Anyon delocalization transitions out of a disordered FQAH insulator

Motivated by the experimental discovery of the fractional quantum anomalous Hall (FQAH) effect, we develop a theory of doping-induced transitions out of the $\nu = 2/3$ lattice Jain state in the presence of quenched disorder. We show that disorder strongly affects the evolution into the conducting phases described in our previous work. The delocalization of charge $2/3$ anyons leads to a chiral topological superconductor through a direct second order transition for a smooth random potential with long-wavelength modulations. The longitudinal resistance has a universal peak at the associated quantum critical point. Close to the transition, we show that the superconducting ground state is an ``Anomalous Vortex Glass (AVG)'' stabilized in the absence of an external magnetic field. For short-wavelength disorder, this transition generically splits into three distinct ones with intermediate insulating topological phases. If instead, the charge $1/3$ anyon delocalizes, then at low doping the result is a Reentrant Integer Quantum Hall state with $\rho_{xy} = h/e^2$. At higher doping this undergoes a second transition to a Fermi liquid metal. We show that this framework provides a plausible explanation for the complex phase diagram recently observed in twisted MoTe$_2$ near $\nu = 2/3$ and discuss future experiments that can test our theory in more detail.