Most natural and artificial materials have crystalline structures from which abundant topological phases emerge1,2,3,4,5,6. However, the bulk–edge correspondence—which has been widely used in experiments to determine the band topology from edge properties—is inadequate in discerning various topological crystalline phases7,8,9,10,11,12,13,14,15,16, leading to challenges in the experimental classification of the large family of topological crystalline materials4,5,6. It has been theoretically predicted that disclinations—ubiquitous crystallographic defects—can provide an effective probe of crystalline topology beyond edges17,18,19, but this has not yet been confirmed in experiments. Here we report an experimental demonstration of bulk–disclination correspondence, which manifests as fractional spectral charge and robust bound states at the disclinations. The fractional disclination charge originates from the symmetry-protected bulk charge patterns—a fundamental property of many topological crystalline insulators (TCIs). Furthermore, the robust bound states at disclinations emerge as a secondary, but directly observable, property of TCIs. Using reconfigurable photonic crystals as photonic TCIs with higher-order topology, we observe these hallmark features via pump–probe and near-field detection measurements. It is shown that both the fractional charge and the localized states emerge at the disclination in the TCI phase but vanish in the trivial phase. This experimental demonstration of bulk–disclination correspondence reveals a fundamental phenomenon and a paradigm for exploring topological materials.
Your institute does not have access to this article
Open Access articles citing this article.
Nature Communications Open Access 19 April 2022
Nature Communications Open Access 26 January 2022
Light: Science & Applications Open Access 05 January 2022
Subscribe to Nature+
Get immediate online access to the entire Nature family of 50+ journals
Subscribe to Journal
Get full journal access for 1 year
only $3.90 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Get time limited or full article access on ReadCube.
All prices are NET prices.
The data that support the findings of this study are available from the corresponding authors on reasonable request.
We use the commercial software COMSOL MULTIPHYSICS to perform the electromagnetic simulations and eigenstate calculations. Requests for computation details can be addressed to the corresponding authors.
Fu, L. Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011).
Slager, R.-J., Mesaros, A., Juričić, V. & Zaanen, J. The space group classification of topological band-insulators. Nat. Phys. 9, 98–102 (2013).
Bradlyn, B. et al. Topological quantum chemistry. Nature 547, 298–305 (2017); correction 582, E14 (2020).
Zhang, T. et al. Catalogue of topological electronic materials. Nature 566, 475–479 (2019).
Vergniory, M. G. et al. A complete catalogue of high-quality topological materials. Nature 566, 480–485 (2019); correction 582, E13 (2020).
Tang, F., Po, H. C., Vishwanath, A. & Wan, X. Comprehensive search for topological materials using symmetry indicators. Nature 566, 486–489 (2019).
Po, H. C., Watanabe, H. & Vishwanath, A. Fragile topology and Wannier obstructions. Phys. Rev. Lett. 121, 126402 (2018).
Cano, J. et al. Topology of disconnected elementary band representations. Phys. Rev. Lett. 120, 266401 (2018).
Liu, S., Vishwanath, A. & Khalaf, E. Shift insulators: rotation-protected two-dimensional topological crystalline insulators. Phys. Rev. X 9, 031003 (2019).
Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017).
Langbehn, J., Peng, Y., Trifunovic, L., von Oppen, F. & Brouwer, P. W. Reflection-symmetric second-order topological insulators and superconductors. Phys. Rev. Lett. 119, 246401 (2017).
Song, Z. D., Fang, Z. & Fang, C. (d − 2)-Dimensional edge states of rotation symmetry protected topological states. Phys. Rev. Lett. 119, 246402 (2017).
Schindler, F. et al. Higher-order topological insulators. Sci. Adv. 4, eaat0346 (2018).
van Miert, G. & Ortix, C. Higher-order topological insulators protected by inversion and rotoinversion symmetries. Phys. Rev. B 98, 081110(R) (2018).
Benalcazar, W. A., Li, T. & Hughes, T. L. Quantization of fractional corner charge in Cn-symmetric higher-order topological crystalline insulators. Phys. Rev. B 99, 245151 (2019).
