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Cited by 33Topological Materials and PhenomenaQuantum and electron transport phenomenaQuantum Mechanics and Non-Hermitian Physics

Non-Abelian generalizations of the Hofstadter model: spin–orbit-coupled butterfly pairs

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Yang, Yi, Zhen, Bo, Joannopoulos, John D., Soljačić, Marin·2020·Light: Science & Applications
DOI 10.1038/s41377-020-00384-7PMC PMC7572376
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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, 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.

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  • The one-dimensional lattice with long-range couplings in Fig. 2(a) is equivalent to a two-dimensional lattice with twisted boundary conditions.

    “The onedimensional lattice with long-range couplings in Fig. 2(a) is equivalent to a two-dimensional lattice with twisted boundary conditions as shown in Fig. 2(b) [52,53].”

    From Non-Abelian lattice gauge fields in photonic synthetic frequency dimensions (2025) · cites this paper as [52]

    The Introduction describes a two-dimensional Hofstadter lattice, and Materials and methods describes square-lattice models with nearest-neighbor coupling; neither addresses the claimed equivalence. Searches for one-dimensional long-range couplings and twisted boundary conditions found no passage establishing that mapping in our copy. No page numbers were available. (LLM, conf=0.74)

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References 86 · Cited by 33

Research works printed but not linked: 46

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Authors
Yang, Y., Zhen, B., Joannopoulos, J. D., & Soljačić, M.
Affiliations
  • Massachusetts Institute of Technology (US)
  • University of Pennsylvania (US)
Corresponding author
Yang, Yi
ORCID
Venue
Light: Science & Applications
Publisher
Springer Science and Business Media LLC
Published
2020-10-19
Volume
9
Issue
1
Pages
177-177
ISSN
2047-7538 (ISSN-L)
DOI
10.1038/s41377-020-00384-7
PMC
PMC7572376
OpenAlex
W3027886207
Cited by
33
References
86
Citations by year
2026: 6 · 2025: 7 · 2024: 6 · 2023: 4 · 2022: 6 · 2021: 3 · 2020: 1
FWCI
1.42
Type
journal article
Language
en
Update notices
correction (2021-04-07)
Open access
Licence
Locations
  • Light Science & Applications (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • DSpace@MIT (Massachusetts Institute of Technology) (submittedVersion, cc-by-nc)
  • Figshare (submittedVersion, cc-by)
  • arXiv (Cornell University)
Funders
  • National Science Foundation (1640012, 1838412)
  • Pittsburgh Foundation
  • Charles E. Kaufman Foundation
  • Multidisciplinary University Research Initiative (N00014-20-1-2325)
  • Office of Naval Research (N00014‐20‐1‐2325, FA9550, N00014-20, N00014)
  • Division of Materials Research (DMR-1838412)
  • Air Force Office of Scientific Research (N00014-20-1-2325, FA9550-, FA9550-20- 1-0115, FA9550‐18‐1‐0133, FA9550-20-1, FA9550-20)
  • Army Research Office (W911NF, W911NF1820048)
  • Air Force Research Laboratory (FA9550)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and electron transport phenomena
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Non-Hermitian Physics
Keywords
Hofstadter model, Yang-Mills fields, spin-orbit coupling, Hofstadter butterfly, loop operators, Dirac point, Weyl points, topological insulators, photonic lattices, integer quantum Hall effect

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