Fig. 3: Quantised orbital Chern insulator at ν = 2 and possible Chern bands.

a, b Landau fan diagram in longitudinal resistance R_xx and transverse resistance R_xy plotted as a function of ν for different B up to 10 T. c Fitting of the Diophantine equation along the minima in R_xx and wedge-like states in R_xy. The slopes of the straight lines give the Chern numbers C. d R_xy data at B = 10 T plotted for a small density range near ν = 2 marked by the blue colourbar on the top axis in panel 3c. R_xy is quantised to h/2e² accompanied by a minima in R_xx at B = 10 T. e Schematic of flat bands with different Chern numbers via valley polarising \({{{{{{{\mathcal{T}}}}}}}}\)-symmetry breaking (TRS) and proximity-induced SOC. The dark and light colours represent each spin-valley flavour’s lower and upper bands, respectively. In the presence of SOC, the spin-valley flavour degeneracy can be completely lifted, leading to 8 isolated bands for appropriate system parameters. Blue(green) and red(yellow) indicate the up and down spin components for valley K(\({K}^{{\prime} }\)). The net Chern number C_net obtained by adding the Chern numbers of the filled bands at ν = 2 (horizontal black dashed line) is +2 in both cases. f Schematic of valley Chern bands giving rise to C_net = + 2 at ν = + 2. The band degeneracy can be lifted completely in the presence of SOC and a spin splitting exchange field that naturally accompanies a ferromagnetic spin polarised phase. We have used the SOC parameters for graphene on WSe₂ following ref. ⁴³ together with the exchange field λ_ex = 0.5meV as summarised in the methods section. The degeneracy split Chern bands lead to finite orbital magnetism M_(orb) that changes with the filling density ν, shown here from −4 to 4 in the right-most sub-panel. We note the changing signs in the total orbital magnetisation due to the relative filling of K↓(\({K}^{{\prime} }\uparrow\)) bands near ν = + 2, suggesting that delicate changes in level ordering with carrier density due to Coulomb interactions can strongly impact the net magnetisation.