The Hofstadter model, well known for its fractal butterfly spectrum, describes two-dimensional electrons under a perpendicular magnetic field, which gives rise to the integer quantum Hall effect. Inspired by the real-space building blocks of non-Abelian gauge fields from a recent experiment, we introduce and theoretically study two non-Abelian generalizations of the Hofstadter model. Each model describes two pairs of Hofstadter butterflies that are spin–orbit coupled. In contrast to the original Hofstadter model that can be equivalently studied in the Landau and symmetric gauges, the corresponding non-Abelian generalizations exhibit distinct spectra due to the non-commutativity of the gauge fields. We derive the genuine (necessary and sufficient) non-Abelian condition for the two models from the commutativity of their arbitrary loop operators. At zero energy, the models are gapless and host Weyl and Dirac points protected by internal and crystalline symmetries. Double (8-fold), triple (12-fold), and quadrupole (16-fold) Dirac points also emerge, especially under equal hopping phases of the non-Abelian potentials. At other fillings, the gapped phases of the models give rise to topological insulators. We conclude by discussing possible schemes for experimental realization of the models on photonic platforms.

霍夫施塔特模型以其分形蝴蝶谱而闻名，它描述了垂直磁场下的二维电子，从而产生了整数量子霍尔效应。受到最近实验中非阿贝尔规范场的实空间构建块的启发，我们引入并从理论上研究了霍夫施塔特模型的两种非阿贝尔推广。每个模型描述了两对自旋轨道耦合的霍夫施塔特蝴蝶。与可以在朗道规范和对称规范中等效研究的原始霍夫施塔特模型相比，由于规范场的非交换性，相应的非阿贝尔概括表现出不同的谱。我们从两个模型的任意循环算子的交换性中推导出两个模型的真正（必要和充分）非阿贝尔条件。在零能量下，模型是无间隙的，并且具有受内部对称性和晶体对称性保护的外尔点和狄拉克点。双（8 倍）、三倍（12 倍）和四极（16 倍）狄拉克点也会出现，特别是在非阿贝尔势的相等跳跃相位下。在其他填充中，模型的间隙相会产生拓扑绝缘体。最后，我们讨论了在光子平台上实验实现模型的可能方案。