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Bulk–disclination correspondence in topological crystalline insulators

Abstract

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.

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Fig. 1: Disclination as a bulk probe of crystalline topology.
Fig. 2: Photonic TCI and disclination.
Fig. 3: Bulk–disclination correspondence.
Fig. 4: Measurement of the disclination states.

Data availability

The data that support the findings of this study are available from the corresponding authors on reasonable request.

Code availability

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.

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Acknowledgements

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.

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Authors

Contributions

J.-H.J. initiated the project. J.-H.J. and Y.P. guided the research. J.-H.J. and Y.L. established the theory. Y.L. performed the numerical calculations and simulations. Z.-K.L. and Y.P. were involved in some of the theory and simulations, respectively. F.-F.L., S.L., X.T., J.-H.J. and Y.P. designed and achieved the experimental setup and the measurements. All the authors contributed to the discussions of the results and the manuscript preparation. J.-H.J., Y.P. and Y.L. wrote the manuscript and the Supplementary Information.

Corresponding authors

Correspondence to Yin Poo or Jian-Hua Jiang.

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The authors declare no competing interests.

Additional information

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

Extended Data Fig. 1 Experimental setup for the LDOS measurements.

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.

Extended Data Fig. 2 Experimental and calculated LDOS for the TCI and NI disclination structures.

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. bf, 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.

Extended Data Fig. 3 Measured phase profiles of the electric fields of disclination states.

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).

Extended Data Fig. 4 Measurements of the disclination states without the PRBs.

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).

Extended Data Fig. 5 No disclination state in the NI disclination.

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.

Supplementary information

Supplementary Information

This file contains Supplementary Notes 1–5, Supplementary Figures 1–6 and Supplementary Tables 1–3.

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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

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