Continuous topological phase transition between $\mathbb{Z}_2$ topologically ordered phases

Abstract

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 $\mathbb{Z}_2$ topological orders in (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 directly continuous transition between them remains a long-standing problem. Motivated by the fact that both phases can arise from condensing distinct anyons in the $\mathbb{Z}_4$ topological order, we introduce a perturbed $\mathbb{Z}_4$ 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 $\mathbb{Z}_4$ anyons, fractionalized from the $\mathbb{Z}_2$ topological orders, become deconfined at the transition between $\mathbb{Z}_2$ topologically ordered phases. Our results open a path toward developing a theoretical framework for topological phase transitions beyond anyon condensation.