Symmetry-protected topological photonic crystal in three dimensions
Open accessAbstract
Crystal symmetries may protect single Dirac cones on the surface of a photonic crystal, creating a photonic analogue of a three-dimensional solid-state topological insulator. Topology of electron wavefunctions was first introduced to characterize the quantum Hall states in two dimensions discovered in 1980 (ref. 1). Over the past decade, it has been recognized that symmetry plays a crucial role in the classification of topological phases, leading to the broad notion of symmetry-protected topological phases2. As a primary example, topological insulators3,4 are distinguished from normal insulators in the presence of time-reversal symmetry (). A three-dimensional (3D) topological insulator3,4,5,6 exhibits an odd number of protected surface Dirac cones, a unique property that cannot be realized in any 2D systems. Importantly, the existence of topological insulators requires Kramers’ degeneracy in spin–orbit coupled electronic materials; this forbids any direct analogue in boson systems7. In this report, we discover a 3D topological photonic crystal phase hosting a single surface Dirac cone, which is protected by a crystal symmetry8,9,10—the nonsymmorphic glide reflection11,12,13 rather than . Such a gapless surface state is fully robust against random disorder of any type14,15. This bosonic topological band structure is achieved by applying alternating magnetization to gap out the 3D ‘generalized Dirac points’ discovered in the bulk of our crystal. The Z2 bulk invariant is characterized through the evolution of Wannier centres16. Our proposal—readily realizable using ferrimagnetic materials at microwave frequencies17,18—expands the scope of 3D topological materials from fermions to bosons.
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Strong 3D topological insulators have topological surface states that form unpaired Dirac points.
“Strong 3D topological insulators, on the other hand, have topological surface states that form unpaired Dirac points16,28–30”
From Observation of an unpaired photonic Dirac point (2020) · cites this paper as [16]
The dispersion of the type-II Dirac point in the kx-ky plane is distinctive from existing type-I Dirac points.
“The dispersion of the type-II DP in the k x -k y plane is distinctive from the existing type-I DPs 29,31,32 (see Fig. 5d).”
From Type-II Dirac photons (2017) · cites this paper as [31]
The discussion of Fig. 1a compares a three-dimensional generalized Dirac point with a regular three-dimensional Dirac point, while the Fig. 2 caption describes anisotropic surface-cone dispersion in the kx–ky plane. Neither passage identifies a type-II Dirac point or compares its in-plane dispersion with type-I Dirac points. Searches for “type-II Dirac point” and “tilted Dirac cone” found no passage addressing that distinction. (LLM, conf=0.74)
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- Authors
- Lu, L., Fang, C., Fu, L., Johnson, S. G., Joannopoulos, J. D., & Soljačić, M.
- Affiliations
- Chinese Academy of Sciences (CN)
- Massachusetts Institute of Technology (US)
- Corresponding author
- Fu, Liang
- Venue
- Nature Physics
- Published
- 2016-01-04
- Volume
- 12
- Issue
- 4
- Pages
- 337-340
- ISSN
- 1745-2473 (ISSN-L)
- 1745-2481
- OpenAlex
- W787176368
- Semantic Scholar
- b7de72bdffb2e026ba770c64e28202c13be8d741
- Cited by
- 329
- References
- 46
- Citations by year
- 2026: 22 · 2025: 25 · 2024: 22 · 2023: 30 · 2022: 38 · 2021: 28 · 2020: 40 · 2019: 39 · 2018: 28 · 2017: 33 · 2016: 23 · 2015: 1
- FWCI
- 19.39
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Nature Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (0001299, 1419807, W911NF-13)
- U.S. Department of Energy (DESC0010526, DESC0001299)
- Materials Research Science and Engineering Center, Harvard University (DMR1419807)
- Solid-State Solar Thermal Energy Conversion, Massachusetts Institute of Technology (DE‐SC0001299)
- Division of Materials Research (DMR-14-19807, 14-19807)
- Basic Energy Sciences (DE‐SC0001299)
- Army Research Office (W911NF–13–D–0001, W911NF-13, W911NF)
- Division of Materials Sciences and Engineering
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Advanced Condensed Matter Physics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Photonic Crystals and Applications
- Keywords
- topological photonic crystals, three-dimensional topological insulators, symmetry-protected topological phases, surface Dirac cones, Z2 topological invariant, ferrimagnets, microwave photonics
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