:
\[
\lambda_3=\phi\approx1.618034,\quad
\lambda_4=\rho\approx1.324718,\quad
\lambda_5\approx1.193859,\quad
\lambda_6\approx1.114798,
\]
and tends to \(1\), corresponding to convergence toward the Fibonacci subshift.

## 2. Base rung \(K=3\): the golden chain in the hierarchy

At the base rung, \(F_3=\{\texttt{SS}\}\), or equivalently the forbidden bit pattern \(\texttt{11}\). The Hamiltonian is
\[
H_3=J_3\sum_{i=1}^{N-1}\Pi^{(\texttt{SS})}_{i,i+1},
\]
with
\[
\Pi^{(\texttt{SS})}_{i,i+1}=\frac12(I-Z_i)\otimes\frac12(I-Z_{i+1}).
\]
Thus \(H_3\) penalizes every occurrence of adjacent \(S\)’s [2511.10672].

Its kernel consists of all binary strings of length \(N\) with no consecutive \(1\)’s. If \(N_L\) is the number of admissible length-\(L\) words, then
\[
N_L=N_{L-1}+N_{L-2},\qquad N_0=1,\quad N_1=2.
\]
Hence
\[
N_L=F_{L+2},\qquad N_L\sim \phi^L/\sqrt5,
\]
so the ground-space growth constant is \(\lambda_3=\phi\).

Within this framework, the criticality of \(K=3\) is explicitly not ordinary quantum criticality. The commuting-projector Hamiltonian \(H_3\) has a discrete spectrum with a unit gap to first excitations, set by \(J_3\), 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
\[
h_K=\log\lambda_K,
\]
so \(h_3=\log\phi\), and every non-redundant added Fibonacci MFF lowers the entropy. Second, \(K=3\) 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 \(H_3\) 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 \(\{1,\tau\}\) and fusion rule
\[
\tau\times\tau=1+\tau.
\]
It is called the “golden chain” because the TL parameter equals the quantum dimension of \(\tau\), namely \(\phi\) [2410.16356].

The TL generators \(e_i\) satisfy
\[
e_i^2=de_i,\qquad
e_i e_{i\pm1} e_i=e_i,\qquad
e_i e_j=e_j e_i \quad \text{for } |i-j|>1,
\]
with loop parameter \(d=\phi\) in the Fibonacci model. In the qubit “inflation-code” realization of the forbidden-word hierarchy, one can choose three-site, non-diagonal operators \(e_i\) acting on the 5-dimensional local sector of valid triples with no \(\texttt{SS}\), so that the TL relations hold exactly with \(d=\phi\). The valid triples are
\[
\{\texttt{LLL},\texttt{LLS},\texttt{LSL},\texttt{SLL},\texttt{SLS}\}.
\]

This exact local rank is decisive. Adding \(\texttt{LLL}\) at \(K=4\) removes one triple and shrinks the local sector to rank \(4\), breaking the Jones–Wenzl/TL constraints. For that reason, exact TL braiding compatibility holds only at \(K=3\).

The conventional Fibonacci anyon golden chain has conformal critical points. The antiferromagnetic coupling flows to the tricritical Ising CFT with central charge \(c=7/10\), while a different sign flows to the 3-state Potts CFT with \(c=4/5\) [2511.10672]. The anyonic Hilbert space may be represented by fusion paths of \(\tau\)-anyons, equivalently the Rydberg blockade-constrained spin-\(1/2\) chain in which neighboring down-spins are disallowed [2410.16356]. 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 \(H_3\) 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 \(K=4\) rung, called the “plastic chain,” defined by
\[
F_4=\{\texttt{SS},\texttt{LLL}\},
\]
that is, by forbidding \(\texttt{11}\) and \(\texttt{000}\). Its Hamiltonian is
\[
H_4
=
J_3\sum_{i=1}^{N-1}\Pi^{(\texttt{SS})}_{i,i+1}
+
J_4\sum_{i=1}^{N-2}\Pi^{(\texttt{LLL})}_{i:i+2},
\]
with
\[
\Pi^{(\texttt{LLL})}_{i:i+2}
=
\frac12(I+Z_i)\otimes\frac12(I+Z_{i+1})\otimes\frac12(I+Z_{i+2}).
\]
If \(a_n\) is the number of valid length-\(n\) words, then for \(n\ge5\),
\[
a_n=a_{n-1}+a_{n-2}-a_{n-4},
\]
with
\[
a_1=2,\quad a_2=3,\quad a_3=4,\quad a_4=5.
\]
The characteristic polynomial is
\[
(x-1)(x^3-x-1),
\]
whose Perron root is the plastic constant \(\rho\approx1.324718\), so
\[
a_n=\Theta(\rho^n),\qquad \lambda_4=\rho.
\]
A closed form exists:
\[
a_n=1+C\rho^n+B\sigma^n+C'\bar{\sigma}^n,
\]
where \(\rho\) is real, \(\sigma,\bar{\sigma}\) are the complex roots of \(x^3-x-1\), and the constants are fixed by the initial conditions [2511.10672].

More generally, the hierarchy defines an entropy staircase
\[
h_K=\log\lambda_K,\qquad \lambda_K>\lambda_{K+1}>\dots\to1,
\]
which converges to the zero-entropy aperiodic fixed point, the Fibonacci subshift. The energy scale for each newly introduced forbidden word is proposed as
\[
J_{M_{F_k}}=\alpha\log(\lambda_{k-1}/\lambda_k),\qquad k=4,\dots,K,
\]
with \(J_{M_{F_3}}\) setting the base scale. With \(\alpha=1\), this gives a “unit-gap” normalization between adjacent plateaus. This suggests an explicit renormalization-group flow from the high-entropy phase at \(K=3\) toward the zero-entropy aperiodic fixed point.