Peterson, C. W., Li, T., Benalcazar, W. A., Hughes, T. L. & Bahl, G. A fractional corner anomaly reveals higher-order topology. Science 368, 1114–1118 (2020).
Rüegg, A. & Lin, C. Bound states of conical singularities in graphene-based topological insulators. Phys. Rev. Lett. 110, 046401 (2013).
Teo, J. C. Y. & Hughes, T. L. Existence of Majorana-fermion bound states on disclinations and the classification of topological crystalline superconductors in two dimensions. Phys. Rev. Lett. 111, 047006 (2013).
Li, T., Zhu, P., Benalcazar, W. A. & Hughes, T. L. Fractional disclination charge in two-dimensional Cn-symmetric topological crystalline insulators. Phys. Rev. B 101, 115115 (2020).
Serra-Garcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342–345 (2018).
Peterson, C. W., Benalcazar, W. A., Hughes, T. L. & Bahl, G. A quantized microwave quadrupole insulator with topological protected corner states. Nature 555, 346–350 (2018).
Imhof, S. et al. Topolectrical circuit realization of topological corner modes. Nat. Phys. 14, 925–929 (2018).
Noh, J. et al. Topological protection of photonic mid-gap defect modes. Nat. Photon. 12, 408–415 (2018).
Xue, H., Yang, Y., Gao, F., Chong, Y. & Zhang, B. Acoustic higher-order topological insulator on a kagome lattice. Nat. Mater. 18, 108–112 (2019).
Ni, X., Weiner, M., Alù, A. & Khanikaev, A. B. Observation of higher-order topological acoustic states protected by generalized chiral symmetry. Nat. Mater. 18, 113–120 (2019).
Zhang, X. et al. Second-order topology and multi-dimensional topological transitions in sonic crystals. Nat. Phys. 15, 582–588 (2019).
Ran, Y., Zhang, Y. & Vishwanath, A. One-dimensional topologically protected modes in topological insulators with lattice dislocations. Nat. Phys. 5, 298–303 (2009).
Juričić, V., Mesaros, A., Slager, R.-J. & Zaanen, J. Universal probes of two-dimensional topological insulators: dislocation and π-flux. Phys. Rev. Lett. 108, 106403 (2012).
Paulose, J., Chen, B. G. & Vitelli, V. Topological modes bound to dislocations in mechanical metamaterials. Nat. Phys. 11, 153–156 (2015).
Li, F.-F. et al. Topological light-trapping on a dislocation. Nat. Commun. 9, 2462 (2018).
van Miert, G. & Ortix, C. Dislocation charges reveal two-dimensional topological crystalline invariants. Phys. Rev. B 97, 201111(R) (2018).
Haldane, F. D. M. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).
Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljacic, M. Observation of unidirectional backscattering immune topological electromagnetic states. Nature 461, 772–775 (2009).
Chen, W.-J. et al. Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide. Nat. Commun. 5, 5782 (2014).
Wu, L.-H. & Hu, X. Scheme for achieving a topological photonic crystal by using dielectric material. Phys. Rev. Lett. 114, 223901 (2015).
Cheng, X. et al. Robust reconfigurable electromagnetic pathways within a photonic topological insulator. Nat. Mater. 15, 542–548 (2016).
Dong, J. W., Chen, X. D., Zhu, H., Wang, Y. & Zhang, X. Valley photonic crystals for control of spin and topology. Nat. Mater. 16, 298–302 (2017).
Barik, S. et al. A topological quantum optics interface. Science 359, 666–668 (2018).
Krasnok, A. E. et al. An antenna model for the Purcell effect. Sci. Rep. 5, 12956 (2015).
Zeng, Y. et al. Electrically pumped topological laser with valley edge modes. Nature 578, 246–250 (2020).
Kraus, J. D. & Marhefka, R. J. Antennas: For All Applications 3rd edn (McGraw-Hill, 2002).
