Lattice symmetry and emergence of antiferromagnetic quantum Hall states

We consider the spinful Harper-Hofstadter model extended by a next-nearest-neighbor (NNN) hopping which opens a gap at half-filling and allows for the realization of a quantum Hall insulator (QHI). The QHI has the Chern number $\mathcal{C}=2$ as both spin components are in the same quantum Hall state. We add to the system a staggered potential $\Delta$ along the $\hat{x}$-direction favoring a normal insulator (NI) and the Hubbard interaction $U$ favoring a Mott insulator (MI). The MI is a Neel antiferromagnet (AF) for small and a stripe AF for large NNN hopping. We investigate the $U$-$\Delta$ phase diagram of the model for both small and large NNN hoppings. We show that while for large NNN hopping there exists a $\mathcal{C}=1$ stripe antiferromagnetic QHI (AFQHI) in the phase diagram, there is no equivalent $\mathcal{C}=1$ Neel AFQHI at the small NNN hopping. We discuss that a $\mathcal{C}=1$ AFQHI can emerge only if the effect of the spin-flip transformation cannot be compensated by a space group operation.