From explicit to spontaneous charge order and the fate of the antiferromagnetic quantum Hall state

The antiferromagnetic quantum Hall insulator (AFQHI), where one of the spin components is in the quantum Hall state and the other in the trivial state, is an established phase emerging as a result of the Hubbard repulsion in spinful quantum Hall systems. The stabilization of the AFQHI requires a charge order preventing the effect of the spin-flip transformation on the electronic state to be compensated by a space-group operation and is often induced via an ionic potential. While one would naively expect the nearest-neighbor (NN) density-density interaction favoring spontaneous charge order to result in qualitatively similar phenomena, an analysis of the Haldane-Hubbard model extended by the NN interaction finds no AFQHI. Here, by considering an extended version of the Harper-Hofstadter-Hubbard model, we go beyond the honeycomb structure and suggest that the realization of the AFQHI generically requires an explicit charge order and cannot be emerged through spontaneous charge order. We unveil how the AFQHI disappears upon approaching from the explicit to the spontaneous charge ordering limit. Our findings shine more light on the stabilization conditions of the AFQHI which can guide the future optical lattice experiments searching for this intriguing magnetic topological insulator phase.