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Cited by 1,971Topological Materials and PhenomenaMetamaterials and Metasurfaces ApplicationsQuantum Mechanics and Non-Hermitian Physics

Photonic topological insulators

Open access
Khanikaev, Alexander B., Hossein Mousavi, S., Tse, Wang-Kong, Kargarian, Mehdi, MacDonald, Allan H., Shvets, Gennady·2013·Nature Materials
DOI 10.1038/nmat3520
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Abstract

Recent progress in understanding the topological properties of condensed matter has led to the discovery of time-reversal-invariant topological insulators. A remarkable and useful property of these materials is that they support unidirectional spin-polarized propagation at their surfaces. Unfortunately topological insulators are rare among solid-state materials. Using suitably designed electromagnetic media (metamaterials) we theoretically demonstrate a photonic analogue of a topological insulator. We show that metacrystals-superlattices of metamaterials with judiciously designed properties-provide a platform for designing topologically non-trivial photonic states, similar to those that have been identified for condensed-matter topological insulators. The interfaces of the metacrystals support helical edge states that exhibit spin-polarized one-way propagation of photons, robust against disorder. Our results demonstrate the possibility of attaining one-way photon transport without application of external magnetic fields or breaking of time-reversal symmetry. Such spin-polarized one-way transport enables exotic spin-cloaked photon sources that do not obscure each other.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper proposes that a two-dimensional bi-anisotropic metamaterial "photonic meta-crystal" can realize a photonic analogue of a time-reversal-invariant topological insulator, with a photonic quantum spin Hall phase supporting helical one-way edge states. Numerical calculations show these spin-polarized edge modes are robust against bends, cavities, and strong lattice disorder.

Key findings

  • Adding an antisymmetric magneto-electric (bi-anisotropy) tensor χ with all other elements zero lifts the TE/TM Dirac degeneracy at the K and K′ points, opening a complete photonic band gap; the effective low-energy Hamiltonian maps onto the Kane–Mele graphene spin-orbit Hamiltonian.
  • The doubly degenerate Dirac bands acquire opposite Chern numbers of ±1 (from equal ±1/2 contributions at K and K′), giving a quantized spin Chern number C_s = 2 and a Z₂ invariant of 1, while the net Chern number vanishes because time-reversal symmetry is preserved.
  • A domain wall between regions of opposite bi-anisotropy sign (χ > 0 and χ < 0) supports a pair of spin-up/spin-down edge states with equal energy and equal field amplitudes, self-rotating in opposite senses and propagating in opposite directions.
  • Edge states at an interface between the topological and a trivial metamaterial (lattice constant scaled by 0.76, matching a trivial band gap of Δ ~ 0.30–0.33 [2π/a]) are transmitted without reflection through sharp zigzag bends, around cavities, and past strongly disordered regions, for both spin-up and spin-down modes.

Methods

  • Design: a hexagonal-lattice photonic meta-crystal of ε/μ-matched uniaxial rods (ε = μ = 14 for rods of radius 0.34a; ε = μ = 1 background), with bi-anisotropy introduced to couple TE and TM polarizations.
  • Analysis: plane-wave expansion for band structure, a six-plane-wave k·p truncation to derive an effective 8-component Hamiltonian, and numerical Chern-number/Berry-connection integration over the Brillouin zone.
  • Edge-state simulations: first-principle finite-element method (FEM) on a 30×1 unit-cell supercell with periodic boundary conditions, plus harmonic propagation problems with selective spin-up/spin-down excitation and modeled defects (bends, cavities, random rod displacements).

Limitations

  • The paper states that a slight ε–μ mismatch can be tolerated only for sufficiently large ε, and that robust edge states survive only as long as the bi-anisotropy gap exceeds the TE/TM spectral detuning; it also notes that high ε and μ values diminish the bi-anisotropy effect (gap width ∝ χ̃/εμ), motivating separation of magnetic and bi-anisotropic elements.
  • Evident from the text: all results are theoretical/numerical (plane-wave expansion and FEM); no experimental demonstration is reported, and the proposed split-ring implementation is described only as a design for the microwave range.
  • Evident from the text: the topological phase is demonstrated for one specific antisymmetric bi-anisotropy form (χ_xy = −χ_yx, all other elements zero), and the authors note a similar phase "should also be observable" in a wider class of bianisotropic systems with quadratic degeneracies, which is not demonstrated here.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1038/nmat3520

Cites

48 of 48 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 17 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § Photonic Analogue of Two-dimensional Topological Insulators and Helical One-Way Edge Transport in Bi-Anisotropic Metamaterials · p. 1

    The helical edge states enable one-way photonic transport that is robust against disorder.

