Abstract
Higher-order topological insulators have attracted significant interest in recent years. However, identifying a universal topological invariant capable of characterizing higher-order topology remains challenging. Here, we propose a entanglement topological invariant designed to characterize second-order topological systems. This entanglement topological invariant captures the entanglement of topological corner states under open boundary conditions by employing a bipartite entanglement entropy method. In several representative models, the entanglement topological invariant assumes a nonzero value exclusively in the presence of second-order topology, with its magnitude exactly matching the number of topologically protected corner states. Consequently, the proposed entanglement topological invariant not only provides a clear criterion for detecting higher-order topology, but also offers a quantitative measure for the related corner states. Our study establishes a universal and precise method for characterizing higher-order topological phases, opening avenues for their fundamental understanding and future investigations.
Similar content being viewed by others
Data availability
The data that support the findings of this study are available in the Supplementary Data file or from the corresponding author upon request.
Code availability
The codes associated with this manuscript are available from the corresponding author upon request.
References
Song, Z., Fang, Z. & Fang, C. (d-2)-Dimensional Edge States of Rotation Symmetry Protected Topological States. Phys. Rev. Lett. 119, 246402 (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).
Schindler, F. et al. Higher-order topological insulators. Sci. Adv. 4, eaat0346 (2018).
Franca, S., van den Brink, J. & Fulga, I. C. An anomalous higher-order topological insulator. Phys. Rev. B 98, 201114 (2018).
Wang, Z. et al. Higher-Order Topology, Monopole Nodal Lines, and the Origin of Large Fermi Arcs in Transition Metal Dichalcogenides XTe2 (X = Mo, W). Phys. Rev. Lett. 123, 186401 (2019).
Ezawa, M. Higher-Order Topological Insulators and Semimetals on the Breathing Kagome and Pyrochlore Lattices. Phys. Rev. Lett. 120, 026801 (2018).
Călugăru, D., Juricić, V. & Roy, B. Higher-order topological phases: A general principle of construction. Phys. Rev. B 99, 041301 (2019).
Trifunovic, L. & Brouwer, P. W. Higher-Order Bulk-Boundary Correspondence for Topological Crystalline Phases. Phys. Rev. X 9, 011012 (2019).
Khalaf, E. Higher-order topological insulators and superconductors protected by inversion symmetry. Phys. Rev. B 97, 205136 (2018).
Zeng, J., Liu, H., Jiang, H., Sun, Q.-F. & Xie, X. C. Multiorbital model reveals a second-order topological insulator in 1H transition metal dichalcogenides. Phys. Rev. B 104, L161108 (2021).
Wang, Z. et al. Realization of a three-dimensional photonic higher-order topological insulator. Nat. Commun. 16, 3122 (2025).
Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall Conductance in a Two-Dimensional Periodic Potential. Phys. Rev. Lett. 49, 405–408 (1982).
Kane, C. L. & Mele, E. J. Quantum Spin Hall Effect in Graphene. Phys. Rev. Lett. 95, 226801 (2005).
Kane, C. L. & Mele, E. J. \({{\mathbb{Z}}}_{2}\) Topological Order and the Quantum Spin Hall Effect. Phys. Rev. Lett. 95, 146802 (2005).
Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61 (2017).
Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators. Phys. Rev. B 96, 245115 (2017).
Ono, S. et al. Difficulties in operator-based formulation of the bulk quadrupole moment. Phys. Rev. B 100, 245133 (2019).
Li, C.-A. et al. Topological insulators in disordered electric quadrupole insulators. Phys. Rev. Lett. 125, 166801 (2020).
Schindler, F. et al. Higher-order topology in bismuth. Nat. Phys. 14, 918 (2018).
Costa, M. et al. Discovery of higher-order topological insulators using the spin Hall conductivity as a topology signature. npj Comput. Mater. 7, 146 (2021).
Miao, C.-M., Sun, Q.-F. & Zhang, Y.-T. Second-order topological corner states in zigzag graphene nanoflake with different types of edge magnetic configurations. Phys. Rev. B 106, 165422 (2022).
Miao, C.-M., Wan, Y.-H., Sun, Q.-F. & Zhang, Y.-T. Engineering topologically protected zero-dimensional interface end states in antiferromagnetic heterojunction graphene nanoflakes. Phys. Rev. B 108, 075401 (2023).
Miao, C.-M., Liu, L., Wan, Y.-H., Sun, Q.-F. & Zhang, Y.-T. General principle behind magnetization-induced second-order topological corner states in the Kane-Mele model. Phys. Rev. B 109, 205417 (2024).
Yang, Y.-B. et al. Type-II quadrupole topological insulators. Phys. Rev. Res. 2, 033029 (2020).
Kitaev, A. & Preskill, J. Topological Entanglement Entropy. Phys. Rev. Lett. 96, 110404 (2006).
Levin, M. & Wen, X.-G. Detecting Topological Order in a Ground State Wave Function. Phys. Rev. Lett. 96, 110405 (2006).
Amico, L., Fazio, R., Osterloh, A. & Vedral, V. Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008).
Calabrese, P. & Cardy, J. Entanglement entropy and conformal field theory. J. Phys. A: Math. Theor. 42, 504005 (2009).
Eisert, J., Cramer, M. & Plenio, M. B. Colloquium: Area laws for the entanglement entropy. Rev. Mod. Phys. 82, 277 (2010).
Fradkin, E. Field Theories of Condensed Matter Physics. Cambridge University Press, Cambridge https://doi.org/10.1017/CBO9781139015509 (2013).
