Constructs non-invertible duality defects for one-form symmetries in 3+1D by partial gauging, derives fusion rules, proves incompatibility with trivial gapped phases, and realizes explicitly in Maxwell theory and lattice models.
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Topological Defects on the Lattice I: The Ising model
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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.
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representative citing papers
Monte Carlo reconstruction via interpolating family and flat histograms computes the Z2-twisted thermodynamic Casimir difference in the critical 3D Ising model as 0.327(2).
Introduces FTH as an extension of TH/SymTFT to type-I and type-II fracton orders, demonstrating boundary switches and dualities for X-cube and Haah's code via stabilizer formalism.
Twin phases are inequivalent phases sharing a generalized charge under symmetry S, enabling stable direct transitions without spontaneous symmetry breaking even after gauging.
String order parameters in 1D gapped phases with invertible or non-invertible symmetries organize into Lagrangian algebras in the Drinfel'd centre via tensor-network module categories.
A new TIM/Ising conformal interface is identified with emergent W3 symmetry, yielding defect spectrum predictions for Rydberg atom experiments.
Any weakly integral fusion category admits a QCA-refined realization on tensor-product Hilbert spaces with QCA and symmetry indices fixed by the categorical data under defect assumptions.
Constructs non-invertible duality defects in (2+1)d QFTs from half-spacetime gauging of 2-group symmetries and derives explicit fusion rules with examples in U(1)^3 gauge theories.
A method is given to construct UV anyonic chain lattice models from SymTFT data realizing IR phases and transitions with non-invertible symmetries, illustrated with Rep(S3).
Higher gauging of 1-form symmetries on surfaces in 2+1d QFT yields condensation defects whose fusion rules involve 1+1d TQFTs and realizes every 0-form symmetry in TQFTs.
The Ising fusion category lattice model features a symmetric critical phase equivalent to the Ising model, a categorical ferromagnetic phase with threefold degeneracy, and a critical categorical antiferromagnetic phase with fourfold degeneracy described by an Ising CFT.
The categorical data of the KW duality defect, including its quantum dimension sqrt(2), is encoded in the entanglement spectrum of the duality-twisted Ising ground state through a Majorana zero mode.
Imposing a duality-inspired fusion rule that forbids the [ε] sector from appearing in the [ε] × [ε] OPE yields numerical bounds on (Δ_σ, Δ_ε) that include the 2d Ising model but exclude the 3d Ising model.
Gauging and duality transformations are equivalent up to constant depth quantum circuits in one-dimensional quantum lattice models, demonstrated via matrix product operators.
Derives additivity and fusion rules for defect g-functions in integrable 2D QFT, with effective amplitudes for non-topological cases and lowered entropy contribution in Ising non-topological fusion.
Lattice realizations of topological defects in non-unitary 2D CFTs are built from modified RSOS models, yielding numerical results that match analytical predictions for spectra and RG flows.
A survey of non-invertible symmetries with constructions in the Ising model and applications to neutral pion decay and other systems.
This review summarizes transformative examples of generalized symmetries in QFT and their applications to anomalies and dynamics.
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Non-Invertible Duality Defects in 3+1 Dimensions
Constructs non-invertible duality defects for one-form symmetries in 3+1D by partial gauging, derives fusion rules, proves incompatibility with trivial gapped phases, and realizes explicitly in Maxwell theory and lattice models.