## Non-interacting phase diagram for α = 1/6 and α = 1/10 at half-filling *E*_{F} = 0 as a function of γ and λ_{x}

_{F}

**Figure 3.** Non-interacting phase diagram for α = 1/6 and α = 1/10 at half-filling *E _{F}* = 0 as a function of γ and λ

_{x}. The QSH effect phase appears symmetrically around maximally spin-mixing γ = 0.25. For our finite size calculations, we identify the metallic regions when the 'gap' size is smaller than the level spacing. Region of the QSH effect and gap sizes are larger for α = 1/6 than for α = 1/10.

**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.