Extended quantum anomalous Hall effect in moiré structures: Phase transitions and transport

Recent experiments on multilayer rhombohedral graphene have unearthed a number of interesting phenomena in the regime where integer and fractional quantum anomalous Hall phenomena were previously reported. Specifically, at low temperature ($T$) and low applied currents, an ``extended'' integer quantum anomalous Hall (EIQAH) state is seen over a wide range of the phase diagram. As the current is increased, at low $T$, the EIQAH state undergoes a phase transition to a metallic state at generic fillings, and to the fractional quantum anomalous Hall (FQAH) state at the Jain fillings. Increasing temperature at the Jain fillings also leads to an evolution out of the EIQAH state to the Jain state. Here we provide an interpretation of many of these observations. We describe the EIQAH state as a crystalline state (either of holes doped into the $\ensuremath{\nu}=1$ state or an anomalous Hall crystal of electrons) that breaks moir\'e translation symmetry. At generic fillings, we show how an electric current-induced depinning transition of the crystalline order leads to peculiar nonlinear current-voltage curves consistent with the experiment. At Jain fillings, we propose that the depinning transition is preempted by an equilibrium transition between EIQAH and Jain FQAH states. This transition occurs due to the large polarizability of the Jain FQAH states, which enables them to effectively lower their energy in an applied electric field compared to the crystal states. We also discuss the finite-temperature evolution in terms of the relative entropies of the crystalline and FQAH states.