Topological Defects on the Lattice: Dualities and Degeneracies

TL;DR

The paper develops a comprehensive framework that uses fusion categories to define topological defect lines on 2D lattice models and to reinterpret these models as boundary data for the Turaev-Viro-Barrett-Westbury state sum. Defects satisfy local commutation relations (defect commutation relations) and fuse according to the underlying fusion category, enabling exact lattice identities and a generalized Kramers-Wannier duality that explains degeneracies in ground and low-lying excited states. The approach connects lattice models (Ising, Potts, ABF/RSOS, parafermions) to conformal field theory via the TVBW TQFT, providing exact results for g-factor ratios, Dehn-twist eigenvalues, and a diagrammatic calculus (F-moves, tetrahedral symbols) that persists on non-integrable models. It also develops boundary states, twists, and tube-category formalisms to extract universal data and boundary-CFT correspondences, with applications to ABF, Potts/clock, minimal models, and potential extensions to Morita-equivalent and orbifold defects. Overall, the work offers a powerful, exactly solvable, and highly general framework for dualities, degeneracies, and boundary phenomena in lattice systems through categorical topology.

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.

Figures (10)

• Figure 1: A fusion diagram, i.e., a labeled planar graph. For simplicity we leave out vertex labels and edge orientations. Using the standard notation of quantum mechanics, the diagram here can be viewed as an inner product between bra labeled $\mu, \gamma, \delta$, a ket labeled $\alpha, \nu, \delta$, and an operator with labels $\alpha, \beta, \gamma, \delta$.
• Figure 2: (a) Possible vertices permitted by the fusion rules Eq. \ref{['eq:Ising_fusion_rules']} in the Ising fusion category. (b) Examples of vertices forbidden by the Ising fusion rules. (c) An example of a fusion diagram in the Ising fusion category, with elements ${\mathds{1}}$, $\sigma$, and $\psi$.
• Figure 3: Notation for each vertex ${\rm v}$ and adjacent edges and faces.
• Figure 4: Defining the vector space of the height models in the transfer matrix/1+1D quantum picture. The heights on the 2D lattice are now written as labels on the trunk of the fusion tree.
• Figure 5: The three types of 0-cells appearing in the TVBW state sum for $\langle \mathcal{F}|D\rangle$. The three 0-cells are indicated with ${\rm v}$. The grey regions are 2-cells which sit inside the surface $Y \times \{ \frac{1}{2} \}$. The red (green) 2-cells are found by extending the nets living on $Y \times \{1 \}$ ($Y \times \{0\}$) into the bulk until they meet $Y \times \{ \frac{1}{2} \}$.