Fibonacci Duality Defect in Critical Lattice Models

• Fibonacci Duality Defect is a non-invertible operator defined in the Fibonacci fusion category, characterized by the fusion rule τ ⊗ τ = 1 ⊕ τ and quantum dimension φ.
• In critical golden chains, the defect operator Y, introduced via a τ-loop, produces an exact finite-size entanglement fingerprint with Pτ/P1 = φ² and ln g = ln φ.
• Fibonacci defects also manifest as topological defect lines in Conway modules and K3 sigma models, linking lattice constructions with modular elliptic genera.

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 (Liang, 1 Jul 2026). 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 (Angius et al., 22 Dec 2025).

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

$d_\tau=\phi=(1+\sqrt{5})/2$0

Its quantum dimensions are

$d_\tau=\phi=(1+\sqrt{5})/2$1

and the total squared dimension is

$d_\tau=\phi=(1+\sqrt{5})/2$2

Equivalently, $d_\tau=\phi=(1+\sqrt{5})/2$3 satisfies

$d_\tau=\phi=(1+\sqrt{5})/2$4

(Liang, 1 Jul 2026).

In the lattice construction relevant for the critical golden chain, the Fibonacci duality defect operator $d_\tau=\phi=(1+\sqrt{5})/2$5 is defined by inserting a $d_\tau=\phi=(1+\sqrt{5})/2$6-loop around the entire periodic chain. Its defining algebraic relation is

$d_\tau=\phi=(1+\sqrt{5})/2$7

so its spectrum consists of the two eigenvalues

$d_\tau=\phi=(1+\sqrt{5})/2$8

If $d_\tau=\phi=(1+\sqrt{5})/2$9 and $Y$0 denote the orthogonal projectors onto the corresponding eigenspaces, then

$Y$1

This relation is the operator-level expression of the non-invertible character of the defect: $Y$2 does not square to the identity, but instead closes on a sum of simple sectors (Liang, 1 Jul 2026).

A useful summary of the basic structures is:

Object	Definition	Key relation
$Y$3	Nontrivial simple object	$Y$4
$Y$5	Quantum dimension of $Y$6	$Y$7
$Y$8	$Y$9-loop defect operator	$\tau$0
$\tau$1	Projectors onto $\tau$2 eigenspaces	$\tau$3

The same category appears in a broader topological-defect-line context. In the Conway-module construction, Fibonacci defects $\tau$4 and $\tau$5 generate a Fibonacci fusion subcategory $\tau$6, and their realization is tied to icosian lattice endomorphisms on the Leech lattice (Angius et al., 22 Dec 2025).

2. Realization in the critical golden chain

The lattice realization is the periodic golden chain in the usual $\tau$7 RSOS/fusion-path basis. On each bond $\tau$8 of a periodic chain of length $\tau$9 one carries an anyon label, and the Hamiltonian is the antiferromagnetic projector onto the vacuum fusion channel of nearest-neighbor $P_\tau/P_1=\phi^2$0 anyons:

$P_\tau/P_1=\phi^2$1

where the Temperley--Lieb generator $P_\tau/P_1=\phi^2$2 projects the two $P_\tau/P_1=\phi^2$3 anyons at sites $P_\tau/P_1=\phi^2$4 to the vacuum and is normalized by

$P_\tau/P_1=\phi^2$5

This is the critical golden chain of Fibonacci anyons (Liang, 1 Jul 2026).

A single bond $P_\tau/P_1=\phi^2$6 may be chosen as a cut of the periodic chain. Across that cut, the total anyonic charge is either $P_\tau/P_1=\phi^2$7 or $P_\tau/P_1=\phi^2$8. The corresponding cut-charge projectors

$P_\tau/P_1=\phi^2$9

enforce the value of the fusion-path label crossing bond $\log g=\log\phi$0, and satisfy

$\log g=\log\phi$1

These projectors are the lattice observables that enter the finite-size entanglement statement (Liang, 1 Jul 2026).