Tsomokos, D. I. et al. Topological order following a quantum quench. Phys. Rev. A 80, 060302 (2009).
Halász, G. B. & Hamma, A. Probing topological order with Rényi entropy. Phys. Rev. A 86, 062330 (2012).
Halász, G. B. & Hamma, A. Topological Rényi Entropy after a Quantum Quench. Phys. Rev. Lett. 110, 170605 (2013).
Depenbrock, S., McCulloch, I. P. & Schollwöck, U. Nature of the Spin-Liquid Ground State of the S=1/2 Heisenberg Model on the Kagome Lattice. Phys. Rev. Lett. 109, 067201 (2012).
Jiang, H.-C., Wang, Z. & Balents, L. Identifying topological order by entanglement entropy. Nat. Phys. 8, 902 (2012).
Isakov, S. V., Hastings, M. B. & Melko, R. G. Topological Entanglement Entropy of a Bose-Hubbard Spin Liquid. Nat. Phys. 7, 772 (2011).
Sankar, S., Sela, E. & Han, C. Measuring Topological Entanglement Entropy Using Maxwell Relations. Phys. Rev. Lett. 131, 016601 (2023).
Karamlou, A. H. et al. Probing entanglement in a 2D hard-core Bose–Hubbard lattice. Nature 629, 561–566 (2024).
Lin, Z.-K. et al. Measuring entanglement entropy and its topological signature for phononic systems. Nat. Commun. 15, 1601 (2024).
Chen, T. et al. Experimental observation of classical analogy of topological entanglement entropy. Nat. Commun. 10, 1557 (2019).
Li, H. & Haldane, F. D. M. Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States. Phys. Rev. Lett. 101, 010504 (2008).
Chandran, A. et al. Bulk-edge correspondence in entanglement spectra. Phys. Rev. B 84, 205136 (2011).
Prodan, E., Hughes, T. L. & Bernevig, B. A. Entanglement Spectrum of a Disordered Topological Chern Insulator. Phys. Rev. Lett. 105, 115501 (2010).
Mao, B.-B. et al. Sampling reduced density matrix to extract fine levels of entanglement spectrum and restore entanglement Hamiltonian. Nat. Commun. 16, 2880 (2025).
Wen, X. et al. Edge theory approach to topological entanglement entropy, mutual information, and entanglement negativity in Chern-Simons theories. Phys. Rev. B 93, 245140 (2016).
Fidkowski, L. Entanglement Spectrum of Topological Insulators and Superconductors. Phys. Rev. Lett. 104, 130502 (2010).
Alexandradinata, A., Hughes, T. L. & Bernevig, B. A. Trace index and spectral flow in the entanglement spectrum of topological insulators. Phys. Rev. B 84, 195103 (2011).
Hughes, T. L., Prodan, E. & Bernevig, B. A. Inversion-symmetric topological insulators. Phys. Rev. B 83, 245132 (2011).
Fang, C., Gilbert, M. J. & Bernevig, B. A. Entanglement spectrum classification of Cn-invariant noninteracting topological insulators in two dimensions. Phys. Rev. B 87, 035119 (2013).
Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum Spin Hall Effect and Topological Insulator in HgTe Quantum Wells. Science 314, 1757 (2006).
Maiellaro, A. et al. Topological squashed entanglement: Nonlocal order parameter for one-dimensional topological superconductors. Phys. Rev. Res. 4, 033088 (2022).
Maiellaro, A. et al. Edge states, Majorana fermions, and topological order in superconducting wires with generalized boundary conditions. Phys. Rev. B 106, 155407 (2022).
Maiellaro, A. et al. Squashed entanglement in one-dimensional quantum matter. Phys. Rev. B 107, 115160 (2023).
Maiellaro, A. et al. Resilience of topological superconductivity under particle current. Phys. Rev. B 107, 064505 (2023).
Liu, L. et al. Engineering second-order topological insulators via coupling two first-order topological insulators. Phys. Rev. B 110, 115427 (2024).
Peschel, I. & Eisler, V. Reduced density matrices and entanglement entropy in free lattice models. J. Phys. A 42, 504003 (2009).
Peschel, I. Calculation of reduced density matrices from correlation functions. J. Phys. A 36, L205 (2003).
Arad, I., Firanko, R. & Jain, R. An Area Law for the Maximally-Mixed Ground State in Arbitrarily Degenerate Systems with Good AGSP. Proc. 56th Annu. ACM Symp. Theory Comput.https://doi.org/10.1145/3618260.3649612 (2024).
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grants No. 12274305, No. 12074097, No. 12374034 and No. 12447147), Natural Science Foundation of Hebei Province (Grant No. A2024205025), the National Key R and D Program of China (Grant No. 2024YFA1409002), the China Postdoctoral Science Foundation (Grant No. 2024M760070), and the Quantum Science and Technology-National Science and Technology Major Project (Grant No. 2021ZD0302403).
Author information
Authors and Affiliations
Contributions
Y.-L.Z. and Y.-T.Z. conceived the work and designed the research strategy. Y.-L.Z. and C.-M.M. carried out the numerical calculations under the supervision of Y.-T.Z. and Q.-F.S. All authors, Y.-L.Z., C.-M.M., J.-J.L., Y.-T.Z., and Q.-F.S. performed the data analysis and wrote the paper together.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Communications Physics thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
About this article
Cite this article
Zhang, YL., Miao, CM., Sun, QF. et al. Characterizing second-order topological insulators via entanglement topological invariant in two-dimensional systems. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02507-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s42005-026-02507-9
