Entanglement-spectrum fingerprint of a non-invertible symmetry: the Kramers--Wannier duality defect on the lattice

Abstract: Non-invertible symmetries are characterized by topological defects of irrational quantum dimension, but their imprint on the entanglement of a quantum many-body state has not been resolved at the level of the spectrum. We show that the categorical data of the canonical example -- the Kramers--Wannier (KW) duality defect of the critical Ising chain, with quantum dimension d_sigma=sqrt(2) -- is encoded in the single-particle entanglement spectrum of its ground state: a maximally mixed Majorana zero mode is the spectral origin of the boundary entropy log g=(1/2)log 2, hence of d_sigma itself. Reading the same duality-twisted ground state along two independent routes -- the transfer-matrix momentum shift and the Casimir curvature of the energy -- pins the twist-field weight h_sigma=1/16 twice over, and the defect Hilbert space organizes into a half-integer sigma-twisted conformal tower. This promotes the boundary entropy from an integrated number to a level-resolved spectral signature of non-invertibility, and supplies an exactly solvable calibration target for tensor-network studies of duality defects that lack a free-fermion shortcut.

• The paper identifies a unique Majorana zero mode in the entanglement spectrum that encodes the irrational quantum dimension of the KW duality defect.
• The paper employs exact free-fermion methods, including diagonalization and Peschel’s technique, to achieve machine-precision insights into spectral features.
• The paper extracts twist-field conformal weight via momentum shift and Casimir energy scaling, establishing a robust diagnostic for non-invertible symmetries.

Entanglement Spectrum as a Probe of Non-Invertible Symmetry: The Kramers--Wannier Duality Defect

Overview

This paper provides an exact, level-resolved analysis of how the categorical invariants associated with non-invertible symmetries, specifically quantum dimensions and conformal data, manifest in the entanglement spectrum of lattice models. Using the Kramers–Wannier (KW) duality defect in the critical one-dimensional Ising chain as a case study, the work demonstrates that the irrational quantum dimension characteristic of non-invertible symmetries is encoded microscopically in a Majorana zero mode within the single-particle entanglement spectrum. The findings establish a precise correspondence between the categorical data of the KW defect and identifiable features in the entanglement properties of a single ground state, providing both a robust diagnostic framework and an exact benchmark for tensor-network and numerical studies in broader, less tractable settings.

Non-Invertible Symmetries and the KW Defect

Non-invertible symmetries, in contrast to traditional group-based symmetries, are represented by fusion categories of topological defect lines. The fusion rules for such objects may lack inverses and are characterized by irrational quantum dimensions, which serve as the primary distinguishing invariants. The Kramers–Wannier duality defect in the Ising chain is a canonical example: it possesses non-group-like fusion rules and a quantum dimension $d_\sigma=2$, fundamentally linked to its non-invertibility.

While previous studies have related certain integrated entanglement quantities, notably the boundary entropy $\log g$, to the presence of such defects, a microscopic mode-by-mode identification within the entanglement spectrum has until now remained elusive.

Lattice Construction and Exact Methods

The analysis is performed on the transverse-field Ising chain at criticality with open boundaries, where the KW defect is implemented via a non-local, duality-twisted boundary condition following Aasen, Mong, and Fendley (AMF). The resulting Hamiltonian,

$$H^d = -\sum_{j=1}^{L-1} Z_j Z_{j+1} - \sum_{j=2}^L X_j - Z_L Y_1,$$

carries one fewer independent Majorana mode ($2L-1$ instead of $2L$), identifying a unique zero-energy Majorana localized at the defect. This serves as the origin of the two-fold ground-state degeneracy and, as the paper rigorously demonstrates, the quantum dimension $d_\sigma$.

Free-fermion techniques, including exact diagonalization and Peschel’s method for entanglement spectra, provide access to all quantities to machine precision. This allows unambiguous assignment of observed spectral features to categorical invariants.

