Figure 1: Band geometry and gap data for the Haldane model.

(a) Honeycomb lattice used to define the Haldane model. Basis vectors a₁, a₂ are shown in red; basis elements are shown by differently coloured/numbered sites. Hopping elements are shown with black and blue edges; arrowheads indicate the chirality convention for complex hoppings. (b) Band geometry over the reciprocal lattice unit cell spanned by b₁, b₂, for parameter values maximizing the gap and minimizing σ_B. Axes of ellipses are proportional to the eigenvectors of the quantum metric g^α(k), and ellipse color is given by the relative deviation of Berry curvature B_α(k) from its Brillouin zone-averaged value. (c) Gap Δ as a function of (φ, M) for N=8 bosons at v=1/2 with an on-site repulsion. (d) Gap as a function of (φ, M) for N=8 fermions at v=1/3 with nearest-neighbour repulsion. Note that this plot differs from Fig. 8 of ref. 32 because we exclude (light grey) parameters failing to meet energetic and entanglement-based criteria for Laughlin-type order. (e) Berry curvature fluctuations σ_B (left scale) and metric fluctuations σ_g and average trace inequality (right scale) as functions of φ at M=0. (f) Reproduction of the gap data from c,d along M=0.