A non-invertible topological line fixes global defect data but does not select a reduced density matrix (RDM) in a spatially twisted sector. For the Kramers-Wannier defect of the critical Ising chain, we choose the equal-weight incoherent mixture of the two charge-sector ground states and reduce it to the ordinary full spin algebra of a complete non-wrapping prefix anchored at the defect endpoint. The finite-size RDM is the convex average of two charge-sector RDMs rather than a sign-zero Gaussian proxy, and an exact descended antiunitary enforces many-body Kramers pairing without determining the entropy. At fixed prefix, the RDMs converge in trace norm to a common Gaussian state as the circumference grows. Convergence is uniform over all ground-doublet preparations, including coherent pure states. The subsequent large-prefix entropy limit follows from the odd-even Toeplitz/Fisher-Hartwig endpoint, yielding half the natural logarithm of two above the homogeneous chain. The matched spin-flip defect has zero excess. The exact joint energy-modified-translation character of the same twisted Hamiltonian records the finite-size roots and their multiplicities, retains chirality in the marked scaling limit, and, using the standard Ising character identities, resolves the four Virasoro towers of the Ising duality-twisted sector.
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.
We construct topological defects in two-dimensional classical lattice models and quantum chains. The defects satisfy local commutation relations guaranteeing that the partition function is independent of their path. These relations and their solutions are extended to allow defect lines to fuse, branch and satisfy all the properties of a fusion category. We show how the two-dimensional classical lattice models and their topological defects are naturally described by boundary conditions of a Turaev-Viro-Barrett-Westbury partition function. These defects allow Kramers-Wannier duality to be generalized to a large class of models, explaining exact degeneracies between non-symmetry-related ground states as well as in the low-energy spectrum. They give a precise and general notion of twisted boundary conditions and the universal behaviour under Dehn twists. Gluing a topological defect to a boundary yields linear identities between partition functions with different boundary conditions, allowing ratios of the universal g-factor to be computed exactly on the lattice. We develop this construction in detail in a variety of examples, including the Potts, parafermion and height models.