# Intensity plot of the single-particle spectral function A(k_y,omega ) = int _{k_x} A(k_x, k_y, omega ) as a function of ω and *k*_{y} for α = 1/6, λ_{x} = 1.5, γ = 0.25 and different interaction strengths *U*

_{y}

**Figure 5.** Intensity plot of the single-particle spectral function A(k_y,\omega ) = \int _{k_x} A(k_x, k_y, \omega ) as a function of ω and *k _{y}* for α = 1/6, λ

_{x}= 1.5, γ = 0.25 and different interaction strengths

*U*. Left panel is for

*U*= 0.5, where the system is in the normal insulating phase, (middle) is for

*U*= 1.0 where we find a metallic phase, and (right) is for

*U*= 3.0, where the system is in the QSH phase. The spectral function visualizes the topological difference, in (left) no edge state is crossing the bulk gap while in (right) a pair of helical edge states is traversing the bulk gap.

**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.