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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, and despite numerous predictions of nontrivial topology in these homobilayers, experimental evidence has remained elusive. Here, we conduct local electronic compressibility measurements of twisted bilayer WSe$_2$ 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. Furthermore, by measuring at a variety of local twist angles, we characterize how the interacting ground states of the underlying honeycomb superlattice depend on the size of the 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.

The emergence of moiré 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 moiré bands whose relative energies, and therefore occupations, are tuned by density and magnetic field. At low densities, competition between moiré 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 moiré systems, and they demonstrate an emergent lattice model system in which both interactions and band dispersion can be experimentally controlled.

Nonabelian anyons offer the prospect of storing quantum information in a topological qubit protected from decoherence, with the degree of protection determined by the energy gap separating the topological vacuum from its low lying excitations. Originally proposed to occur in quantum wells in high magnetic fields, experimental systems thought to harbor nonabelian anyons range from p-wave superfluids to superconducting systems with strong spin orbit coupling. However, all of these systems are characterized by small energy gaps, and despite several decades of experimental work, definitive evidence for nonabelian anyons remains elusive. Here, we report the observation of arobust, incompressible even-denominator fractional quantum Hall phase in a new generation of dual-gated, hexagonal boron nitride encapsulated bilayer graphene samples. Numerical simulations suggest that this state is in the Pfaffian phase and hosts nonabelian anyons, and the measured energy gaps are several times larger than those observed in other systems. Moreover, the unique electronic structure of bilayer graphene endows the electron system with two new control parameters. Magnetic field continuously tunes the effective electron interactions, changing the even-denominator gap non-monotonically and consistent with predictions that a transition between the Pfaffian phase and the composite Fermi liquid (CFL) occurs just beyond the experimentally explored magnetic field range. Electric field, meanwhile, tunes crossings between levels from different valleys. By directly measuring the valley polarization, we observe a continuous transition from an incompressible to a compressible phase at half-filling mediated by an unexpected incompressible, yet polarizable, intermediate phase. Valley conservation implies this phase is an electrical insulator with gapless neutral excitations., Considerably expanded and rewritten. New experimental data from two additional devices, including transport data and thermodynamic gap measurements up to 35T. Theoretical model updated to include screening from filled Landau levels. Experimental and numerical methods section greatly expanded. Paper, including Methods and supplementary data totals 27 pages with 20 figures, 3 tables