Fractional Quantum Anomalous Hall Effect and
Fractional Chern Insulators

The fractional quantum anomalous Hall effect (FQAHE), the analog of the fractional quantum Hall effect at zero magnetic field, is predicted to exist in topological flat bands under spontaneous time-reversal-symmetry breaking . The demonstration of FQAHE could lead to non-Abelian anyons which form the basis of topological quantum computation . In this talk, I will report the observation of integer and fractional QAH effects in a rhombohedral pentalayer graphene/hBN moiré superlattice. At zero magnetic field, we observed plateaus of quantized Hall resistance at filling factors v = 1, 2/3, 3/5, 4/7, 4/9, 3/7 and 2/5 of the moiré superlattice. These features are accompanied by clear dips in the longitudinal resistance Rxx. In addition, we observed signatures of the composite Fermi liquid (CFL) at half-filling. By tuning the gate displacement field D and v, we observed phase transitions from CFL and FQAH states to other correlated electron states. Our graphene system provides an ideal platform for exploring charge fractionalization and (non-Abelian) anyonic braiding at zero magnetic field, especially considering a lateral junction between FQAHE and superconducting regions in the same device.

The emergence of topological moiré flat bands provides exciting opportunities to realize the lattice analogs of both the integer and fractional quantum Hall effects without the need for an external magnetic field. These effects are known as the integer and fractional quantum anomalous Hall (IQAH and FQAH) effects. In this talk, I will mainly present electrical transport studies of moiré superlattices built on 2D transition metal dichalcogenide (TMDc) semiconductors. We have successfully achieved highly tunable topological phases in both TMDc heterobilayer and TMDc homobilayer moiré superlattices. Specifically, we have observed a robust IQAH effect and signatures of quantum spin Hall effect in AB-stacked WSe2/MoTe2. Furthermore, both the IQAH effect and the long-sought FQAH effect have been realized in twisted bilayer MoTe2. The band topology in TMDc moiré superlattices is highly tunable by external electric fields, which enable us to realize novel topological quantum phase transitions. Our studies pave the path for the investigation of fractionally charged excitations and anyonic statistics at zero magnetic field based on 2D moiré materials. [1] F. Xu et al. PRX 13, 031037 (2023). [2] T. Li et al. Nature 600, 641-646 (2021). [3] T. Li et al. Nature 597, 350-354 (2021).

Fractional Chern insulators (FCIs), initially conceptualized as lattice counterparts of fractional quantum Hall (FQH) states in the absence of a magnetic field, stand as a captivating frontier in condensed matter physics, illustrating a complex interplay between robust electron-electron interactions, band topology, and quantum geometry. This presentation provides an overview of selected recent key developments in the realm of FCIs, synthesizing cutting-edge theoretical advancements and remarkable experimental findings. After over a decade of theoretical exploration, engineered moiré superlattice materials have recently emerged as an experimental platform for FCI physics. Milestone experimental results include weak-field FQH states in magic angle twisted bilayer graphene on boron nitride, originating from lattice FCIs rather than continuum Landau level states. Even more spectacular recent observations have identified authentic FCIs exhibiting a (zero-field) fractional quantum anomalous Hall effect in both twisted transition metal dichalcogenides and multilayer graphene. The primary focus of this presentation is to elucidate the distinctions between moiré Chern bands and Landau levels, arising due to symmetry-breaking competing states as a consequence of the underlying quantum geometry in the moiré lattice setting. This platform also opens intriguing avenues for numerous new phenomena beyond Landau level physics, including FCIs in bands with higher Chern numbers, high-temperature FCIs, fractional anomalous Hall crystals combining charge density wave (CDW) order with anyon excitations, non-Abelian genons accompanying topological lattice defects, spin singlet and multicomponent FCIs, as well as a plethora of competing ordered single and multi-band states. This offers an exciting prospect where fundamental theory evolves through direct interaction with experiments, with first principles calculations serving as a bridge linking these two realms.