Pozar, D. M. Microwave Engineering 3rd edn (Wiley, 2005).
Y.P., F.-F.L., S.L. and X.T. thank the National Natural Science Foundation of China (NSFC) (61671232 and 61771237), the Project Supported by the Fundamental Research Funds for the Central Universities (14380160 and 14380147) and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. Y.L., Z.-K.L. and J.-H.J. are supported by the Jiangsu Province Specially-Appointed Professor Funding, the National Natural Science Foundation of China (grant number 12074281) and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. J.-H.J. thanks Y. Jing for helpful discussions.
The authors declare no competing interests.
Peer review information Nature thanks Carmine Ortix and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data figures and tables
a, Schematic of the measured region (one-fifth of the whole disclination structure) as illustrated for the TCI disclination structure. The purple dots represent the dielectric pillars. Each unit cell is divided into 24 small triangular regions (illustrated in detail in the 32 unit cell). The blue dots represent the measurement points in the small triangular regions. Each region has one measurement point. b, Schematic of the measurement region for the NI disclination with the same triangular divisions. c, Schematic of the sub-miniature version A (SMA) monopole antenna mounted on the lower plate of the parallel-plates-defined 2D photonic systems. The diameter Φ of the monopole antenna is 1.24 mm. For microwave photons, the aluminium plate acts as a perfect electric conductor (PEC). A hole is drilled on the lower plate with a diameter Φ of 4 mm to insert the probing antenna. We fill the remaining space in the hole with polytetrafluoroetylene (PTFE) (relative permittivity ε = 2.1) to achieve impedance matching with the coaxial cable, which has a characteristic impedance of 50 ohm. The length of the monopole cylindrical antenna is l2 and its diameter is 1.2 mm. The length between the A plane and the B plane is l1 = 6 mm.
a, Schematic of the disclination structure of which a one-fifth sector (with 15 unit cells, enclosed by the dashed lines) is measured and calculated for the LDOS and the spectral charge. Note that both unit cells 21 and 41 are separated into two halves. b–f, Comparison between the experimental and calculated LDOS for the 11 (b), 21 (c), 32 (d), 51 (e) and 53 (f) unit cells for both the TCI and NI disclination structures. In the calculation of the LDOS based on Supplementary Note 4, the Lorentz broadening parameter Γ is set as 30 MHz.
a, Phase profiles from eigenstate calculations. b, Phase profiles from experimental measurements. These results are consistent with the symmetric and antisymmetric wavefunctions with respect to the mirror plane of the setup (Extended Data Fig. 4a).
a, Illustration of the experimental system. A PhC disclination made of Al2O3 pillars (pink) is cladded by parallel metal plates (grey) above and below, and is surrounded by microwave absorption sponges (blue). Inset: top view of the sample. The white dashed line denotes the mirror symmetry plane of the disclination. b, Eigenstates spectrum (top) and the simulated (middle) and measured (bottom) transmission between source A and detector B for the PhC with d = a/2 (no central pillar). Grey regions denote the bulk band regions. c, d, Electric-field profiles of disclination states from eigenstate calculations (c) and measurements (d). The positive and negative signs indicate whether the sources C and D (green stars) have the same sign or opposite signs. e, f, The corresponding phase distributions of the electric-field from calculations (e) and measurements (f).
a, Top view of the NI disclination with 75 unit cells where d = 0.23a for each unit cell. b, Photonic spectrum and transmission for the NI disclination. Top: photonic spectrum from eigenstate calculations. Middle: simulated transmission from the source A to the detector B (positions of A and B are illustrated in Extended Data Fig. 4a). Lower: measured transmission from the source A to the detector B.
About this article
Cite this article
Liu, Y., Leung, S., Li, FF. et al. Bulk–disclination correspondence in topological crystalline insulators. Nature 589, 381–385 (2021). https://doi.org/10.1038/s41586-020-03125-3
Light: Science & Applications (2022)
Nature Communications (2022)
Nature Physics (2022)
Nature Communications (2022)
Acta Mechanica Sinica (2022)