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

TL;DR

This paper introduces two non-Abelian generalizations of the Hofstadter model, revealing complex spectra with Weyl, Dirac points, and topological phases, and discusses potential experimental realizations.

Contribution

The paper develops and analyzes two novel non-Abelian Hofstadter models with unique spectral properties and topological phases, extending the understanding of gauge field effects in quantum systems.

Findings

Models exhibit distinct spectra due to non-commutative gauge fields.

Presence of Weyl, Dirac points, and topological insulator phases.

Potential for experimental realization in photonic systems.

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.