Topological order subject to local errors I: a systematic study of error-induced phases

Topologically ordered ground states can serve as a quantum memory that is robust against local errors. The encoded information can be protected using a specific quantum error correction algorithm up to a finite error threshold. In this talk, we show the mixed state describing topologically ordered states corrupted by local errors can exhibit distinct phases characterized by its capability to encode information. Remarkably, these error-induced phases cannot be detected by observables in a single-copy density matrix and instead are only probed by nonlinear functions of the density matrix. To characterize such phases, we introduce an errorfield double formalism that identifies the density matrix with a pure state in the double Hilbert space. We further use the path-integral formulation of this pure state and map the error-induced phases to (1+1)D boundary phases of a topologically ordered system. In the concrete examples of the Abelian topological order, the Toric code and the double semion model, subject to incoherent errors, we provide a systematic study of the possible phases.