# Hofstadter butterfly: energy spectrum (black) as a function of magnetic flux α = *p*/*q*

**Figure 2.** Hofstadter butterfly: energy spectrum (black) as a function of magnetic flux α = *p*/*q*. We sample 500 different rational flux values α = {1/2, 1/3, ..., 14/46, 15/46}. In (b) and (c) we observe that the system is predominantly metallic. The stripy appearance is due to the fact that we take α to be a rational number. The colour inside the gap denotes the nature of the gapped phase: QSH (blue) and NI (green). Panel (a) is for γ = λ_{x} = 0, (b) for γ = 0.1, λ_{x} = 0.5, and (c) is for γ = 0.25 and λ_{x} = 1.0. Note a QSH phase (blue) that appears at *E _{F}* = 0 for α = 1/6 in panel (c) indicated by a yellow arrow.

**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.