Unconventional topological mixed-state transition and critical phase induced by self-dual coherent errors

A topological phase can undergo a phase transition driven by anyon condensation. A potential obstruction to such a mechanism could arise if there exists a symmetry between anyons that have nontrivial mutual statistics. Here we consider the toric code subjected to errors that tend to proliferate anyons with nontrivial mutual statistics. Using the Cauchy-Schwarz inequality, we show that in the presence of electromagnetic duality and a partial-transpose symmetry, a decoherence-induced phase transition out of the topological phase must be rather unconventional and lie beyond the standard rules of anyon condensation. To explore such physics, we first subject the toric code to a self-dual quantum channel where Kraus operators are proportional to $X+Z$. We find that the topological phase is stable up to the maximal error rate, when viewing the density matrix as a pure state in the double Hilbert space. To access an unconventional transition, we then consider a perturbed toric code subjected to the self-dual channel and find numerical evidence that beyond a critical error rate, the topological phase is destroyed, resulting in a critical phase where anyons are only power-law condensed.