---
title: Fibonacci Duality Defect in Critical Lattice Models
url: https://www.emergentmind.com/topics/fibonacci-duality-defect
type: topic
---

# Fibonacci Duality Defect in Critical Lattice Models

Searching arXiv for the cited paper and closely related background on the critical golden chain and Fibonacci/non-invertible defects.
The Fibonacci duality defect is a non-invertible defect associated with the Fibonacci fusion category, characterized by the fusion rule $\tau \otimes \tau = 1 \oplus \tau$ and quantum dimension $d_\tau=\phi=(1+\sqrt{5})/2$. In the critical golden chain, it is realized by a defect operator $Y$ defined by inserting a $\tau$-loop around a periodic chain, and it exhibits an exact finite-size entanglement fingerprint: for the even-length antiferromagnetic ground state, the cut-charge probabilities satisfy $P_\tau/P_1=\phi^2$ and $\log g=\log\phi$ without finite-size extrapolation [2607.01151]. In a distinct but related setting, Fibonacci defects also occur as topological defect lines in the Conway module $V^{f\natural}$ and in particular K3 non-linear sigma models, where they are realized by icosian lattice endomorphisms and have defect-twined elliptic genera matching across the two frameworks [2512.19640].

## 1. Categorical data and defining relations

The relevant fusion category has two simple objects, $1$ and $\tau$, with
$$
\tau\otimes\tau = 1 \oplus \tau,
$$
together with
$$
1\otimes\tau \cong \tau\otimes1 \cong \tau,\qquad 1\otimes1 \cong 1.
$$
Its quantum dimensions are
$$
d_1=1,\qquad d_\tau=\phi\equiv \frac{1+\sqrt{5}}{2},
$$
and the total squared dimension is
$$
D^2=d_1^2+d_\tau^2=1+\phi^2.
$$
Equivalently, $d_\tau$ satisfies
$$
d_\tau^2=d_\tau+1,\qquad \phi^2=\phi+1
$$
[2607.01151].

In the lattice construction relevant for the critical golden chain, the Fibonacci duality defect operator $Y$ is defined by inserting a $\tau$-loop around the entire periodic chain. Its defining algebraic relation is
$$
Y^2=I+Y,
$$
so its spectrum consists of the two eigenvalues
$$
\lambda_+=\phi,\qquad \lambda_-=-\frac{1}{\phi}.
$$
If $E_+$ and $E_-$ denote the orthogonal projectors onto the corresponding eigenspaces, then
$$
E_+ + E_- = I,\qquad
Y=\phi E_+-\frac{1}{\phi}E_-.
$$
This relation is the operator-level expression of the non-invertible character of the defect: $Y$ does not square to the identity, but instead closes on a sum of simple sectors [2607.01151].

A useful summary of the basic structures is:

| Object | Definition | Key relation |
|---|---|---|
| $\tau$ | Nontrivial simple object | $\tau\otimes\tau=1\oplus\tau$ |
| $d_\tau$ | Quantum dimension of $\tau$ | $d_\tau=\phi$ |
| $Y$ | $\tau$-loop defect operator | $Y^2=I+Y$ |
| $E_\pm$ | Projectors onto $Y$ eigenspaces | $Y=\phi E_+-(1/\phi)E_-$ |

The same category appears in a broader topological-defect-line context. In the Conway-module construction, Fibonacci defects $W^A$ and $W^B$ generate a Fibonacci fusion subcategory $\mathrm{Fib}\subset \mathrm{Top}$, and their realization is tied to icosian lattice endomorphisms on the Leech lattice [2512.19640].

## 2. Realization in the critical golden chain

The lattice realization is the periodic golden chain in the usual $A_4$ RSOS/fusion-path basis. On each bond $i$ of a periodic chain of length $L$ one carries an anyon label, and the Hamiltonian is the antiferromagnetic projector onto the vacuum fusion channel of nearest-neighbor $\tau$ anyons:
$$
H=-\sum_{i=1}^L P_i,\qquad P_i=\frac{e_i}{\phi},
$$
where the Temperley--Lieb generator $e_i$ projects the two $\tau$ anyons at sites $i,i+1$ to the vacuum and is normalized by
$$
e_i^2=\phi e_i.
$$
This is the critical golden chain of Fibonacci anyons [2607.01151].

A single bond $\ell$ may be chosen as a cut of the periodic chain. Across that cut, the total anyonic charge is either $1$ or $\tau$. The corresponding cut-charge projectors
$$
\Pi_a:\mathcal H\to\mathcal H,\qquad a\in\{1,\tau\},
$$
enforce the value of the fusion-path label crossing bond $\ell$, and satisfy
$$
\Pi_1+\Pi_\tau=I.
$$
These projectors are the lattice observables that enter the finite-size entanglement statement [2607.01151].

The lattice theorem does not proceed through scaling spectra or infrared CFT data. Instead, it identifies an exact categorical fingerprint directly at finite lattice size. This is significant because non-invertible defects are usually diagnosed through scaling spectra or infrared CFT data, whereas here the defect is detected by exact operator identities on the finite-dimensional fusion-path Hilbert space [2607.01151].

