Instability of topological order to nucleation of multiple non-Abelian anyon species

Topological quantum computation relies on braiding of non-Abelian anyons to enact fault-tolerant gates. However, nucleating a finite density of anyons—e.g., by imperfect ground-state preparation or environmental perturbations—potentially induces condensation transitions that obliterate the host topological phase. We analyze the stability of topological order to the proliferation of multiple species of non-Abelian anyons that exhibit nontrivial mutual braiding statistics and can fuse to generate parasitic Abelian anyons. We focus on D₄ topological order, recently experimentally realized, and address this question by considering ground-state wavefunctions corrupted by non-unitary operators that nucleate multiple anyon species. An exact mapping between these deformed wavefunctions and two-dimensional local stat-mech models (capturing fusion and braiding statistics) enables characterization of the response of D₄ via Monte Carlo simulations. We find that distinct anyon condensation transitions correspond to the spontaneous breaking of a non-Abelian symmetry and that non-trivial braiding between proliferated anyons can indeed reduce the system’s stability. We characterize the universality and nature of the various transitions, and provide a systematic derivation of the relevant symmetry group using projected entangled pair states. Our analysis sheds light on the stability of D₄ topological order under a general noise model.