We show that an electrostatic superlattice, applied to an appropriate 2D material with a Berry curvature hotspot, can be used to engineer topological minibands with highly tunable properties. In particular, we give a general recipe for designing topological flat bands that are favorable for realizing fractional Chern insulators at partial filling.

In this talk, I will first review our observation of FCI states at low magnetic fields in MATBG aligned with hBN enabled by high-resolution local compressibility measurements. Our findings highlight the interplay between symmetry, topology, and interactions leading to a competition between ordered electronic solid phases and fractionalized electronic liquid phases. Next, I will discuss the impact of the pervasive supermoiré pattern in multilayer moiré heterostructures, also present in MATBG aligned with hBN, on the landscape of many-body phases in twisted trilayer graphene.

Twisted bilayer MoTe$_2$ is a promising platform to investigate the interplay between band topology and many-body interaction. In this talk, I will present our theoretical study of its interaction-driven quantum phase diagrams based on a three-orbital model, which can be viewed as a generalization of the Kane-Mele-Hubbard model with one additional orbital and long-range Coulomb repulsion. We predict a cascade of phase transitions tuned by the twist angle $\theta$. At the hole filling factor $\nu=1$ (one hole per moir\'e unit cell), the ground state can be in the multiferroic phase with coexisting spontaneous layer polarization and magnetism, the quantum anomalous Hall phase, and finally the topologically trivial magnetic phases, as $\theta$ increases from $1.5^{\circ}$ to $5^{\circ}$. At $\nu=2$, the ground state can have a second-order phase transition between an antiferromagnetic phase and the quantum spin Hall phase as $\theta$ passes through a critical value. The dependence of the phase boundaries on model parameters such as the gate-to-sample distance, the dielectric constant, and the moir\'e potential amplitude is examined. The predicted phase diagrams can guide the search for topological phases in twisted transition metal dichalcogenide homobilayers.

From recent studies on flat bands in moiré structures, many fascinating novel quantum phenomena have been unearthed or proposed. In this talk, we delve into the expansive question of potential routes towards topological exact flat bands and their classifications. Our study reveals that there exist various potential pathways towards exact topological flat bands, not limited to K-valley 2D materials and not necessarily requiring bilayer or twisting. For instance, we demonstrate that in materials featuring quadratic band touching points, an exact topological flat band can be achieved with only a single-layer 2D material subjected to periodic strains. These exact flat bands showcase novel traits akin to those found in magic-angle twisted-bilayer graphene. Apart from this similarity, these flat bands attain a significantly more uniform Berry curvature distribution, positioning them as a promising platform for the pursuit of novel fractional topological states. Moreover, we will also show that this example is not an isolated occurrence and can be extended to various setups. We additionally present a classification for these topological flat bands, based on their topology, space-group symmetry, and degree of degeneracy.

The interplay between spontaneous symmetry breaking and topology can result in exotic quantum states of matter. A celebrated example is the quantum anomalous Hall (QAH) effect, which exhibits an integer quantum Hall effect at zero magnetic field due to topologically nontrivial bands and intrinsic magnetism. In the presence of strong electron-electron interactions, fractional-QAH (FQAH) effect at zero magnetic field can emerge, which is a lattice analog of fractional quantum Hall effect without Landau level formation. In this talk, I will present experimental observation of FQAH effect in twisted MoTe2 bilayer, using combined magneto-optical and -transport measurements. In addition, we find an anomalous Hall state near the filling factor -1/2, whose behavior resembles that of the composite Fermi liquid phase in the half-filled lowest Landau level of a two-dimensional electron gas at high magnetic field. Direct observation of the FQAH and associated effects paves the way for researching charge fractionalization and anyonic statistics at zero magnetic field.

