Theory of Twist Liquids: Gauging an Anyonic Symmetry

TL;DR

This paper develops a comprehensive framework for gauging symmetries in topological phases, leading to the creation of twist liquids with complex non-Abelian anyon structures, and provides explicit models and examples.

Contribution

It introduces a general gauging framework for topological phases with symmetries, characterizes the resulting twist liquids, and offers solvable lattice models for phase transitions.

Findings

Gauging symmetries produces non-Abelian twist liquids.

Explicit models for gauging $ ext{Z}_2$, $SO(2N)_1$, and $SU(3)_1$ states.

Demonstrates phase transitions via solvable lattice models.

Abstract

Topological phases in (2+1)-dimensions are frequently equipped with global symmetries, like conjugation, bilayer or electric-magnetic duality, that relabel anyons without affecting the topological structures. Twist defects are static point-like objects that permute the labels of orbiting anyons. Gauging these symmetries by quantizing defects into dynamical excitations leads to a wide class of more exotic topological phases referred as twist liquids, which are generically non-Abelian. We formulate a general gauging framework, characterize the anyon structure of twist liquids and provide solvable lattice models that capture the gauging phase transitions. We explicitly demonstrate the gauging of the $\mathbb{Z}_2$-symmetric toric code, $SO(2N)_1$ and $SU(3)_1$ state as well as the $S_3$-symmetric $SO(8)_1$ state and a non-Abelian chiral state we call the "4-Potts" state.