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

TL;DR

This paper introduces a 3D non-Abelian Hofstadter model with spin-orbit coupling, revealing complex topological phases and a necessary condition for non-Abelian behavior based on gauge phases.

Contribution

It presents a novel 3D non-Abelian Hofstadter model with a derived non-Abelian condition and identifies various topological insulating phases.

Findings

Identification of weak and strong topological insulators in the model

Derivation of a necessary and sufficient non-Abelian condition

Reduction of 3D problem to coupled 2D non-Abelian Hofstadter problems

Abstract

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, 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.