Recent experiments have revealed the possibility of achieving sizeable Coulomb interaction driven gapped phases in dual gated rhombohedral multilayer graphene devices where carrier densities and perpendicular electric fields can be simultaneously controlled. Of particular importance are the pentalayer [1] and tetralayer devices [2] whose spontaneous gaps of the order of a few tens of meV are attributable to layer pseudospin polarization of the states near the Dirac point in chiral 2DEG systems [3,4,5]. The layer pseudospin polarization can take place in a variety of ways leading to different Hall conductivities depending on the signs of the mass terms for each one of the spin-valley flavors. By means of mean-field Hartree-Fock approach, using both limited range U-V and longer-range Coulomb tails, we examine and compare the self-consistent solutions corresponding to the different pseudospin magnetic phases. Addition of a spin-orbit coupling term or moire potentials are expected to sensitively alter the ground-state of the system, allowing for example to split the folded moire bands such that they are prone to CDW gaps with fractional filling densities [6]. We comment on the energetics involved in the spontaneous spin-valley polarization when we add a finite carrier doping in the presence of perpendicular electric fields in a moire-less system [7].

Correlation and frustration play essential roles in physics, giving rise to novel quantum phases. A typical frustrated system is correlated bosons on moat bands, which could host topological orders with long-range quantum entanglement. However, the realization of moat-band physics is still challenging. Here, we explore moat-band phenomena in shallowly inverted InAs/GaSb quantum wells, where we observe an unconventional time-reversal-symmetry breaking excitonic ground state under imbalanced electron and hole densities. Theoretically, we show that strong frustration from density imbalance leads to a moat band for excitons, resulting in a time-reversal-symmetry breaking excitonic topological order, which explains all our experimental observations. More recently by new experiments we have confirmed the moat band physics in strongly spin-orbit coupled InAs/GaSb quantum wells. These developments open a new direction for realizing bosonic fractional quantum Hall effects in the presence of zero magnetic field. Wang, R., Sedrakyan, T.A., Wang, B.G., Du, L.J. & Du, R.R. Excitonic topological order in imbalanced electron–hole bilayers. Nature 619, 57–62 (2023). https://doi.org/10.1038/s41586-023-06065-w

Fractional Chern insulators arising in lattice models share many aspects with fractional quantum Hall states, but also differ in some respects. We focus our discussion on those aspects, in particular competition and coexistence with other phases. States with conventional Landau order due to symmetry breaking can on one hand suppress topological order by taking over, but can also allow it to exit.

I will analyze recent QAHE/FQAHE breakthrough experiments on moire TMD layers from the perspective of the QHE/FQHE paradigm, and suggest some future directions for research.

Recent experimental observations of the fractional quantum anomalous Hall effect in spin/valley polarized moiré systems call for a more expansive theoretical exploration of strongly correlated physics in partially filled topological bands. In this work we study a state that we refer to as the conjugate-composite Fermi liquid (cCFL), which arises when a pair of Chern bands with opposite Chern numbers are both half-filled. We demonstrate that the cCFL is the parent state of various interesting phenomena. As an example, we demonstrate that with the existence of an inplane spin order, the cCFL could be driven into a quantum bad metal phase, in the sense that it is a metallic phase whose zero temperature longitudinal resistivity is finite, but far greater than the Mott-Ioffe-Regal limit, i.e. ρe ≫ h/e^2. The bad metal phase is also accompanied with a new Wiedemann-Franz law, meaning the thermal conductivity is proportional to the electrical resistivity rather than conductivity. Other proximate phases of the cCFL such as superconductivity and a chiral spin liquid phase can occur when the composite fermions (CF) form the inter-valley CF-exciton condensate.

