## Spatial staggering of the magnetization amplitude *S*_{tot}(*x*) as a function of *U* for α = 1/6

**Figure 7.** Spatial staggering of the magnetization amplitude *S*_{tot}(*x*) as a function of *U* for α = 1/6. Staggering is defined by the difference of the maximal and the minimal *S*_{tot} within one period of the spin order. The staggering is indicated by the coloured background in the spin boxes of figure 6. We observe that the staggering decreases for increasing *U*, because the fermions become more and more localized and quantum fluctuations are suppressed. This also occurs for the other values of α that we investigated.

**Abstract**

Motivated by the recent progress in engineering artificial non-Abelian gauge fields for ultracold fermions in optical lattices, we investigate the time-reversal-invariant Hofstadter–Hubbard model. We include an additional staggered lattice potential and an artificial Rashba-type spin–orbit coupling term available in experiment. Without interactions, the system can be either a (semi)-metal, a normal or a topological insulator, and we present the non-Abelian generalization of the Hofstadter butterfly. Using a combination of real-space dynamical mean-field theory (RDMFT), analytical arguments, and Monte-Carlo simulations we study the effect of strong on-site interactions. We determine the interacting phase diagram, and discuss a scenario of an interaction-induced transition from a normal to a topological insulator. At half-filling and large interactions, the system is described by a quantum spin Hamiltonian, which exhibits exotic magnetic order due to the interplay of Rashba-type spin–orbit coupling and the artificial time-reversal-invariant magnetic field term. We determine the magnetic phase diagram: both for the itinerant model using RDMFT and for the corresponding spin model in the classical limit using Monte-Carlo simulations.