Fig. 1: Van Hove singularities and relevant interactions in the square Hofstadter model.
From: Unconventional self-similar Hofstadter superconductivity from repulsive interactions
VHSs are shown at a zero, c π, and d 2π/3-flux, and e–g the corresponding peaks in the density of states at indicated fillings. Due to the MTG symmetry, the magnetic Brillouin zone (MBZ) splits into q (energy degenerate) reduced magnetic Brillouin zones (rMBZ) labeled with ℓ = 0, …, q − 1. In each band there are a total of 2q VHSs occurring at momenta \({{{{{{{{\bf{K}}}}}}}}}_{\ell,{{{{{{{\rm{v}}}}}}}}}=\left((1+{{{{{{{\rm{v}}}}}}}})\frac{\pi }{q},({{{{{{{\rm{v}}}}}}}}+2p\ell )\frac{\pi }{q}\right)\), such that there is a pair of VHSs in each rMBZ labeled with a VHS index v = 0, ± 1, with the identification of VHS ℓ, 1 and ℓ + 1, − 1. Arrows in a and the Feynman diagrams in b show the types of interaction processes considered in the RG analysis: intra-VHS processes g1 and \({g}_{{1}^{{\prime} }}\) (red and light red); inter-VHS forward scattering g2 (blue); exchange g3 (magenta); and pair-hopping g4. The VHS index color-coded in a and b as red/green/blue for v = 1, 0, − 1, respectively, and the black diagram shows the additional rMBZ indices ℓ, m, n = 0, …, q − 1 carried by the coupling constants \({g}_{mn}^{(\ell )}\), ℓ denoting the total momentum of the interacting pair.
