Theory of Twist Liquids: Gauging an Anyonic Symmetry

TL;DR

<3-5 sentence high-level summary>This work develops a general framework to gauge anyonic symmetries in 2+1D topological phases, producing twist liquids via a defect-fusion-category input and a Drinfeld center (relative where needed) construction, and captures the phase transition with exactly solvable Levin–Wen string-net models. It shows that gauging an anyonic symmetry G increases the total quantum dimension by a factor |G| (D_TL = D_0 |G|) while preserving the chiral central charge, and that twist liquids in general are not discrete gauge theories due to irrational defect quantum dimensions. The authors apply the formalism to a broad set of examples—toric code, SU(3)_1, SO(N)_1, SO(8)_1 with S_3 triality, and the non-Abelian chiral 4-Potts state—demonstrating rich families of Ising, SU(2), parafermion, and Potts-like twist liquids and revealing obstructions and multiple outcomes tied to cohomology and SP T stacking. The framework furnishes a powerful route to generate exotic non-Abelian anyons from Abelian parents and offers insights for edge-CFT orbifolds and potential quantum computation platforms.</p>

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.

Figures (16)

• Figure 1: Wen plaquette-model (for our purposes this is essentially equivalent to the toric code) with a dislocation (yellow pentagon).
• Figure 2: (a) A quasiparticle changing type when traveling across a branch cut terminated at a twist defect $\sigma$. (b) The Wilson loop $\Theta$ that distinguishes defect species.
• Figure 3: Hexagon equation.
• Figure 4: Twist defects (crosses) connected by arbitrary branch cuts (curvy lines) where a passing anyon changes type ${\bf a}\to M{\bf a}$ according to an anyonic symmetry $M$.
• Figure 5: Fusion of a pair of (bare) defects associated to opposite anyonic symmetries $M$ and $M^{-1}$.