## 3. Exact cut-charge theorem and entanglement fingerprint

The central finite-size operator identity is
$$
E_+\Pi_a E_+ = \frac{d_a^2}{D^2}E_+,\qquad a\in\{1,\tau\}.
$$
Using $d_1=1$ and $d_\tau=\phi$, this becomes
$$
E_+\Pi_1 E_+ = \frac{1}{D^2}E_+,\qquad
E_+\Pi_\tau E_+ = \frac{\phi^2}{D^2}E_+.
$$
The proof is described as purely based on fusion-path combinatorics and cyclic $F$-symbol traces [2607.01151].

For the normalized even-length antiferromagnetic ground state $|\Omega\rangle$, the probability, or Schmidt weight, of finding charge $a$ across the cut is defined by
$$
P_a\equiv \langle\Omega|\Pi_a|\Omega\rangle.
$$
Once the ground state is shown to lie in $\mathrm{Ran}\,E_+$, the projector identity gives
$$
P_a=\frac{d_a^2}{D^2}.
$$
Therefore
$$
\frac{P_\tau}{P_1}=\phi^2.
$$
If the defect boundary entropy $g$ is defined by
$$
g^2=\frac{P_\tau}{P_1},
$$
then
$$
g=\phi,\qquad \ln g=\ln\phi.
$$
These equalities hold in exact finite size, with no extrapolation in $L$ [2607.01151].

The result is a sharp lattice-level boundary entropy for a non-Abelian duality defect. The terminology “categorical fingerprint” refers to the fact that the weights are fixed directly by the categorical dimensions $d_a$, not by a scaling-limit fit or by approximate finite-size numerics. A plausible implication is that, in this setting, the entanglement cut resolves the defect already at the category-theoretic layer.

## 4. Sector selection, positivity, and even-length ground states

The finite-size theorem requires identifying the $Y$-sector containing the physical ground state. In the fusion-path $(A_4$ RSOS$)$ basis, all off-diagonal matrix elements of $H$ are non-positive, and the action of $\{e_i\}$ is transitive on admissible paths for even $L$. By the Perron--Frobenius theorem, the unique ground-state wavefunction $|\Omega\rangle$ can therefore be chosen with strictly positive components in that basis [2607.01151].

Because
$$
[H,Y]=0,
$$
the ground state lies entirely in either the $\lambda_+=\phi$ sector or the $\lambda_-=-1/\phi$ sector. The sector selection is then fixed by an explicit sign argument on the matrix elements $\langle \text{path}|E_+|\text{all-}\tau\rangle$: $E_+|\text{all-}\tau\rangle$ has all nonnegative components and a nonzero overlap with $|\text{all-}\tau\rangle$. Since $|\text{all-}\tau\rangle$ also has positive overlap with $|\Omega\rangle$, one concludes that
$$
\langle\Omega|E_+|\Omega\rangle\neq 0,
$$
hence
$$
E_+|\Omega\rangle=|\Omega\rangle,\qquad Y|\Omega\rangle=\phi|\Omega\rangle.
$$
This is the step that connects the abstract defect eigenspace decomposition to the physical ground state of the even-length chain [2607.01151].

The theorem is explicitly stated for the even-length antiferromagnetic ground state. The emphasis on even $L$ is therefore structural rather than incidental: the transitivity and positivity input used in the Perron--Frobenius argument is formulated for even length. A common misconception is to conflate the exact cut-charge statement with a generic property of all finite chains; the theorem, as stated, is specific to the even-length antiferromagnetic ground state.

## 5. Relation to affine-TL packets and tricritical-Ising resolution

The continuum tricritical Ising CFT $M(5,4)$ with $c=7/10$ has six Virasoro primaries. In the standard scaling-limit RSOS/affine-TL approach, one groups finite-$L$ TL-modules into “$A_4$ packets” whose characters branch into six towers:
$$
\chi^{\mathrm{affTL}}_{r,n}(q)\to \sum_{(r,n)} \chi^{\mathrm{Vir}}_{r,n}(q).
$$
However, none of that Virasoro branching data enters the finite-size proof of the cut-projector identity or the Perron--Frobenius sector argument [2607.01151].

The conceptual hierarchy is presented as
$$
\{1,\tau\}\ \text{cut charges}\ \Rightarrow\ \text{affine-TL packets}\ \Rightarrow\ \text{six CFT primaries}.
$$
The exact theorem lives entirely in the first layer. Its proof uses:
- the local Fibonacci fusion rule $[(Y)_{11}=0]$,
- the cyclic $F$-symbol trace $\mathrm{Tr}\,Y=(-1/\phi)^L$,
- a block-graph rank identity giving $\mathrm{Ran}\,E_+$ as the graph of an isometry,
- Perron--Frobenius positivity selecting the $E_+$ sector for even $L$ [2607.01151].

