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We theoretically introduce and study a three-dimensional Hofstadter modelwith linearly varying non-Abelian gauge potentials along all three dimensions.The model can be interpreted as spin-orbit coupling among a trio of Hofstadterbutterfly pairs since each Cartesian surface ($xy$, $yz$, or $zx$) of the modelreduces to a two-dimensional non-Abelian Hofstadter problem. By evaluating thecommutativity among arbitrary loop operators around all axes, we derive itsgenuine (necessary and sufficient) non-Abelian condition, namely, at least twoout of the three hopping phases should be neither 0 nor $\pi$. Under differentchoices of gauge fields in either the Abelian or the non-Abelian regime, bothweak and strong topological insulating phases are identified in the model.

Three-dimensional non-Abelian generalizations of the Hofstadter model: spin-orbit-coupled butterfly trios