    “and that these edge states enable one-way photonic transport that is robust against disorder.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § 2. Photonic topological insulator · p. 4

    The band structure around the K and K' points is identical because of time-reversal symmetry of the system.

    “Note that the band structure around the K and K points is identical because of time-reversal symmetry of the system.”
    Quote in the paperNot a substantial claim
  • Findingown data
    § 2. Photonic topological insulator · p. 4

    A pair of degenerate Dirac points corresponding to TE and TM polarizations appears for the second and third doubly degenerate bands crossing at the K point.

    “In addition, a pair of degenerate Dirac points corresponding to TE and TM polarizations appears for the second and third doubly degenerate bands crossing at the K point (Figs. 1a and 1b).”
    Quote in the paperSubstantial claim
  • Findingown data
    § 2. Photonic topological insulator · p. 4

    In the optically passive (xi = 0) crystal, the band structure contains two identical overlaid spectra corresponding to degenerate TE and TM modes.

    “One can see that the band structure contains two identical overlaid spectra corresponding to the degenerate TE and TM modes, so that any linear combination of these modes is also a solution of the eigenvalue problem.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The electric permittivity and magnetic permeability of all constituents of the structure are matched to restore photon polarization degeneracy.
  • Findingreasoning
    § 2. Photonic topological insulator · p. 4

    The TE-TM degeneracy at the Dirac point is lifted by the bi-anisotropy, producing a photonic band gap.

    “The TETM degeneracy at the Dirac point will be lifted thus producing a photonic band gap.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The bi-anisotropy tensor is taken to be anti-symmetric with xi_xy = -xi_yx not equal to 0 and all other elements equal to 0.
  • May combine two claims. 11
  • The quote only partly states this claim. 1
  • Classified as probably not a substantial claim. 3

  • May combine two claims.

    Findingnone given
    § 5. Summary · p. 10

    These edge states are reciprocal (bi-directional), are related to each other by time-reversal symmetry, and are robust against different types of defects and disorder.

    “These states are reciprocal (bi-directional) and are related to each other by time-reversal symmetry and robust against different types of defects and disorder.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 3. Edge states · p. 7

    The edge modes exist regardless of the configuration of the interface for any separations and for both commensurate and incommensurate junctions of the crystals.

    “Moreover, it was found that these modes exist regardless of the configuration of the interface for any separations and for both commensurate and incommensurate junctions of the crystals.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 3. Edge states · p. 6

    In close proximity to the K-point, the pair of spin-up and spin-down edge states respect time-reversal symmetry with the same energy and the same field amplitudes.

    “In close proximity to the K -point, the pair of spin-up ( ) and spin-down ( ) edge states are found to respect time-reversal symmetry with the same energy ( || ) || and the same field amplitudes | ( )| | ( )|”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 5. Summary · p. 10

    Topologically protected spin-polarized one-way edge states exist at interfaces formed between metamaterials with reversed bi-anisotropy and at interfaces between topologically trivial and nontrivial structures.

    “Topologically protected spin-polarized one-way edge states were shown to exist at interfaces formed between metamaterials with reversed bi-anisotropy and at interfaces between topologically trivial and nontrivial structures.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 3. Edge states · p. 7

    A forward propagating spin-up edge state is perfectly transmitted by both sharp corners of a zigzag-shaped interface regardless of stacking configurations, with no reflection into the backward-propagating spin-down edge mode.

    “As one can see from the top subplot in Fig. 2b, the forward propagating spin-up edge state is perfectly transmitted by both sharp corners of the zigzag-shaped interface regardless of the different stacking configurations at every segment along the zigzag.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 4. Possible experimental implementation · p. 8

    A slight epsilon/mu mismatch can be tolerated for sufficiently large xi, and robust edge states survive as long as the gap due to the bi-anisotropy exceeds the spectral detuning between the TE and TM bands caused by a finite epsilon/mu mismatch.

