Fig. 1.

(A) Schematic depiction of the predicted energy gap to $2e$ excitations (red) and twice the energy of $1e$ excitations (green) at half filling of the Hofstadter–Hubbard model with ${\mathrm{\Phi }}_{▵}=\pi /2$ flux per triangle as a function of interaction strength U in the 2D limit. Proposed phase diagram in the plane of U and chemical potential μ. The star indicates a topological quantum phase transition from an integer quantum Hall (IQH) insulator to chiral spin liquid (CSL) at half filling. The dashed lines show the chemical potential required to add carriers and produce a chiral superconductor (SC) with topologically protected edge modes. (B) Finite region of the two-dimensional model with ${\mathrm{\Phi }}_{▵}=\pi /2$ flux per triangle, repulsive on-site interactions U, and Bravais vectors ${\mathit{a}}_{1,2}$ indicated. The hoppings may be chosen to be $C_6$-invariant and imaginary (each arrow indicates an amplitude $-it$), with a four-site unit cell (sublattices A-D in red). (C) Left: band structure at ${\mathrm{\Phi }}_{▵}=\pi /2$ consisting of two bands related by particle–hole symmetry $\mathcal{P}$. Right: two-dimensional color map of energy as a function of momentum for the upper energy band over the Brillouin zone of the $2\times 2$ unit cell (note the bands are two-fold degenerate at every momentum).