Topological Defects on the Lattice: Dualities and Degeneracies

Abstract

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.

87 pages + appendices, 250 diagrams

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• Condensate induced transitions between topologically ordered phases
• Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases
• Turaev-Viro invariants as an extended TQFT
• String-net model of Turaev-Viro invariants
• Topological order from quantum loops and nets
• Cardy states, defect lines and chiral operators of coset CFTs on the lattice
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Cited by in corpus (2)

• Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond $N$-ality
• Non-invertible topological defects in 4-dimensional ${\mathbb{Z}}_2$ pure lattice gauge theory