Decoherence represents the process that connects the quantum realm and the classical realm. Decoherence can also be viewed as a quantum system being weakly-measured by the environment. In this talk we will investigate the archetypal invertible topological order, the Chern insulator, and its behaviors under decoherence. We demonstrate that the fidelity between the decohered Chern insulator mixed density matrix and a trivial state is mapped to the partition function of a conformal field theory, and the nature of the conformal field theory strongly relies on the decoherence channel, which can be extracted through finite size scaling. We will also investigate the effect of weak-measurement on systems with higher form symmetries. We demonstrate that a state which spontaneously breaks the higher form symmetry can be altered through weak-measurement, which can potentially drive a phase transition that significantly weakens the spontaneous symmetry breaking. Intriguingly, this phase transition also enjoys a line of fixed points with self-duality.