Entanglement Spectrum and Quantum Dimension

The primary result is the explicit identification of the irrational quantum dimension in the entanglement spectrum. The entanglement spectrum of a block containing the defect always features a single maximally mixed Majorana mode with $\xi=0$ (corresponding to occupation $\nu=0$), quantized to double-precision accuracy. This mode contributes exactly $\ln 2$ to the block entanglement entropy—precisely the value that underlies the boundary entropy $\log g = \frac{1}{2} \ln 2$ per boundary. The contrast with invertible $\log g$0 (symmetry-twist) defects is pronounced: the latter yield no such zero mode, and their boundary entropies vanish.

This result elevates the irrational quantum dimension from an integrated “number” to a microscopic, spectral signature—a maximally mixed entanglement mode uniquely associated with non-invertibility.

Conformal Data: Twist-Field Weight and Tower Structure

The work further establishes that the twist-field conformal weight $\log g$1 can be extracted in two independent, level-resolved ways:

• Transfer-Matrix Momentum Shift: Diagonalization of the defect-compatible translation operator yields momentum eigenvalues showing that the ground-state sector momentum shift is $\log g$2, unambiguously assigning $\log g$3.
• Casimir Energy Curvature: Fitting finite-size scalings of ground-state energies for the defect chain, the Casimir term precisely yields $\log g$4 to five-digit accuracy, contrasting with the vanishing value for the homogeneous (defect-free) chain.

Additionally, the many-body spectrum forms a half-integer (Ramond-like) conformal tower, matching the theoretical expectation for the $\log g$5 sector, in sharp distinction to the $\log g$6-offset Neveu–Schwarz tower of the homogeneous chain.

Robustness and Uniqueness of the Fingerprint

The quantization and uniqueness of the entanglement spectrum fingerprint are validated through explicit controls:

• Invertible $\log g$7 defects (disorder lines) lack both the zero mode and the boundary entropy contribution, confirming that the maximally mixed mode is a signature of non-invertibility, not generic to defects.
• The location, degeneracy, and contribution of the entanglement zero mode are invariant with respect to system size and block position, up to boundaries, and are insensitive to the particulars of the free-fermion representation.

Implications and Future Developments

The findings have several significant implications for condensed matter theory and quantum information:

• Benchmark for Tensor-Network Methods: The exact, machine-precision numbers obtained serve as a reference for tensor-network or DMRG approaches in systems lacking a free-fermion solution. The presence, quantization, and separation of the entanglement zero mode provide concrete numerical targets and error bounds.
• Extension to Non-Solvable and Higher-Dimensional Cases: While the Ising chain is exactly solvable, the diagnostic framework of spectrum-resolved entanglement applies equally to non-Abelian anyon chains and higher-dimensional lattice models with categorical (e.g., Fibonacci) defects, as noted in concurrent work cited by the author. There, the irrational quantum dimension $\log g$8 is similarly detected without recourse to free fermions.
• Categorical Data from Entanglement: The work strengthens the paradigm in which bulk categorical/topological data (SymTFT invariants) can be extracted from boundary or cut entanglement properties, sharpening the utility of the entanglement spectrum for the identification and classification of quantum phases and defects.

On a foundational level, the result clarifies which microscopic entanglement features are robust indicators of categorical symmetry properties, providing a rigorous answer in a context where analytic solutions are available.

Conclusion

This paper establishes that the categorical invariants of a non-invertible symmetry defect, notably its irrational quantum dimension and associated conformal data, are encoded at the spectral level in the entanglement structure of a simple lattice Hamiltonian. The maximally mixed Majorana zero mode in the entanglement spectrum serves as the microscopic carrier for the quantum dimension, while independent conformal signatures recover the twist sector in energy and momentum space. The work offers rigorous diagnostics and error benchmarks for future numerical investigations in more general topological and non-invertible defect settings, both in one dimension and beyond (2607.01137).