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

We theoretically introduce and study a three-dimensional Hofstadter model with linearly varying non-Abelian gauge potentials along all three dimensions. The model can be interpreted as spin-orbit coupling among a trio of Hofstadter butterfly pairs since each Cartesian surface (xy, yz, or zx) of the model reduces to a two-dimensional non-Abelian Hofstadter problem. By evaluating the commutativity among arbitrary loop operators around all axes, we derive its genuine (necessary and sufficient) non-Abelian condition, namely, that at least two out of the three hopping phases should be neither 0 nor pi. Under different choices of gauge fields in either the Abelian or the non-Abelian regime, both weak and strong topological insulating phases are identified in the model.

Automated summary

In this theoretical study, a three-dimensional Hofstadter model with linearly varying non-Abelian gauge potentials along all dimensions is introduced and analyzed. The model can be interpreted as spin-orbit coupling among a trio of Hofstadter butterfly pairs. It is found that under different choices of gauge fields, both weak and strong topological insulating phases can be identified in the model.