Moire material has been of intense interest for the range of correlated electron phenomena they exhibit and for their high degree of tenability. Recently, fractional Chern insulators (FCI) have been observed in various moire materials, including twisted TMD and graphene systems, opening promising future route towards experimental realization of non-abelian anyon and universal quantum computation. This talk presents an exact geometric criteria for the stability of FCI in general flatbands and discuss its implications. The common wisdom to find FCI is to engineer material to approach the limit with uniform Berry curvature. This talk will disprove such common lore by showing a new theory (ideal flatband) which emphasizes the fundamental importance of quantum metric. We prove ideal flatbands support exact zero energy ground state FCI for arbitrary amount of Berry curvature fluctuation for all nontrivial Chern number, and prove ideal quantum geometry alone can imply universal wavefunction and exactly closed density algebra. The ideal flatband is motivated by the chiral model of twisted bilayer graphene, but is a much general concept. It has direct implication for recently experimental observed FCI and composite Fermi liquid phases in moire materials.

The experimental discoveries of fractional quantum anomalous Hall effects under zero magnetic fields in both transition metal dichalcogenide and graphene moir\'e superlattices have aroused significant research interest. In this work, we theoretically study the fractional quantum anomalous Hall states (also dubbed as Chern insulator states) in a moir\'e superlattice consisted of pentalayer graphene which is nearly aligned with hexagonal boron nitride (hBN). Including remote-band renormalization effects and self-consistent screening of vertical displacement fields, within the Hartree-Fock framework, we obtain a spin-valley polarized integer Chern insulator state with Chern number $1$ at filling factor of $1$ for displacement field $D\lessapprox\,0.1\,$V/nm, which competes with another Chern-number-zero Hartree-Fock state. Either of the two competing Hartree-Fock states would give rise to an isolated flat band. We then explore the interacting ground states of the two types of topologically distinct isolated flat bands at hole doping levels of 1/3, 2/5, 3/5, and 2/3 (corresponding electron dopings of 2/3, 3/5, 2/5, 1/3) in the parameter space of displacement field $D$ and background dielectric constant $\epsilon_r$. Our exact-diagonlization calculations suggest that the system stays in fractional Chern insulator (FCI) state at 2/3 electron doping when $0.9\,\textrm{V/nm}\leq\!D\!\leq 0.92\,\textrm{V/nm}$ and $5\lessapprox\epsilon_r\lessapprox 6$; while no robust FCI state is obtained at 1/3 electron doping in the experimentally relevant parameter regime ($0.7\,\textrm{V/nm}\leq\!D\!\leq 1.1\,\textrm{V/nm}$ and $4\leq\epsilon_r\leq 8$). These numerical results are quantitatively consistent with experimental observations. We have also obtained composite-fermion type FCI ground states at 3/5 electron filling for $0.8\,\textrm{V/nm}\lessapprox\!D\!\lessapprox 0.9\,\textrm{V/nm}$ and $5\lessapprox\epsilon_r\lessapprox 6$.

This talk will show our recent theoretical and computational investigations into moiré superlattices using machine-learning based approaches. We start by demonstrating that a deep neural network guided by first-principles data can be used to examine moiré structural reconstruction in various homobilayers and heterobilayers of transition metal dichalcogenides. Going beyond the capacity of direct DFT calculations, our machine-learning enabled workflow discovers salient structural features and key topological characters controlled by twist angles, layer composition, and other tuning knobs. This knowledge can be used to inform accurate continuum model, and to predict new forms of moiré potential and moiré topology. Finally, we connect our theoretical discoveries to experimental results and explore potential applications.

The recent experimental observation of the fractional Chern insulator (FCI) in twisted MoTe2 moiré superlattice and pentalayer graphene on hBN has generated tremendous excitement in the community. The condition for realizing the FCI in toy models with topological flat bands in terms of the quantum geometry of the wave functions is fairly well understood theoretically. However, identifying experimentally relevant models for FCI remains a challenge. For instance, why does twisted bilayer graphene not support FCI at zero magnetic field while twisted MoTe2 can? Here, we provide a simple argument to account for the absence of FCI in twisted bilayer graphene at zero magnetic fields in terms of the emergent gauge field. We further give a general condition for stabilizing the FCI in the moiré superlattice. If time permits, we will also discuss electrical current control of topology in the moiré superlattice.