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.
Abstract In this paper and its sequel, we construct topologically invariant defects in two-dimensional classical lattice models and quantum spin chains. We show how defect lines commute with the transfer matrix/Hamiltonian when they obey the defect commutation relations , cousins of the Yang–Baxter equation. These relations and their solutions can be extended to allow defect lines to branch and fuse, again with properties depending only on topology. In this part I, we focus on the simplest example, the Ising model. We define lattice spin-flip and duality defects and their branching, and prove they are topological. One useful consequence is a simple implementation of Kramers–Wannier duality on the torus and higher genus surfaces by using the fusion of duality defects. We use these topological defects to do simple calculations that yield exact properties of the conformal field theory describing the continuum limit. For example, the shift in momentum quantization with duality-twisted boundary conditions yields the conformal spin 1/16 of the chiral spin field. Even more strikingly, we derive the modular transformation matrices explicitly and exactly.
We determine exact finite-volume spin form factors and two-point correlations for a massive Kramer--Wannier interface in the scaling Ising model, separating ordered and disordered phases. The interface admits two complementary lattice realizations: a self-dual inhomogeneous Ising chain whose continuum limit has a sign-changing Majorana mass profile, and, after exchanging Euclidean space and time, a mixed thermal average amenable to Toeplitz/Fredholm methods. Combining these descriptions, we obtain the exact Neveu-Schwarz-Ramond spin matrix elements and a factorized Fredholm determinant for the interface correlator. Defect TCSA and independent fermionic spin-chain calculations confirm the predicted spectrum and matrix elements.
The critical two-dimensional classical Ising model on the square lattice has two topological conformal defects: the
Z2 symmetry defect ${D}_{\ensuremath{\epsilon}}$ and the Kramers-Wannier duality defect ${D}_{\ensuremath{\sigma}}$. These two defects implement antiperiodic boundary conditions and a more exotic form of twisted boundary conditions, respectively. On the torus, the partition function
ZD of the critical Ising model in the presence of a topological conformal defect
D is expressed in terms of the scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$ and conformal spins ${s}_{\ensuremath{\alpha}}$ of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising conformal field theory. This characteristic conformal data ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{D}$ can be extracted from the eigenvalue spectrum of a transfer matrix
MD for the partition function
ZD. In this paper, we investigate the use of tensor network techniques to both represent and coarse grain the partition functions ${Z}_{{D}_{\ensuremath{\epsilon}}}$ and ${Z}_{{D}_{\ensuremath{\sigma}}}$ of the critical Ising model with either a symmetry defect ${D}_{\ensuremath{\epsilon}}$ or a duality defect ${D}_{\ensuremath{\sigma}}$. We also explain how to coarse grain the corresponding transfer matrices ${M}_{{D}_{\ensuremath{\epsilon}}}$ and ${M}_{{D}_{\ensuremath{\sigma}}}$, from which we can extract accurate numerical estimates of ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\epsilon}}}$ and ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\sigma}}}$. Two key ingredients of our approach are (i) coarse graining of the defect
D, which applies to any (i.e., not just topological) conformal defect and yields a set of associated scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$, and (ii) construction and coarse graining of a generalized translation operator using a local unitary transformation that moves the defect, which only exist for topological conformal defects and yields the corresponding conformal spins ${s}_{\ensuremath{\alpha}}$.
The Rényi entropies of a non-unitary critical system are not fixed by a single power of the central charge but by the full four-point function of twist fields in a cyclic orbifold, constrained by null-vector equations. For the Yang–Lee universality class—the minimal model with —Dupic, Estienne and Ikhlef (DEI) computed several of these twist correlators in closed form. We provide an independent lattice realization and calibration of those exact correlators on two distinct critical lattice models: the critical Restricted Solid-On-Solid (RSOS) model with and the quantum Ising chain in an imaginary longitudinal field. Both are diagonalised exactly, and the lattice-to-continuum correspondence is certified on three independent objects: the low-lying spectrum reproduces the tower with effective central charge and no exceptional-point degeneracy on the RSOS side; the single-interval vacuum and ground-state twist correlators reproduce the exact two- and four-point functions, including a quantitative determination of the sub-leading fusion channel; and the vacuum two-interval four-twist function is reproduced with a crossing self-consistency of . Along the way we correct a typographical error in the DEI Temperley–Lieb Hamiltonian. We define precisely the boundary-height-sector-resolved trace functional that the lattice certifies, and we do not address the microscopic-realization dependence of the physical two-interval mutual information, which involves a choice of observable subalgebra and is left to a separate study.
Universal information extracted from a critical quantum state depends on how the state is probed. Spatial entanglement, participation statistics, and stabilizer Rényi entropy correspond to different replica geometries and need not isolate the same boundary data. We show that the complete distribution of Pauli magnitudes in critical Ising chains carries an exact geometric fingerprint of boundary termination, distinguishing lattice realizations that flow to the same infrared boundary fixed point. This structure is organized by discrete Selberg ensembles associated with the classical root systems
A,
B,
C, and
D: three open chains sharing the free--free Ising boundary condition retain distinct
BL,
CL, and
DL trigonometric wall geometries. The standard symmetry-breaking representatives instead produce rectangular, pinned, and character-inserted
C ensembles. Their four boundary moments contain exactly two independent multiplicative combinations invariant under local endpoint normalizations. One of these reduces exactly at finite rank to a moment of the fundamental symplectic character and provides a lattice target for a defect--measurement-boundary junction amplitude. A conditional Gaussian-character law predicts its values throughout the unlocked phase, while exact-arithmetic finite-rank results at the
α=4 transition depart sharply from the Gaussian continuation. These results show that the full Pauli spectrum resolves boundary information invisible to the infrared boundary fixed point alone and provide concrete targets for replicated boundary conformal field theory.
Defects associated with non-invertible symmetries have attracted significant attention in recent years. Among them, Kramers-Wannier (KW) duality defects have been investigated in both classical statistical systems and quantum Hamiltonian models. Aasen et al. analyzed duality defects in the 2D Ising model and in statistical models built from fusion categories, while Koide et al. later constructed a duality defect in 4D lattice gauge theory. In this work, we extend these developments by providing a systematic construction of KW duality defects/KW defects for a broad class of models formulated within the chain complex framework. Our construction employs the strange correlator, an overlap between a topologically ordered state and a product state, to realize these KW defects.