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We study novel itinerant phases that can be accessed by doping a fractional quantum anomalous Hall (FQAH) insulator, with a focus on the experimentally observed Jain states at lattice filling 𝜈 =𝑝/(2⁢𝑝 +1). Unlike in the lowest Landau level, where charge motion is confined into cyclotron orbits, the charged excitations in the FQAH occupy Bloch states with well-defined crystal momenta. At a nonzero doping density, this feature enables the formation of itinerant states of the doped anyons just beyond the FQAH plateau region. Focusing on the vicinity of 𝜈 =2/3, we describe a few possible itinerant states, including a topological superconductor with chiral neutral fermion edge modes as well as a more exotic pair density wave (PDW) superconductor with non-Abelian topological order. A Fermi liquid metal with a doping-induced period-3 charge density wave also occurs naturally in our analysis. This Fermi liquid (as well as the PDW) arises from pairing instabilities of a composite Fermi liquid metal that can emerge near filling 2/3. Though inspired by the theory of anyon superconductivity, we explain how our construction is qualitatively different. At a general Jain filling 𝜈 =𝑝/(2⁢𝑝 +1), the same analytical framework leads to a wider variety of phases, including higher-charge superconductors and generalized composite Fermi liquids. We predict unusual physical signatures associated with each phase and analyze the crossover between different temperature regimes. These results provide a proof-of-principle that exotic itinerant phases can be stabilized by correlations intrinsic to the FQAH setup. 

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 𝜈=1 state or an anomalous Hall crystal of electrons) that breaks moiré 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. 

Recent experiments showing an integer quantum anomalous Hall effect in pentalayer rhombohedral graphene have been interpreted in terms of a valley-polarized interaction-induced Chern band. The resulting many-body state can be viewed as an anomalous Hall crystal (AHC), with a further coupling to a weak moiré potential. We explain the origin of the Chern band and the corresponding AHC in the pentalayer system. To describe the competition between AHC and Wigner crystal (WC) phases, we propose a simplified low-energy description that predicts the Hartree-Fock phase diagram to good accuracy. This theory can be fruitfully viewed as “superconducting ring” in momentum space, where the emergence of Chern number is analogous to the flux quantization in a Little-Parks experiment. We discuss the possible role of the moiré potential, and emphasize that even if in the moiréless limit, the AHC is not favored (beyond Hartree-Fock) over a correlated Fermi liquid, the moiré potential will push the system into a “moiré-enabled AHC”. We also suggest that there is a range of alignment angles between R5G and hBN where a 𝐶=2 insulator may be found at integer filling.