Topological Defects on the Lattice: Dualities and Degeneracies

TL;DR

This paper develops a framework for topological defects in 2D lattice models, revealing their fusion properties, dualities, and implications for degeneracies and boundary conditions, with applications to various models.

Contribution

It introduces a comprehensive construction of topological defects satisfying fusion category properties and connects them to boundary conditions and dualities in lattice models.

Findings

Defects satisfy local commutation relations ensuring path independence.

Dualities are generalized, explaining degeneracies in ground states.

Exact ratios of universal g-factors are computed using defect boundary conditions.

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.