This separation is important because the six-primary tricritical-Ising resolution is finer than the two-charge categorical resolution. The exact result $P_\tau/P_1=\phi^2$ is not derived from six-primary projectors, Virasoro characters, or scaling-limit assumptions. It is a purely finite-lattice categorical result, sharp and non-perturbative. This directly addresses a potential confusion in the literature: the two-charge fingerprint and the six-primary continuum decomposition are compatible, but they are not logically equivalent.

## 6. Conway-module and K3 realizations

A distinct realization of Fibonacci duality defects occurs in the Conway module $V^{f\natural}$, where topological defect lines preserving the relevant supersymmetry are described through Leech-lattice endomorphisms. The candidate Fibonacci defect $W\equiv W^A$ is defined by a lattice endomorphism
$$
\rho(W):\Lambda\to\Lambda
$$
constructed using the “icosian” realization of $\Lambda$. Writing
$$
\Lambda\otimes\mathbb R\approx (\mathbb H\oplus \iota\mathbb H)^3,\qquad \iota^2=+1,
$$
and setting
$$
\theta=\iota(1-\sqrt{5})/2\in \mathbb H\oplus \iota\mathbb H,
$$
the map
$$
\rho(W):(x_1,x_2,x_3)\mapsto (x_1+\theta x_1,\;x_2+\theta x_2,\;x_3+\theta x_3)
$$
satisfies
$$
\rho(W)^2=1+\rho(W),\qquad \rho(W)^t=\rho(W).
$$
On $\Lambda\otimes\mathbb R$ it splits into two $12$-planes,
$$
\Lambda\otimes\mathbb R = V^A\oplus V^B,
$$
with
$$
\rho(W)|_{V^A}=\phi\cdot \mathrm{id},\qquad
\rho(W)|_{V^B}=(1-\phi)\cdot \mathrm{id}.
$$
These are the $\tau$-twist and $\tau$-cotwist eigenspaces. The subgroup commuting with $\rho(W)$ is the icosian subgroup
$$
2\cdot (A_5\times J_2)\subset Co_0\cong \mathrm{Aut}_\tau(V^{f\natural}),
$$
and conjugation by an involution in the normalizer $2\cdot(A_5\times J_2):2$ exchanges $V^A$ and $V^B$, yielding a second Fibonacci defect $W^B$ [2512.19640].

The associated braided data are explicitly specified. The unique non-trivial $F$-symbol and the $R$-symbols are
$$
F^{\tau\tau\tau}_\tau=
\begin{pmatrix}
\phi^{-1} & \phi^{-1/2}\\[6pt]
\phi^{-1/2} & -\phi^{-1}
\end{pmatrix},
\qquad
R^{\tau\tau}_1=e^{-4\pi i/5},\qquad
R^{\tau\tau}_\tau=e^{3\pi i/5},
$$
and they satisfy the pentagon and hexagon equations of a unitary braided category [2512.19640].

The Conway-module realization also supplies defect-twined elliptic genera. The $W$-twined graded trace is
$$
\phi^W(V^{f\natural})(\tau,z)
=
\mathrm{Tr}_{V^{f\natural}_{tw}}
W(-1)^F q^{L_0-\frac12} y^{J_0^3},\qquad y=e^{2\pi i z},
$$
with
$$
\phi^W(V^{f\natural})(\tau,z)
=
\frac12\,\phi(V^{f\natural})(\tau,z)
+\sqrt{5}\,\theta_1(\tau,z)^2\,\frac{f(\tau)}{\eta(\tau)^6},
$$
where
$$
\phi(V^{f\natural})(\tau,z)=\frac{2\,\theta_1(\tau,z)^2}{\eta(\tau)^6},
\qquad
f(\tau)= \frac{\eta(\tau)^5}{\eta(5\tau)} + 25\,\frac{\eta(5\tau)^5}{\eta(\tau)}
$$
is the unique weight-$2$ modular form for $\Gamma_0(5)$. Its $q$-expansion begins
$$
\phi^W=\sqrt{5}\,(2y^{-1}-8+2y)+O(q).
$$
In two K3 Gepner models of type $(3)^2(8)$ and $(2)(3)(18)$, one constructs explicit topological defect lines $W$ preserving $N=(4,4)$ and spectral flow, and their elliptic genera are computed in closed form and exactly reproduce the Conway-module expression [2512.19640].

Within this framework, Fibonacci defects exist only in subcategories $\mathrm{Top}_\Pi$ where $\Pi=V^A$ or $V^B$ is a $12$-plane in $\Lambda\otimes\mathbb R$ with icosian stabilizer. The paper further conjectures that every four-plane $\Pi\subset \Lambda\otimes\mathbb R$ stabilized by a subgroup isomorphic to $2\cdot(A_5\times J_2)$ realizes a K3 NLSM carrying a Fibonacci defect with coincident $\phi^W$; this is explicitly presented as a conjecture rather than a theorem [2512.19640].

Source: https://www.emergentmind.com/topics/fibonacci-duality-defect