    “We note that according to our simulations, a slight -mismatch can be tolerated for sufficiently large , and robust edge states survive as long as the gap due to the bi-anisotropy exceeds the spectral detuning between the TE and TM bands caused by a finite value of .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 2. Photonic topological insulator · p. 6

    The system exhibits a topologically nontrivial phase that is the photonic analogue of the quantum spin Hall effect, characterized by a quantized spin Chern number C_s = 2 and a Z2 invariant of 1.

    “the system exhibits a topologically nontrivial phase which is the photonic analogue of the quantum spin Hall effect [ 43], characterized by a quantized spin Chern number [ 44] = ( ) = 2 and a Z 2 invariant [ 1, 5] of 1.”
    Quote in the paperSubstantial claim0 of 4 sources checked
    • cites [43] Gansel, J. K. et al. Gold helix photonic metamaterial as broadband circular polarizer. Science 325, 1513–1515 (2009). source

      Not checked in the free run.

    • cites [44] Saenz, E. et al. Modeling of spirals with equal dielectric, magnetic, and chiral susceptibilities. Electromagnetics 28, 476–493 (2008). source

      Not checked in the free run.

    • cites [1] Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005). source

      Not checked in the free run.

    • cites [5] Roy, R. Z2 classification of quantum spin Hall systems: An approach using time-reversal invariance. Phys. Rev. B 79, 195321 (2009). source

      Not checked in the free run.

  • May combine two claims. The quote only partly states this claim.

    Findingown data
    § 3. Edge states · p. 7

    An edge mode demonstrates complete tolerance to strong local distortion of the crystalline lattice introduced by randomly shifting rods and avoids back-scattering by detouring around the disordered region.

    “However, even in this case the edge mode demonstrates complete tolerance and avoids back-scattering by detouring around the disordered region.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 1. Introduction · p. 1

    Enhanced spin-orbit coupling can be achieved in bi-anisotropic materials with magneto-electric coupling.

    “More generally, enhanced spin-orbit coupling can be achieved in bi-anisotropic materials with magneto-electric coupling [ 28, 29].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [28] Zharov, A. A., Shadrivov, I. V. & Kivshar, Y. S. Nonlinear properties of left-handed metamaterials. Phys. Rev. Lett. 91, 037401 (2003). source

      Not checked in the free run.

    • cites [29] Klein, M. W., Enkrich, C., Wegener, M. & Linden, S. Second-harmonic generation from magnetic metamaterials. Science 313, 502–504 (2006). source

      Not checked in the free run.

  • May combine two claims.

    Interpretationown data
    § 2. Photonic topological insulator · p. 5

    The proposed photonic meta-crystal preserves time-reversal symmetry and realizes a quantum spin Hall response with spin-polarized one-way photonic edge transport, unlike earlier photonic crystal structures that break time-reversal symmetry.

    “In contrast, our photonic meta-crystal structure preserves time-reversal symmetry and realizes quantum spin Hall response with spin-polarized one-way photonic edge transport.”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [13] Haldane, F. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008). source

      Not checked in the free run.

    • cites [14] Wang, Z., Chong, Y., Joannopoulos, J. & Soljaćić, M. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys. Rev. Lett. 100, 013905 (2008). source

      Not checked in the free run.

    • cites [16] Smith, D. R., Padilla, W. J., Vier, D. C., Nemat-Nasser, S. C. & Schultz, S. Composite medium with simultaneously negative permeability and permittivity. Phys. Rev. Lett. 84, 4184–4187 (2000). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationcitation
    § 4. Possible experimental implementation · p. 9

    A topological phase similar to that reported here should also be observable in a wider class of bianisotropic systems with quadratic degeneracies.

    “Finally we mention that a topological phase similar to that reported here should also be observable in a wider class of bianisotropic systems with quadratic degeneracies [ 14, 46].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [14] Wang, Z., Chong, Y., Joannopoulos, J. & Soljaćić, M. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys. Rev. Lett. 100, 013905 (2008). source

      Not checked in the free run.

    • cites [46] Haldane, F. D. M. Model for a quantum Hall effect without landau levels: Condensed-matter realization of the parity anomaly. Phys. Rev. Lett. 61, 2015–2018 (1988). source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § 3. Edge states · p. 6

    The edge states are time-reversed partners that self-rotate in opposite senses and propagate along the edge in opposite directions.

