Non-Abelian Generalizations of the Hofstadter model: Spin-orbit-coupled Butterfly Pairs

Abstract

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 [Science, 365, 1021 (2019)], 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 ${\mathbb{Z}}_2$ topological insulators. We conclude by discussing possible schemes for the experimental realizations of the models in photonic platforms.

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• Non-Abelian braiding on photonic chips
• Non-Abelian lattice gauge fields in the photonic synthetic frequency dimension
• Artificial non-Abelian lattice gauge fields for photons in the synthetic frequency dimension
• Synthetic non-Abelian gauge fields for non-Hermitian systems
• Chiral Dynamics of Ultracold Atoms under a Tunable SU(2) Synthetic Gauge Field
• Non-Abelian generalization of non-Hermitian quasicrystal: PT-symmetry breaking, localization, entanglement and topological transitions
• Higher-Order Topological Insulators via Momentum-Space Nonsymmorphic Symmetries
• Chiral limits and effect of light on the Hofstadter butterfly in twisted bilayer graphene
• Synthetic non-Abelian Electric Fields and Spin-Orbit Coupling in Photonic Synthetic Dimensions
• Implementing non-Abelian Hatano-Nelson model in electric circuits
• Simulating Higher-Order Topological Insulators in Density Wave Insulators
• Three-dimensional non-Abelian generalizations of the Hofstadter model: spin-orbit-coupled butterfly trios