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

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 indicates 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.

Figures (5)

• Figure 1: $\mathbb{Z}_2$ sign of the $S$ and $T$ transformations.
• Figure 2: Schematic phase diagram of the lattice model Eq. (\ref{['eq:LH']}) for $n>2$, with schematic RG flows derived from the effective field theory Eq. (\ref{['eq:WF']}). The relevant RG flow towards the $O(3)$ Wilson-Fisher fixed point is highlighted in orange. The gauge structure of each phase is shown explicitly. The 3D XY and the 3D Ising transitions are marked in red and yellow, respectively. From left to right the three colored planes are phase diagrams for sufficiently small, mediate, and sufficiently large $\kappa$, respectively. The relative position of the $O(3)$ Wilson-Fisher fixed point and the phase diagram for intermediate $\kappa$ is for illustrative purposes only and may differ from the actual situation.
• Figure S1: Genus-$g$ Riemann surface. Homology cycles $\alpha_i,~\beta_i$ with $i=1,2,...,g$ are marked by dashed loops, where $\alpha_i$ and $\beta_j$ intersect if and only if $i=j$.
• Figure S2: Numerical phase diagram in the $(J_v/T,J_s/T)$ parameter space for various $\kappa$ obtained from Monte Carlo simulation for the gauge theory \ref{['Z_monopole_suppression']}. $|\mathbf{v}|$ and $s$ are the order parameters for the $U(1)$ and $\mathbb{Z}_2$ symmetries, respectively. The vertical dashed line in the upper left panel denotes the critical value for the 3D XY transition, $J_v/T = 0.45422$, and the horizontal dashed line in the upper right panel denotes that for the 3D Ising transition, $J_s/T = 0.22165$.
• Figure S3: Mean-field phase diagram of the complex $O(2)$ theory Eq. (\ref{['eq:CL']}) with $\lambda=1.5$ and $\Lambda=0.2$ when all the possible gauge fields are deconfined. The gauge structure of each phase is shown explicitly. The 3D XY and the 3D Ising transitions are marked in red and yellow, respectively. The direct transition between the $O(2)$ Coulomb phase and the $D(\mathbb{Z}_n)$ phase, marked in pink, could be in a coexistence of the 3D XY and 3D Ising classes.