Entanglement fingerprint of a non-invertible symmetry: exact Fibonacci cut charges on the lattice

Abstract: Non-invertible defects are usually diagnosed through scaling spectra or infrared CFT data. We show that the Fibonacci duality defect of the critical golden chain already carries an exact categorical fingerprint at finite lattice size. The even-length antiferromagnetic ground state has fixed cut-charge weights, giving P_tau/P_1=phi^2 and log g=log phi without finite-size extrapolation. The proof is a finite-dimensional operator identity for the sandwiched cut projectors, combined with a Perron-Frobenius sector theorem for the even-length ground state. This gives a sharp lattice-level boundary entropy for a non-Abelian duality defect. We also separate this exact two-charge result from the finer six-primary tricritical-Ising resolution: the latter is located by the standard scaling-limit Virasoro branching of A_4 affine-TL packets, and is not an assumption in the finite-size theorem.

• The paper establishes an exact operator identity that reveals finite-size entanglement cut charges corresponding to Fibonacci non-invertible defects on the lattice.
• It rigorously proves that the Schmidt spectrum decomposes into two sectors with weights exactly matching the categorical quantum dimensions 1 and φ for even-length chains.
• The methodology bridges lattice anyon models with continuum defect CFTs, suggesting broader applications in quantum computation and diagnostics of modular categories.

Entanglement Cut-Resolved Fingerprinting of Non-Invertible Symmetry in the Fibonacci Golden Chain

Overview and Background

The paper "Entanglement fingerprint of a non-invertible symmetry: exact Fibonacci cut charges on the lattice" (2607.01151) addresses a long-standing open problem in the diagnosis of non-invertible symmetries by establishing an exact, finite-size entanglement fingerprint for the Fibonacci non-invertible defect in the critical golden chain anyonic model. Distinct from previous approaches that rely on CFT scaling limits or boundary partition function computations, this work proves that the lattice—before reaching the thermodynamic or scaling limit—already encodes the categorical quantum dimensions of the Fibonacci category as sharply quantized cut-charge weights in the ground state entanglement.

Non-invertible symmetries, characterized by categorical fusion rather than group-like invertibility, exhibit distinct defect fusion and entanglement structures. In the Fibonacci chain, this manifests in non-invertible duality defect operators that satisfy the fusion rule $Y^2 = 1 + Y$, leading to quantum dimensions $d_1 = 1$, $d_\tau = \varphi = (1+\sqrt{5})/2$ and total quantum dimension $D^2 = 1+\varphi^2$. The critical golden chain model with periodic boundary conditions flows to the tricritical Ising CFT ($c = 7/10$), but at any finite lattice size the system supports only two topological superselection sectors ($\{1,\,\tau\}$) across any entanglement cut, not the full set of six CFT primaries.

Main Theorem and Finite-Size Cut-Charge Resolution

The central result is an exact operator identity on the finite periodic Fibonacci chain at even length $L$, which underpins the following claims:

• In the exact periodic ground state, tracing over half the system with a spatial cut decomposes the Schmidt spectrum into exactly two blocks corresponding to cut charges $a \in \{1, \tau\}$, regardless of lattice size.
• The probability weights for finding each sector are precisely given by the categorical quantum dimensions:
  • $P_1 = 1/D^2 \approx 0.2764$
  • $P_\tau = \varphi^2 / D^2 \approx 0.7236$
• The ratio $d_1 = 1$0, the square of the golden ratio, holds with machine precision for all checked even $d_1 = 1$1; no extrapolation or scaling is required.
• The corresponding defect boundary (Affleck-Ludwig) entropy is $d_1 = 1$2.

The proof leverages:

• A finite-dimensional operator identity for sandwiched cut projectors involving the topological symmetry operator $d_1 = 1$3.
• Construction of projectors $d_1 = 1$4 onto the $d_1 = 1$5 sector and explicit evaluation in the fusion-path basis.
• Perron-Frobenius arguments to select the unique, positive ground state supported in the $d_1 = 1$6 sector for even $d_1 = 1$7.
• Direct confirmation by constructing basis states numerically for moderate $d_1 = 1$8 (8–14), which verifies the analytic predictions to within $d_1 = 1$9.

Crucially, this exact result does not rely on knowledge of the Virasoro primary-sector resolution of the CFT or any scaling limit arguments. The separation between the two-charge resolution at the lattice level and the six-primary decomposition in the scaling limit is both precise and emphasized.

Physical and Categorical Implications

The findings provide a microscopic, operator-level bridge between lattice anyon algebra and continuum defect CFT physics:

• The non-Abelian quantum dimension $d_\tau = \varphi = (1+\sqrt{5})/2$0 is directly and exactly observable in the Schmidt sector entanglement weights on any finite lattice, not only as the limiting value of a defect entropy.
• The hierarchy is clear:

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

• In contrast to the well-understood Abelian case (e.g., Majorana zero modes in Ising chains), here the nontrivial categorical structure is retained without a free-fermion representation.
• The exactness and operator nature of the result suggest a generic approach for extracting categorical data (quantum dimensions, fusion channels) from finite-size ground states in other models with non-invertible symmetries. The Supplemental Material includes a rank-3 $d_\tau = \varphi = (1+\sqrt{5})/2$2 benchmark as a category-level check beyond the Fibonacci case.

Numerical Results and Strong Claims

• Cut-sector weights $d_\tau = \varphi = (1+\sqrt{5})/2$3 match their categorical dimension predictions exactly for all $d_\tau = \varphi = (1+\sqrt{5})/2$4.
• The ratio $d_\tau = \varphi = (1+\sqrt{5})/2$5 remains invariant to machine precision, confirmed by finite-matrix construction.
• All observed deviations from the theoretical values are below $d_\tau = \varphi = (1+\sqrt{5})/2$6.

No properties of the continuum CFT—such as the Kac table, modular $d_\tau = \varphi = (1+\sqrt{5})/2$7 matrix, or partition function boundary conditions—are assumed or required in the proof of the finite-size result.

Implications for Entanglement Studies and Future Directions

From both theoretical and practical standpoints, these results highlight new directions:

• Microscopically accessible diagnostics: The categorical entanglement signature of non-invertible defects can be extracted from moderate lattice sizes, opening up new regime for numerical simulations of non-Abelian anyons and topological defects in DMRG, MPS, and tensor network approaches.
• Hierarchy of entanglement resolutions: The work enforces a careful delineation between categorical (finite-size) and conformal (scaling-limit) data. This separation clarifies the precise observable layer accessible at any fixed size, with full CFT branching appearing only at the continuum limit.
• Generalization and benchmarking: The methodology may be extended rigorously to higher-rank modular categories (such as $d_\tau = \varphi = (1+\sqrt{5})/2$8 chains), where more refined categorical charge-sector fingerprints could be established. The presented $d_\tau = \varphi = (1+\sqrt{5})/2$9 benchmark suggests a practical path for such generalizations.
• Implications for quantum computation: Finite-size, exact charge identification could be utilized in the design and analysis of topological qubits and defect-based quantum memory, where categorical symmetries play a direct role.

Conclusion

This work provides an exact, operator-level identification of topological cut-charge fingerprints arising from non-invertible Fibonacci duality defects in the golden chain at finite size, bypassing the need for scaling extrapolation or full CFT primary-sector resolution. The precise match between entanglement cut weights and categorical quantum dimensions demonstrates that categorical data is directly encoded in interacting lattice ground states, establishing a new paradigm for diagnosis and numerical study of non-invertible symmetry defects (2607.01151). The methods and separation of resolution layers presented here invite further application to more complex modular categories and lattice models supporting non-invertible symmetries.