The same paper reports small-instance D-Wave annealing snapshots. For \(K=3\), the problem is a trivial quadratic instance with \(100\%\) success and recovery of all \(F_{N+2}\) ground states. For \(K=4\), cubic penalties open a clean unit spectral gap and yield moderate success, for example approximately \(12\%\) at \(N=12\) over \(5000\) reads. For \(K\ge5\), forward annealing becomes fragile because higher-degree penalties make HOBO\(\to\)QUBO reduction and embedding dominate, whereas reverse annealing from near-feasible states robustly recovers \(>99\%\) ground-state success. The same progression clarifies the local-rank obstruction: \(d_3(3)=5\), while \(d_3(K)=4\) for all \(K\ge4\), 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 \(\tau\) with
\[
\tau\otimes\tau=\mathbf1\oplus\tau,
\]
and bonds may prefer either the trivial channel \(1\) or the \(\tau\) channel. Bonds preferring \(1\) are called antiferromagnetic (AFM), while bonds preferring \(\tau\) are called ferromagnetic (FM).

The disordered projector Hamiltonian is
\[
H=\sum_i J_i\left(1-P_i^{\Sigma_i}\right),
\]
where \(J_i>0\) are random couplings and \(\Sigma_i\in\{A,F\}\) specifies whether bond \(i\) favors the AFM or FM channel. In a sign convention useful for local derivations,
\[
H=-J_1P_1^A-J_2P_2^A-J_3P_3^A,
\]
with \(J_i>0\) denoting AFM bonds and \(J_i<0\) denoting FM bonds.

Strong-disorder real-space RG decimates the strongest bond \(J_*\). For AFM decimation, the two anyons fuse to the trivial sector and are removed, generating
\[
J_{\text{eff}}=\frac{2}{\tau^2}\frac{J_LJ_R}{J_*}.
\]
For FM decimation, the pair fuses into an effective \(\tau\) cluster, and neighboring couplings renormalize as
\[
\tilde J_L=-\frac1\tau J_L,\qquad
\tilde J_R=-\frac1\tau J_R.
\]
Thus FM decimation flips neighboring bond signs and reduces their magnitudes by a factor \(1/\tau\).

Using
\[
\beta=\ln\frac{\Omega}{|J|},\qquad
\Gamma=\ln\frac{\Omega_0}{\Omega},
\]
the flow exhibits two infinite-randomness fixed points. The AFM random-singlet fixed point has
\[
P_\Gamma(\beta)=\frac1\Gamma e^{-\beta/\Gamma},\qquad N=0,\qquad \psi=\frac12.
\]
The mixed fixed point has equal AFM and FM distributions,
\[
P_\Gamma(\beta)=N_\Gamma(\beta)=\frac1\Gamma e^{-2\beta/\Gamma},\qquad \psi=\frac13.
\]
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:
\[
\frac{d\delta}{d\Gamma}=+2\delta
\]
about the AFM random-singlet fixed point, and
\[
\frac{d\delta}{d\Gamma}=-5\delta
\]
about the mixed fixed point.

Entanglement scaling provides the effective central charge. In the AFM random-singlet phase,
\[
S(L)\sim \frac{\ln\varphi}{3}\ln L,
\qquad
c_{\rm eff}^{\rm RS}=\ln\varphi\approx0.481.
\]
In the mixed phase,
\[
S(L)\approx \frac{0.702}{3}\ln L+\text{const},
\qquad
c_{\rm eff}^{\rm mixed}\approx0.702.
\]
Because \(c_{\rm eff}\) increases along the RG flow from the random-singlet fixed point to the mixed fixed point, the paper concludes that there is no \(c\)-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 \(H_3\) [2410.16356]. The simple objects are
\[
\mathrm{Obj}(H_3)=\{1,a,a^2,\rho,a\rho,a^2\rho\},\qquad a^3=1,
\]
with nontrivial fusion rules including
\[
\rho\times a^2=a\rho,\qquad
\rho\times\rho=1+\rho+a\rho+a^2\rho.
\]
The constrained fusion-path Hilbert space \(V^L\) is obtained by projecting \((\mathbb C^6)^{\otimes L}\) onto the allowed nearest-neighbor sector; of the \(36\) possible pairs, \(15\) are allowed, and the dimension grows as
\[
d_L\sim \psi^L,\qquad \psi=\frac{3+\sqrt{13}}2.
\]

The first integrable Hamiltonian is a projector onto the identity fusion channel \(P_1\) for pairs of \(\rho\) anyons. Writing \(Q_2=-\sum e_i\), its local generators satisfy
\[
e_i^2=\delta e_i,\qquad
e_i e_{i\pm1}e_i=e_i,\qquad
[e_i,e_j]=0 \quad (|i-j|>1),
\]
with
\[
\delta=\psi=\frac{3+\sqrt{13}}2\approx3.3028.
\]
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 \(P_1\) model is gapless with dynamical critical exponent \(z\neq1\), and its half-chain entanglement does not show the standard \(S(\ell\approx L/2)\sim (c/3)\log L\) scaling. A second integrable Haagerup model breaks the \(Y_\rho\) topological symmetry while retaining \(Y_a\), is gapless with \(\Delta E(L)\sim1/(L-5/2)\), and has
\[
S(\ell=L/2)\approx(c/3)\log L+s_0,
\]
with best fits giving \(c\approx3/2\) for \(L>20\).

These comparisons delimit the scope of the term “critical golden chain.” In the Fibonacci setting, the clean anyonic chain has relativistic CFT criticality with \(z=1\), the forbidden-word \(K=3\) 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.

Source: https://www.emergentmind.com/topics/critical-golden-chain