Energy spectrum for a cylinder geometry as a function of <em>k<sub>y</sub></em>

<p><strong>Figure 1.</strong> Energy spectrum for a cylinder geometry as a function of <em>k<sub>y</sub></em>. Bulk spectrum is shown in blue, edge states in red. Artificial magnetic flux is α = 1/10, and γ and λ<sub><em>x</em></sub> take values as indicated. Doubly degenerate helical edge states (red) are traversing the bulk gaps.</p> <p><strong>Abstract</strong></p> <p>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.</p>