Fig. 4: Properties of the gap functions.
From: Unconventional self-similar Hofstadter superconductivity from repulsive interactions
a Gap functions at the VHS obtained from the RG analysis for q = 2 at perfect nesting (left) and for q = 3 at dph = 0.8 in the top and bottom Hofstadter bands (right). In both cases the gap function changes sign between the two VHSs v = 0, 1. Here we focus on pairing with zero total momentum ℓ = 0, with pairings for ℓ ≠ 0 determined by MTG symmetries. b Real-space structure of the gap function for q = 2 even under \({\hat{T}}_{1}\) and \({\hat{T}}_{2}\) and odd under \({\hat{C}}_{4}\), shown within a single magnetic unit cell (the pattern repeats in all cells). c Profile of the gap function \({{{\Delta }}}_{{R}_{x}\hat{{{{{{{{\bf{x}}}}}}}}},s;0,{s}^{{\prime} }}\) for q = 3 as a function of the horizontal magnetic unit cell separation Rx between Cooper pairs (with lattice constant a = 1). Note that the gap function oscillates between each unit cell and decays as \(1/{R}_{x}^{2}\) at long distances. See the Supplementary Material for more details. d The projection onto the Fermi surface of the gap function for q = 2 shown in b as a function of the angle θp along the Fermi surface within the rMBZ (note that \({{{\Delta }}}_{m}^{(\ell )}\) are equal within each patch m). Note that the gap crosses zero, indicating nodes in the fermionic spectrum. e, f The projection onto the Fermi surface of the model gap function for q = 3 for the top (red) and bottom (blue) bands that agrees with the gap function found in the RG analysis (color online). Note that the magnitude of the gap function never vanishes as shown in e, implying that the fermionic spectrum is fully gapped (the sharp features at θp = 0, π are due to the corners of the Fermi surface). The phase of the projected gap functions, however, winds by ± 4π around the Fermi surface in the top and bottom bands respectively, as shown in f, implying each \({{{\Delta }}}_{m}^{(\ell )}\) contributes ± 2 to the Chern number. Plots (c–e) are given in arbitrary units as the magnitude of the gap function is not determined within the weak-coupling theory.
