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Motivated by the recent discovery of fractional quantum anomalous Hall states in moir\'e systems, we consider the possibility of realizing non-Abelian phases in topological minibands. We study a family of moir\'e systems, skyrmion Chern band (SCB) models, which can be realized in two-dimensional semiconductor/magnetic skyrmion heterostructures and also capture the essence of twisted transition metal dichalcogenide (TMD) homobilayers. We show using many-body exact diagonalization that, in spite of strong Berry curvature variations in momentum space, the non-Abelian Moore-Read state can be realized at half filling of the second miniband. These results demonstrate the feasibility of non-Abelian fractionalization in moir\'e systems without Landau levels and shed light on the desirable conditions for their realization. In particular, we highlight the prospect of realizing the Moore-Read state in twisted semiconductor bilayers., Comment: V1: 5 pages, 4 figures. V2: 5 pages, 5 figures. Added exact diagonalization results for twisted MoTe2. V3: 5 pages, 5 figures (main) + 13 pages, 9 figures (SM). Added results on Pfaffian vs anti-Pfaffian

We show that topological flat minibands can be engineered in a class of narrow gap semiconductor films using only an external electrostatic superlattice potential. We demonstrate that, for realistic material parameters, these bands are capable of hosting correlated topological phases such as integer and fractional quantum anomalous Hall states and composite Fermi liquid phases at zero magnetic field. Our results provide a path towards the realization of fractionalized topological states in a broad range of materials.

Semiconductor moir\'e superlattices provide a versatile platform to engineer new quantum solids composed of artificial atoms on moir\'e sites. Previous studies have mostly focused on the simplest correlated quantum solid - the Fermi-Hubbard model - where intra-atom interactions are simplified to a single onsite repulsion energy U. These studies have revealed novel quantum phases ranging from Mott insulators to quantum anomalous Hall insulators at a filling of one electron per moir\'e unit cell. New types of quantum solids should arise at even higher filling factors where the multi-electron configuration of moir\'e artificial atoms provides new degrees of freedom. Here we report the experimental observation of Wigner molecular crystals emerging from multi-electron artificial atoms in twisted bilayer WS2 moir\'e superlattices. Moir\'e artificial atoms, unlike natural atoms, can host qualitatively different electron states due to the interplay between quantized energy levels and Coulomb interactions. Using scanning tunneling microscopy (STM), we demonstrate that Wigner molecules appear in multi-electron artificial atoms when Coulomb interactions dominate. Three-electron Wigner molecules, for example, are seen to exhibit a characteristic trimer pattern. The array of Wigner molecules observed in a moir\'e superlattice comprises a new crystalline phase of electrons: the Wigner molecular crystal. We show that these Wigner molecular crystals are highly tunable through mechanical strain, moir\'e period, and carrier charge type. Our study presents new opportunities for exploring quantum phenomena in moir\'e quantum solids composed of multi-electron artificial atoms.

The fractional quantum anomalous Hall effect (FQAHE), the analog of the fractional quantum Hall effect1 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. So far, FQAHE has been observed only in twisted MoTe2 (t-MoTe2) at moire filling factor v > 1/2. Graphene-based moire superlattices are believed to host FQAHE with the potential advantage of superior material quality and higher electron mobility. Here we report the observation of integer and fractional QAH effects in a rhombohedral pentalayer graphene/hBN moire superlattice. At zero magnetic field, we observed plateaus of quantized Hall resistance Rxy = h/(ve^2) at filling factors v = 1, 2/3, 3/5, 4/7, 4/9, 3/7 and 2/5 of the moire superlattice respectively. These features are accompanied by clear dips in the longitudinal resistance Rxx. In addition, at zero magnetic field, Rxy equals 2h/e^2 at v = 1/2 and varies linearly with the filling factor-similar to the composite Fermi liquid (CFL) in the half-filled lowest Landau level at high magnetic fields. 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., Comment: Nature, in press

Remarkable recent experiments have observed fractional quantum anomalous Hall (FQAH) effects at zero field and unusually high temperatures in twisted semiconductor bilayer $t$MoTe$_2$. Intriguing observations in these experiments such as the absence of integer Hall effects at twist angles where a fractional Hall effect is observed, do however remain unexplained. The experimental phase diagram as a function of twist angle remains to be established. By comprehensive numerical study, we show that band mixing has large qualitative and quantitative effects on the energetics of competing states and their energy gaps throughout the twist angle range $\theta\leq 4^\circ$. This lays the ground for the detailed realistic study of a rich variety of strongly correlated twisted semiconductor multilayers and an understanding of the phase diagram of these fascinating systems., Comment: 5 + 2 pages, 7 Figures

Recent experiments on the twisted semiconductor bilayer system $t$MoTe$_2$ have observed integer and fractional quantum anomalous Hall effects, which occur in topological moir\'e bands at zero magnetic field. Here, we present a global phase diagram of $t$MoTe$_2$ throughout the filling range $0< n\leq 1$ substantiated by exact diagonalization calculations. At a magic angle, we find that the system resembles the lowest Landau level (LLL) to a remarkable degree, exhibiting an abundance of incompressible fractional quantum anomalous Hall states and compressible anomalous composite Fermi liquid states. Away from the magic angle, particle-hole symmetry is strongly broken. Some LLL-like features remain robust near half-filling, while others are replaced, predominantly by charge density waves near $n=0$ and anomalous Hall Fermi liquids near $n=1$. Among LLL-like phases, we find the anomalous composite Fermi liquid at $n=\frac{1}{2}$ to be most robust against deviations from the magic angle. Within the band-projected model, we show that strong particle-hole asymmetry above the magic angle results from interaction-enhanced quasiparticle dispersion near $n=1$. Our work sets the stage for future exploration of LLL-like and beyond-LLL phases in fractional quantum anomalous Hall systems., Comment: New numerical results and supplementary material added. Accepted to Phys. Rev. B. 8 + 8 pages, 7 + 9 figures