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 (Liang, 1 Jul 2026).

3. Exact cut-charge theorem and entanglement fingerprint

The central finite-size operator identity is

$\log g=\log\phi$2

Using $\log g=\log\phi$3 and $\log g=\log\phi$4, this becomes

$\log g=\log\phi$5

The proof is described as purely based on fusion-path combinatorics and cyclic $\log g=\log\phi$6-symbol traces (Liang, 1 Jul 2026).

For the normalized even-length antiferromagnetic ground state $\log g=\log\phi$7, the probability, or Schmidt weight, of finding charge $\log g=\log\phi$8 across the cut is defined by

$\log g=\log\phi$9

Once the ground state is shown to lie in $V^{f\natural}$0, the projector identity gives

$V^{f\natural}$1

Therefore

$V^{f\natural}$2

If the defect boundary entropy $V^{f\natural}$3 is defined by

$V^{f\natural}$4

then

$V^{f\natural}$5

These equalities hold in exact finite size, with no extrapolation in $V^{f\natural}$6 (Liang, 1 Jul 2026).

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 $V^{f\natural}$7, 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 $V^{f\natural}$8-sector containing the physical ground state. In the fusion-path $V^{f\natural}$9 RSOS$1$0 basis, all off-diagonal matrix elements of $1$1 are non-positive, and the action of $1$2 is transitive on admissible paths for even $1$3. By the Perron--Frobenius theorem, the unique ground-state wavefunction $1$4 can therefore be chosen with strictly positive components in that basis (Liang, 1 Jul 2026).

Because

$1$5

the ground state lies entirely in either the $1$6 sector or the $1$7 sector. The sector selection is then fixed by an explicit sign argument on the matrix elements $1$8: $1$9 has all nonnegative components and a nonzero overlap with $\tau$0. Since $\tau$1 also has positive overlap with $\tau$2, one concludes that

$\tau$3

hence

$\tau$4

This is the step that connects the abstract defect eigenspace decomposition to the physical ground state of the even-length chain (Liang, 1 Jul 2026).

The theorem is explicitly stated for the even-length antiferromagnetic ground state. The emphasis on even $\tau$5 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 $\tau$6 with $\tau$7 has six Virasoro primaries. In the standard scaling-limit RSOS/affine-TL approach, one groups finite-$\tau$8 TL-modules into “$\tau$9 packets” whose characters branch into six towers:

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

However, none of that Virasoro branching data enters the finite-size proof of the cut-projector identity or the Perron--Frobenius sector argument (Liang, 1 Jul 2026).

The conceptual hierarchy is presented as

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

The exact theorem lives entirely in the first layer. Its proof uses:

• the local Fibonacci fusion rule $\tau\otimes\tau = 1 \oplus \tau,$2,
• the cyclic $\tau\otimes\tau = 1 \oplus \tau,$3-symbol trace $\tau\otimes\tau = 1 \oplus \tau,$4,
• a block-graph rank identity giving $\tau\otimes\tau = 1 \oplus \tau,$5 as the graph of an isometry,
• Perron--Frobenius positivity selecting the $\tau\otimes\tau = 1 \oplus \tau,$6 sector for even $\tau\otimes\tau = 1 \oplus \tau,$7 (Liang, 1 Jul 2026).

