Topological Defects on the Lattice I: The Ising model

TL;DR

The paper develops a lattice formulation of topological defects for the Ising model, introducing spin-flip and duality defects and showing their defect lines obey a fusion-category framework. By enforcing defect commutation relations, the authors demonstrate path independence and construct twisted boundaries, Dehn twists, and F-moves that yield exact lattice realizations of Kramers–Wannier duality, modular transformations, and conformal data. They compute the conformal spin 1/16 of the chiral spin field directly from the lattice and derive exact Ising CFT modular matrices, linking lattice models to continuum theories. The results establish a robust lattice–CFT dictionary via fusion categories, enabling duality and modular invariance analyses on torus and higher-genus surfaces and setting the stage for generalizations to height models in Part II.

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.

Figures (9)

• Figure 1: Ising model with a sample configuration on the square lattice. The height variables $h = 0,1$, corresponding to spins $\sigma = +1,-1$ respectively, live on what we call the "original" lattice. The original lattice and its dual also form a square lattice drawn in the figure.
• Figure 2: A defect line by choosing a closed path along the edges of the lattice, and then inserting a seam of parallelograms. (a) and (b) are horizontal and vertical spin-flip defects respectively; the stars label the locations of the spins. We have omitted the dots (weight per site) for clarity.
• Figure 3: Illustrations of a horizontal duality defect line in (a1) and (a2) and a vertical duality defect line in (b1) and (b2), with the spins denoted by stars. The duality defect line splices together the lattice with the dual lattice. The only difference between (a1) and (a2) is the position of the spins, and likewise for (b1) and (b2).
• Figure 4: The two types of domain walls present with periodicity in the horizontal direction and a vertical defect. The duality defect and the domain wall each separate the ordered phase with the disordered phases. Squares with hatching have spectral parameter $u$ while blank ones have $u'$; the direction of hatching indicates the locations of the spins. Top and bottom differ by the locations of the spins, and dots are omitted for clarity.
• Figure 5: On the left, a schematic view of a trivalent junction of vertex lines. On the right, how this is defined precisely by introducing a triangular face.