Magnetic order of the itinerant model at <em>U</em> = 5 as a function of γ ∈ [0, 0.25] for different values of α = 1/3, 1/6, 1/4, 1/5 (from top to bottom) obtained within RDMFT

<p><strong>Figure 6.</strong> Magnetic order of the itinerant model at <em>U</em> = 5 as a function of γ ∈ [0, 0.25] for different values of α = 1/3, 1/6, 1/4, 1/5 (from top to bottom) obtained within RDMFT. We generally find Néel order around γ = 0 and collinear order around γ = 0.25. In between for 0.05 γ 0.20, we observe different kinds of spiral order with periodicities of three, four, six, eight, and ten sites due to a competition of a Dzyaloshinskii–Moriya (DM) and an anisotropic XXZ Néel type spin interaction, see equation (<a href="http://iopscience.iop.org/0953-4075/46/13/134004/article#jpb458197eqn04" target="_blank">4</a>). Phase boundaries are indicated by solid vertical lines. Black dots denote the actual values of γ that we simulated using RDMFT. The magnetization diagrams show the direction of magnetic order in <em>S<sub>y</sub></em>–<em>S<sub>z</sub></em> plane. The background colour indicates the magnetization amplitude <em>S</em><sub>tot</sub>(<em>x</em>). The magnetization diagrams are extracted from the specific parameter point that is indicated by the dashed circle. If, due to quantum fluctuations, the classical spin Hamiltonian exhibits a different phase boundary than the itinerant model, we denote the phase boundary of the spin Hamiltonian by a vertical dashed line with the label <em>H</em><sub>spin</sub>.</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>