Recent experiments have produced evidence for fractional quantum anomalous Hall (FQAH) states at zero magnetic field in the semiconductor moir\'e superlattice system $t$MoTe$_2$. Here we argue that a composite fermion description, already a unifying framework for the phenomenology of 2d electron gases at high magnetic fields, provides a similarly powerful perspective in this new context. To this end, we present exact diagonalization evidence for composite Fermi liquid states at zero magnetic field in $t$MoTe$_2$ at fillings $n=\frac{1}{2}$ and $n=\frac{3}{4}$. We dub these non-Fermi liquid metals anomalous composite Fermi liquids (ACFLs), and we argue that they play a central organizing role in the FQAH phase diagram. We proceed to develop a long wavelength theory for this ACFL state that offers concrete experimental predictions upon doping the composite Fermi sea, including a Jain sequence of FQAH states and a new type of commensurability oscillations originating from the superlattice potential intrinsic to the system., Comment: Phys. Rev. Lett. journal version. Editors' Suggestion and Featured in Physics. main: 5 pages, 3 figures. SM: 5 pages, 3 figures

We demonstrate via exact diagonalization that AA-stacked TMD homobilayers host fractional quantum anomalous Hall (FQAH) states with fractionally quantized Hall conductance at fractional fillings $n=\frac{1}{3},\, \frac{2}{3}$ and zero magnetic field. While both states are most robust at angles near $\theta\approx 2^{\circ}$, the $n=\frac{1}{3}$ state gives way to a charge density wave with increasing twist angle whereas the $n=\frac{2}{3}$ state survives across a much broader range of twist angles. We show that the competition between FQAH states and charge density wave or metallic phases is primarily controlled by the wavefunctions and dispersion of the underlying Chern band, respectively. Additionally, Ising ferromagnetism is found across a broad range of fillings where the system is insulating or metallic alike. The spin gap is enhanced at filling fractions where integer and fractional quantum anomalous Hall states are formed., Comment: Journal version (PRB, Editor's suggestion). Supplementary Material added. Main: 10 pages, 9 figures. SM: 8 pages, 5 figures

Semiconductor moir\'e superlattices have been shown to host a wide array of interaction-driven ground states. However, twisted homobilayers have been difficult to study in the limit of large moir\'e wavelength, where interactions are most dominant. Here, we conduct local electronic compressibility measurements of twisted bilayer WSe$_2$ (tWSe$_2$) at small twist angles. We demonstrate multiple topological bands which host a series of Chern insulators at zero magnetic field near a 'magic angle' around $1.23^\circ$. Using a locally applied electric field, we induce a topological quantum phase transition at one hole per moir\'e unit cell. Our work establishes the topological phase diagram of a generalized Kane-Mele-Hubbard model in tWSe$_2$, demonstrating a tunable platform for strongly correlated topological phases., Comment: This is the author's version of the work. It is posted here by permission of the AAAS for personal use, not for redistribution. The definitive version was published in Science 384 (doi below)

Moir\'e engineering in atomically thin van der Waals heterostructures creates artificial quantum materials with designer properties. We solve the many-body problem of interacting electrons confined to a moir\'e superlattice potential minimum (the moir\'e atom) using a 2D fermionic neural network. We show that strong Coulomb interactions in combination with the anisotropic moir\'e potential lead to striking ``Wigner molecule" charge density distributions observable with scanning tunneling microscopy.

Semiconductor moir\'e superlattices comprise an array of artificial atoms and provide a highly tunable platform for exploring novel electronic phases. We introduce a theoretical framework for studying moir\'e quantum matter that treats intra-moir\'e-atom interactions exactly and is controlled in the limit of large moir\'e period. We reveal an abundance of new physics arising from strong electron interactions when there are multiple electrons within a moir\'e unit cell. In particular, at filling factor $n=3$, the Coulomb interaction within each three-electron moir\'e atom leads to a three-lobed ``Wigner molecule''. When their size is comparable to the moir\'e period, the Wigner molecules form an emergent Kagome lattice. Our work identifies two universal length scales characterizing the kinetic and interaction energies in moir\'e materials and demonstrates a rich phase diagram due to their interplay., Comment: Accepted by PRL. main 5 pages, 4 figures; supplement 12 pages, 4 figures

The emergence of moir\'e materials with flat bands provides a platform to systematically investigate and precisely control correlated electronic phases. Here, we report local electronic compressibility measurements of a twisted WSe$_2$/MoSe$_2$ heterobilayer which reveal a rich phase diagram of interpenetrating Hofstadter states and electron solids. We show that this reflects the presence of both flat and dispersive moir\'e bands whose relative energies, and therefore occupations, are tuned by density and magnetic field. At low densities, competition between moir\'e bands leads to a transition from commensurate arrangements of singlets at doubly occupied sites to triplet configurations at high fields. Hofstadter states (i.e., Chern insulators) are generally favored at high densities as dispersive bands are populated, but are suppressed by an intervening region of reentrant charge-ordered states in which holes originating from multiple bands cooperatively crystallize. Our results reveal the key microscopic ingredients that favor distinct correlated ground states in semiconductor moir\'e systems, and they demonstrate an emergent lattice model system in which both interactions and band dispersion can be experimentally controlled.