Thermodynamics and phase transitions in the fractional quantum anomalous hall effect in pentalayer graphene

The fractional quantum anomalous hall effect (FQAHE) was observed in a pentalayer rhombohedral graphene-hBN moire superlattice, with plateaus in Rxy and corresponding dips in Rxx at moire filling factors of 2/5, 3/7, 4/9, 5/11, 5/9, 4/7, 3/5, and 2/3. Additionally, a device demonstrating the FQAHE also shows signatures of the extended quantum anomalous Hall (EQAH) states where over a wide range of electron density Rxy is quantized at h/e^2 along with vanishing Rxx. We investigate how Rxx and Rxy evolve when tuning the displacement field at the fractional fillings of the moire. The Rxx vs. Rxy plots are strongly reminiscent of the semi-circle law of the ordinary FQHE and demonstrate phase transitions between a FQAH state or a composite Fermi liquid, the EQAH state, and a Fermi liquid. The D-parameterized semicircle-like curves and their temperature dependence are modeled well by a three-phase effective medium approximation. Furthermore, we use thermal activation to extract the gap sizes of the FQAH states at both zero magnetic field and at a large external magnetic field and find qualitative similarity between zero and finite field.