Topological defects on the lattice: I. The Ising model

By D. Aasen, Roger S. K. Mong, P. Fendley. 2016

251 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/191056c4d63bcbe3530cac0de5418c171ad51fa8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. Norbert Schuch
2. Frank Verstraete
3. I. Cirac
4. David Pérez-García
5. Ryan Thorngren
6. Yifan Wang
7. J. Haegeman
8. J. Dubail

In Condensed Matter Physics

1. Frank Verstraete
2. Norbert Schuch
3. I. Cirac
4. David Pérez-García
5. Yifan Wang
6. Ryan Thorngren
7. J. Haegeman
8. Roman Orus

In Geometry and Topology

1. Frank Verstraete
2. I. Cirac
3. David Pérez-García
4. Yifan Wang
5. Norbert Schuch
6. Ryan Thorngren

Citations in the corpus

• Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases (2019, background): “These symmetries may also be realized by matrix-product operators (MPOs) [8,11,54], and were even found in some related 2d statistical mechanical models [2].”
• Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, methodology): “In a similar vein, it is possible to construct the classical statistical mechanical lattice models using this construction; this gives rise to the famous RSOS models of Andrews, Baxter and Forester (Aasen et al., 2016; Andrews et al., 1984), which are known to yield lattice critical systems correspo”