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

Yi Yang; Bo Zhen; John D. Joannopoulos; Marin Soljačić

doi:10.1038/s41377-020-00384-7

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

Original Articles

- Non-Abelian generalizations of the Hofstadter model: spin–orbit-coupled butterfly pairs
- Non-Abelian generalizations of the Hofstadter model: spin–orbit-coupled butterfly pairs
- LSA 2020年9卷第6期页码：1810-1822
- Affiliations：
  
  1.Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
  2.Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
- Author bio：
  
  Yi Yang (yiy@mit.edu)
- Funds：
- DOI：10.1038/s41377-020-00384-7
  中图分类号：
- 纸质出版日期：2020，
  
  网络出版日期：2020-10-19，
  
  收稿日期：2020-03-04，
  
  修回日期：2020-07-28，
  
  录用日期：2020-08-02
- Accepted：
Scan QR Code
Non-Abelian generalizations of the Hofstadter model: spin–orbit-coupled butterfly pairs[J]. LSA, 2020,9(6):1810-1822.

Yang, Y. et al. Metasurface interferometry toward quantum sensors. Light: Science & Applications, 9, 1810-1822 (2020).
Non-Abelian generalizations of the Hofstadter model: spin–orbit-coupled butterfly pairs[J]. LSA, 2020,9(6):1810-1822. DOI： 10.1038/s41377-020-00384-7.

Yang, Y. et al. Metasurface interferometry toward quantum sensors. Light: Science & Applications, 9, 1810-1822 (2020). DOI： 10.1038/s41377-020-00384-7.

摘要

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.

关键词

Keywords

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