Three-dimensional non-Abelian generalizations of the Hofstadter model: Spin-orbit-coupled butterfly trios
Open accessAbstract
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, that at least two out of the three hopping phases should be neither 0 nor $\ensuremath{\pi}$. 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.
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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 [53]
The Introduction describes a three-dimensional cubic-lattice model whose Cartesian surfaces reduce to two-dimensional models, not an equivalence between a one-dimensional long-range lattice and a two-dimensional lattice. The source paper’s Fig. 2 caption describes band structures and inversion eigenvalues; searches for long-range couplings and twisted boundary conditions found no passage addressing the claimed equivalence. Page numbers were unavailable in our copy. (LLM, conf=0.74)
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References 35 · Cited by 4
Research works printed but not linked: 31
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- Authors
- Liu, V., Yang, Y., Joannopoulos, J. D., & Soljačić, M.
- Affiliations
- Massachusetts Institute of Technology (US)
- Research Laboratory of Electronics and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
- Venue
- Physical Review B
- Publisher
- American Physical Society (APS)
- Published
- 2021-09-15
- Volume
- 104
- Issue
- 11
- ISSN
- 2469-9950 (ISSN-L)
- 2469-9969
- OpenAlex
- W3174887448
- Cited by
- 4
- References
- 35
- Citations by year
- 2026: 1 · 2025: 1 · 2023: 1 · 2022: 1
- FWCI
- 0.22
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Physical review. B./Physical review. B (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- DSpace@MIT (Massachusetts Institute of Technology) (submittedVersion, cc-by-nc)
- Figshare (submittedVersion, cc-by)
- arXiv (Cornell University)
- Funders
- Office of Naval Research (N00014-20-1-2325)
- Army Research Office (W911NF-18-2-0048)
- 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 › Condensed Matter Physics › Advanced Condensed Matter Physics
- Keywords
- Hofstadter model, non-Abelian gauge potentials, spin-orbit coupling, Yang-Mills fields, loop operators, topological insulators, weak topological insulator, strong topological insulator
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