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

Quantum double model provides a systematic way to realize a large class of non-Abelian topological ordered phases with a finite non-Abelian group as the input. We show that a typical type of quantum double phases constructed from diheral groups can be unified by a topological BF theory with gauge group S[O(2)×O(2)], which naturally arises from Higgsing a parent O(2) gauge theory. By comparing the anyon statistics encoded in modular S and T matrices, we show that Dijkgraaf-Witten twists on top of these quantum double phases can be also captured by the BF theory, which consequently gives rise to a one-to-one classification of twisted dihedral quantum double phases. We further construct lattice models to realize the Higgsing picture, where the dihedral quantum double phase indeed shows up on the phase diagram. Based on a renormalization group analysis, we argue that the dihedral quantum double phase can undergo a direct transition to a gapless phase with deconfined U(1) gauge field through a multi-critical point with emergent O(3) symmetry. This multi-critical point resembles the concept of deconfined quantum criticalilty in literatures.