## Interacting phase diagram at α = 1/6 as a function of interaction *U* and staggering λ_{x} for various values of spin-mixing γ

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**Figure 4.** Interacting phase diagram at α = 1/6 as a function of interaction *U* and staggering λ_{x} for various values of spin-mixing γ. We find the (semi)-metallic regime (Metal), magnetically ordered state (Mag), NI and QSH state.

**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.