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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. 

Incompressible insulating phases of electronic systems at partial filling of a lattice are often associated with charge ordering that breaks lattice symmetry. The resulting phases have an enlarged unit cell with an effective integer filling. Here we explore the possibility of insulating states—which we dub “quantum charge liquids” (QCLs)—at partial lattice filling that preserve lattice translation symmetry. Such QCL phases must necessarily either have gapped fractionally charged excitations and associated topological order or have gapless neutral excitations. We establish some general constraints on gapped fermionic QCL phases that restrict the nature of their topological order. We prove a number of results on the minimal topological order that is consistent with the lattice filling. In particular we show that, at rational fillings 𝜈=𝑝/𝑞 with 𝑞 an even integer, the minimal ground-state degeneracy on a torus of the fermionic QCL is 4⁢𝑞2, four times larger than that of the bosonic QCL at the same filling. We comment on models and physical systems which may host fermionic QCL phases and discuss the phenomenology of these phases. 

Quantum spin liquids are exotic phases of quantum matter especially pertinent to many modern condensed matter systems. Dirac spin liquids (DSLs) are a class of gapless quantum spin liquids that do not have a quasi-particle description and are potentially realized in a wide variety of spin 1 / 2 magnetic systems on 2 d lattices. In particular, the DSL in square lattice spin- 1 / 2 magnets is described at low energies by ( 2 + 1 ) d quantum electrodynamics with N f = 4 flavors of massless Dirac fermions minimally coupled to an emergent U ( 1 ) gauge field. The existence of a relevant, symmetry-allowed monopole perturbation renders the DSL on the square lattice intrinsically unstable. We argue that the DSL describes a stable continuous phase transition within the familiar Neel phase (or within the Valence Bond Solid (VBS) phase). In other words, the DSL is an "unnecessary" quantum critical point within a single phase of matter. Our result offers a novel view of the square lattice DSL in that the critical spin liquid can exist within either the Neel or VBS state itself, and does not require leaving these conventional states. 

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. 

Remarkable recent experiments on the moiré structure formed by pentalayer rhombohedral graphene aligned with a hexagonal boron nitride substrate report the discovery of a zero field fractional quantum Hall effect. These “(fractional) quantum anomalous Hall” [(F)QAH] phases occur for one sign of a perpendicular displacement field, and correspond, experimentally, to full or partial filling of a valley polarized Chern-1 band. Such a band is absent in the noninteracting band structure. Here we show that electron-electron interactions play a crucial role, and present microscopic theoretical calculations demonstrating the emergence of a nearly flat, isolated, Chern-1 band and FQAH phases in this system. We also study the four- and six-layer analogs and identify parameters where a nearly flat isolated Chern-1 band emerges which may be suitable to host FQAH physics. 

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. 

The recent observation of fractional quantum anomalous Hall (FQAH) states in tunable moiré materials encourages study of several new phenomena that may be uniquely accessible in these platforms. Here, we show that an isolated localized anyon of the FQAH state will nucleate a “halo” of charge-density-wave (CDW) order around it. We demonstrate this effect using a recently proposed quantum Ginzburg-Landau theory that describes the interplay between the topological order of the FQAH state and the broken-symmetry order of a CDW. The spatial extent of the CDW order will, in general, be larger than the length scale at which the fractional charge of the anyon is localized. The strength and the decay length of the CDW order around anyons induced by doping or the magnetic field differ qualitatively from those nucleated by a random potential. Our results leverage a precise mathematical analogy to earlier studies of the superfluid-CDW competition of a system of lattice bosons which has been used to interpret the observed CDW halos around vortices in high-Tc superconductors. We show that measurement of these patches of CDW order can give an indirect route to measuring the fractional charge of the anyon. Such a measurement may be possible by scanning tunneling microscopy in moiré systems.