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Efficient algorithms for the surface density of states in topological photonic and acoustic systems

Nature Computational Science volume 5, pages 1192–1201 (2025)Cite this article

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A preprint version of the article is available at arXiv.

Abstract

Topological photonics and acoustics have attracted wide research interest for their ability to manipulate light and sound at surfaces. The supercell technique is the conventional standard approach used to calculate these boundary effects, but, as the supercell grows in size, this method requires increasingly large computational resources. Additionally, it falls short in differentiating the surface states at opposite boundaries and, due to finite-size effects, from bulk states. Here, to overcome these limitations, we provide two complementary efficient methods for obtaining the ideal topological surface states of semi-infinite systems of diverse surface configurations. The first is the cyclic reduction method, which is based on iteratively inverting the Hamiltonian for a single unit cell, and the other is the transfer matrix method, which relies on eigenanalysis of a transfer matrix for a pair of unit cells. Numerical benchmarks, including gyromagnetic photonic crystals, valley photonic crystals, spin-Hall acoustic crystals and quadrupole photonic crystals, jointly show that both methods can effectively sort out the boundary modes via the surface density of states, at reduced computational cost and increased speed. Our computational schemes enable direct comparisons with near-field scanning measurements, thereby expediting the exploration of topological artificial materials and the design of topological devices.

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Fig. 1: Various photonic or acoustic crystal structures that support topological surface states.
Fig. 2: Topological surface states of representative surface configurations in photonic crystals.
Fig. 3: Topological surface and corner states of three representative configurations.
Fig. 4: Accuracy and convergence analyses of the CRM, TMM and SCM.

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Data availability

All data in this study were generated by running our codes (ref. 58). Source data are provided with this paper.

Code availability

Source codes associated with this manuscript are available on Zenodo (ref. 58) and via GitHub at https://github.com/YixinSha/SDOS.

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Acknowledgements

Y.Y. acknowledges support from the National Natural Science Foundation of China Excellent Young Scientists Fund (12222417), the Hong Kong Research Grants Council through the Early Career Scheme (27300924), a Strategic Topics Grant (STG3/E-704/23-N), the Collaborative Research Fund (C7015-24GF), the Areas of Excellence Scheme (AoE/P-604/25-R), the Startup Fund of The University of Hong Kong, Ms. Belinda Hung, the Asian Young Scientist Fellowship, the Croucher Foundation, the New Cornerstone Science Foundation through the Xplorer Prize, and the Alibaba DAMO Academy Young Fellow Award. M.-Y.X. acknowledges support from the National Natural Science Foundation of China (62231001 and 62171005). L.L. acknowledges support from the National Natural Science Foundation of China (12025409), the Chinese Academy of Sciences through the Project for Young Scientists in Basic Research (YSBR-021) and through the IOP-HKUST-Joint Laboratory for Wave Functional Materials Research.

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Authors and Affiliations

  1. Department of Physics and HK Institute of Quantum Science and Technology, The University of Hong Kong, Hong Kong, China

    Yi-Xin Sha & Yi Yang

  2. State Key Laboratory of Optical Quantum Materials, The University of Hong Kong, Hong Kong, China

    Yi-Xin Sha & Yi Yang

  3. State Key Laboratory of Photonics and Communications, School of Electronics, Peking University, Beijing, China

    Ming-Yao Xia

  4. Institute of Physics, Chinese Academy of Sciences/Beijing National Laboratory for Condensed Matter Physics, Beijing, China

    Ling Lu

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Contributions

Y.-X.S. and L.L. conceived the idea. Y.-X.S. developed the codes, performed the simulations, and drafted the manuscript. All authors discussed the results and revised the paper. Y.Y. supervised the project.

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Correspondence to Yi-Xin Sha or Yi Yang.

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Nature Computational Science thanks the anonymous reviewer(s) for their contribution to the peer review of this work. Primary Handling Editor: Jie Pan, in collaboration with the Nature Computational Science team.

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Sha, YX., Xia, MY., Lu, L. et al. Efficient algorithms for the surface density of states in topological photonic and acoustic systems. Nat Comput Sci 5, 1192–1201 (2025). https://doi.org/10.1038/s43588-025-00898-3

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