    “This confirms that the edge states are indeed time-reversed partners that satisfy ( ) = ( ) , self-rotate in opposite senses, and propogate along the edge in opposite directions.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given
    § 5. Summary · p. 10

    The proposed system can be implemented without magneto-optical or gyromagnetic materials, which are invariably lossy.

    “Our proposed system is particularly attractive because it can be implemented without magneto-optical or gyromagnetic materials which are invariably lossy.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationown data
    § 3. Edge states · p. 7

    The edge modes exhibit a high degree of tolerance to local configurations of the interface, confirming their topological origin.

    “Thus the modes exhibit a high degree of tolerance to local configurations of the interface, confirming their topological origin.”
    Quote in the paperSubstantial claim
14 of 14

  • The authors state this as an assumption.

    The electric permittivity and magnetic permeability of all constituents of the structure are matched to restore photon polarization degeneracy.

    “To restore the photon polarization degeneracy, we assume matched values of the electric permittivity and magnetic permeability of all the structure's constituents.”

  • The authors state this as an assumption.

    The bi-anisotropic constitutive relations are treated to first order in the bi-anisotropy tensor.

    “To first order in ̂ , these constitutive relations can be rewritten using Maxwell's equations in reciprocal space to obtain a new set of relations”

  • The authors state this as an assumption.

    The bi-anisotropy tensor is taken to be anti-symmetric with xi_xy = -xi_yx not equal to 0 and all other elements equal to 0.

    “introduce an anti-symmetric magneto-electric tensor = ≠ 0 with all other elements =0”
3 of 3

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • The first observation of topological protection in a photonic system demonstrated lack of scattering in a microwave magneto-optic photonic crystal with a strong external static magnetic field.

    “The first observation of topological protection in a photonic system 12 demonstrated the lack of scattering in a microwave, magneto-optic photonic crystal with a strong, external, static magnetic field, similarly to that prescribed by Raghu and Haldane 9 .”

    From Photonic Floquet topological insulators (2013) · cites this paper as [9]

  • Raghu and Haldane proposed that gyromagnetic materials could realize a photonic analogue of the quantum Hall effect and topological protection of light.

    “In the domain of optics, Raghu and Handane proposed in 2008 that by using gyromagnetic materials, a photonic analogue of the quantum Hall effect could be realized and topological protection of light could be achieved 9 .”

    From Photonic Floquet topological insulators (2013) · cites this paper as [9]

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. Serdyukov, A. N., Semchenko, I. V., Tretyakov, S. A. & Sihvola, A. Electromagnetics of Bi-Anisotropic Materials: Theory and Applications (Gordon and Breach Science, 2001).

Cited by

Sorted by citations

Authors
Khanikaev, A. B., Hossein Mousavi, S., Tse, W.-K., Kargarian, M., MacDonald, A. H., & Shvets, G.
Affiliations
The University of Texas at Austin (US)
Corresponding author
  • Khanikaev, Alexander B.
  • Shvets, Gennady
ORCID
Venue
Nature Materials
Publisher
Springer Science and Business Media LLC
Published
2012-12-16
Volume
12
Issue
3
Pages
233-239
ISSN
  • 1476-1122 (ISSN-L)
  • 1476-4660
DOI
10.1038/nmat3520
OpenAlex
W2053033894
Semantic Scholar
67ef8663d6b59756b8b4b45bf9f4014146e57797
Cited by
1,971
References
48
Citations by year
2026: 98 · 2025: 163 · 2024: 171 · 2023: 183 · 2022: 185 · 2021: 183 · 2020: 183 · 2019: 197 · 2018: 187 · 2017: 140 · 2016: 115 · 2015: 97 · 2014: 34 · 2013: 34 · 2012: 1
FWCI
27.48
Type
journal article
Language
en
Open access
Licence
Locations
  • Nature Materials (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
  • Physical Sciences › Materials Science › Electronic, Optical and Magnetic Materials › Metamaterials and Metasurfaces Applications
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Non-Hermitian Physics
Keywords
photonic topological insulators, metamaterials, metacrystals, helical edge states, time-reversal symmetry, superlattices, disorder robustness

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