Continuous topological phase transition between Z 2 topologically ordered phases

Topological phase transitions beyond anyon condensation remain poorly understood. A notable example is the transition between the toric code (TC) and double semion (DS) phases, which has two distinct {Z}₂ topological orders in (2+1) dimensions [ (2+1)D ]. Previous studies reveal that the transition between them can be either first order or via an intermediate phase, thus the existence of a direct continuous transition between them remains a long-standing problem. Motivated by the fact that both phases can arise from condensing distinct anyons in the {Z}₄ topological order, we introduce a perturbed {Z}₄ quantum double (QD) model to study the TC-DS transition. We confirm the existence of a continuous (2+1)D XY* transition between the TC and DS phases by mapping it to a two-coupled quantum Ising model. Importantly, using the condensation order parameters and the area law coefficients of the Wilson loops, we further reveal that {Z}₄ anyons, fractionalized from the {Z}₂ topological orders, become deconfined at the transition between {Z}₂ topologically ordered phases. Our results open a path toward developing a theoretical framework for topological phase transitions beyond anyon condensation.