Magnetic order of the itinerant model at U = 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
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Figure 6. Magnetic order of the itinerant model at U = 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 (4). 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 Sy–Sz plane. The background colour indicates the magnetization amplitude Stot(x). 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 Hspin.
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.