This separation is important because the six-primary tricritical-Ising resolution is finer than the two-charge categorical resolution. The exact result $\tau\otimes\tau = 1 \oplus \tau,$8 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 $\tau\otimes\tau = 1 \oplus \tau,$9, where topological defect lines preserving the relevant supersymmetry are described through Leech-lattice endomorphisms. The candidate Fibonacci defect $d_\tau=\phi=(1+\sqrt{5})/2$00 is defined by a lattice endomorphism

$d_\tau=\phi=(1+\sqrt{5})/2$01

constructed using the “icosian” realization of $d_\tau=\phi=(1+\sqrt{5})/2$02. Writing

$d_\tau=\phi=(1+\sqrt{5})/2$03

and setting

$d_\tau=\phi=(1+\sqrt{5})/2$04

the map

$d_\tau=\phi=(1+\sqrt{5})/2$05

satisfies

$d_\tau=\phi=(1+\sqrt{5})/2$06

On $d_\tau=\phi=(1+\sqrt{5})/2$07 it splits into two $d_\tau=\phi=(1+\sqrt{5})/2$08-planes,

$d_\tau=\phi=(1+\sqrt{5})/2$09

with

$d_\tau=\phi=(1+\sqrt{5})/2$10

These are the $d_\tau=\phi=(1+\sqrt{5})/2$11-twist and $d_\tau=\phi=(1+\sqrt{5})/2$12-cotwist eigenspaces. The subgroup commuting with $d_\tau=\phi=(1+\sqrt{5})/2$13 is the icosian subgroup

$d_\tau=\phi=(1+\sqrt{5})/2$14

and conjugation by an involution in the normalizer $d_\tau=\phi=(1+\sqrt{5})/2$15 exchanges $d_\tau=\phi=(1+\sqrt{5})/2$16 and $d_\tau=\phi=(1+\sqrt{5})/2$17, yielding a second Fibonacci defect $d_\tau=\phi=(1+\sqrt{5})/2$18 (Angius et al., 22 Dec 2025).

The associated braided data are explicitly specified. The unique non-trivial $d_\tau=\phi=(1+\sqrt{5})/2$19-symbol and the $d_\tau=\phi=(1+\sqrt{5})/2$20-symbols are

$d_\tau=\phi=(1+\sqrt{5})/2$21

and they satisfy the pentagon and hexagon equations of a unitary braided category (Angius et al., 22 Dec 2025).

The Conway-module realization also supplies defect-twined elliptic genera. The $d_\tau=\phi=(1+\sqrt{5})/2$22-twined graded trace is

$d_\tau=\phi=(1+\sqrt{5})/2$23

with

$d_\tau=\phi=(1+\sqrt{5})/2$24

where

$d_\tau=\phi=(1+\sqrt{5})/2$25

is the unique weight-$d_\tau=\phi=(1+\sqrt{5})/2$26 modular form for $d_\tau=\phi=(1+\sqrt{5})/2$27. Its $d_\tau=\phi=(1+\sqrt{5})/2$28-expansion begins

$d_\tau=\phi=(1+\sqrt{5})/2$29

In two K3 Gepner models of type $d_\tau=\phi=(1+\sqrt{5})/2$30 and $d_\tau=\phi=(1+\sqrt{5})/2$31, one constructs explicit topological defect lines $d_\tau=\phi=(1+\sqrt{5})/2$32 preserving $d_\tau=\phi=(1+\sqrt{5})/2$33 and spectral flow, and their elliptic genera are computed in closed form and exactly reproduce the Conway-module expression (Angius et al., 22 Dec 2025).

Within this framework, Fibonacci defects exist only in subcategories $d_\tau=\phi=(1+\sqrt{5})/2$34 where $d_\tau=\phi=(1+\sqrt{5})/2$35 or $d_\tau=\phi=(1+\sqrt{5})/2$36 is a $d_\tau=\phi=(1+\sqrt{5})/2$37-plane in $d_\tau=\phi=(1+\sqrt{5})/2$38 with icosian stabilizer. The paper further conjectures that every four-plane $d_\tau=\phi=(1+\sqrt{5})/2$39 stabilized by a subgroup isomorphic to $d_\tau=\phi=(1+\sqrt{5})/2$40 realizes a K3 NLSM carrying a Fibonacci defect with coincident $d_\tau=\phi=(1+\sqrt{5})/2$41; this is explicitly presented as a conjecture rather than a theorem (Angius et al., 22 Dec 2025).