Collapse of the Hofstadter Butterfly from Long-Range Interactions

Experimental advances in moiré systems allow for the realization of large lattice constants with large magnetic fluxes per unit cell, facilitating the observation of the Hofstadter Butterfly. In addition to Chern bands predicted by the Hofstadter model, correlated states, such as symmetry-broken Chern insulators and in-field fractional Chern insulators have been observed in twisted bilayer graphene and related systems under nonzero magnetic fields.

Motivated by these findings, we study the Hofstadter model on a square lattice with exactly solvable, but long-range, Hatsugai-Kohmoto (HK) interactions. We incorporate Peierls-like phases to preserve the gauge invariance of HK interaction—an aspect overlooked in previous studies. In the strong-coupling limit we demonstrate that the non-locality of HK interactions leads to the collapse of all gapped IQHE and QAHE phases, leaving the half-filled Mott insulator as the only gapped phase. However, at intermediate interaction strengths, we observe interaction-driven gapped phases, including quantum Hall ferromagnetic states with odd Chern numbers, partially consistent with prior results on the Hofstadter-Hubbard model.