Topological BF Theory construction of twisted dihedral quantum double phases from spontaneous symmetry breaking

TL;DR

This paper introduces a systematic topological BF theory framework to construct nonabelian dihedral quantum double phases from a continuous $O(2)$ gauge field, linking theory with microscopic models and potential experimental realization.

Contribution

It presents a novel topological BF theory approach for nonabelian dihedral phases, including a microscopic lattice model and analysis of phase transitions with emergent symmetries.

Findings

Complete anyon data reproduction via BF theory

Microscopic model with Higgsing to dihedral quantum double phase

Phase transition analysis suggests stable multicritical points with emergent symmetry

Abstract

Nonabelian topological orders host exotic anyons central to quantum computing, yet established realizations rely on case-by-case constructions that are often conceptually involved. In this work, we present a systematic construction of nonabelian dihedral quantum double phases based on a continuous $O(2)$ gauge field. We first formulate a topological $S[O(2)\times O(2)]$ BF theory, and by identifying the Wilson loops and twist operators of this theory with anyons, we show that our topological BF theory reproduces the complete anyon data, and can incorporate all Dijkgraaf--Witten twists. Building on this correspondence, we present a microscopic model with $O(2)$ lattice gauge field coupled to Ising and rotor matter whose Higgsing yields the desired dihedral quantum double phase. A perturbative renormalization group analysis further suggests a direct transition from this phase to a $U(1)$ Coulomb or chiral topological phase at a stable multicritical point with emergent $O(3)$ symmetry. Our proposal offers an alternative route to nonabelian topological order with promising prospects in synthetic gauge field platforms.