GoKawiila-c, Self-consistent HF band eigenvalues along the high symmetry line K t – Г – M in the moiré Brillouin zone. The HF self-consistency is reached on a 15 × 15 k-mesh while band eigenvalues are evaluated and plotted on a finer k-mesh. For all the HF calculations, U ξ = 24 meV and k B T HF = 0.0026 meV. The dashed horizontal line indicates the Fermi energy. All the plotted bands are included in the HF self-consistency. ∆ U is the potential difference between adjacent graphene layers. (a) v = 1 ∆ U = 13 meV: the occupied active HF band is polarized to the flavor (valley η, spin s) = (+, ↑) and has Chern number C = 4. (b) v = 1 ∆ U = 16 meV: the occupied active HF band is (+, ↑)-polarized with C = 3. (c) v = 3 ∆ U = 11 meV: the 3 occupied active HF bands are partially polarized toward (+, ↑), which can be decomposed into an unpolarized two-band background plus one (+, ↑)-polarized band, yielding C = 4. d-f, Total Berry curvature Ω(k) summed over the occupied active HF bands, and the data are extracted from the same HF calculations of (a-c), respectively. Note that, depending on the initial conditions, the HF iteration can converge to other spin/valley-polarized solutions with identical energies, indicating spontaneous breaking of time-reversal symmetry and spin SU(2) symmetry.
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