An algorithm to generate two-dimensional critical lattice models using competing anyon condensation

Nature Physics (2026) Cite this article The critical behaviour at a second-order phase transition is often described by a conformal field theory. The restrictions imposed by conformal symmetry give rise to a number of important theoretical techniques that have made these theories central to the understanding of critical phenomena. However, efforts to classify conformal field theories have run into the challenge of identifying lattice models that have a corresponding critical point. Here we introduce an algorithm that we call a conformal field theory factory for methodically generating two-dimensional lattice models that would flow to conformal field theories in the infrared limit. We realize these lattice models by engineering the boundary conditions of three-dimensional topological orders described by string-net models. The critical points are induced by a commensurate condensation of non-commuting anyons. Our structured method generates an infinite family of critical lattice models, including previously unknown critical points. We recover known conformal field theories that preserve the Haagerup symmetries and identify three further candidate theories. The critical couplings of our models are precisely encoded in algebraic data associated with the string-net models, thereby establishing a scheme for discovering and potentially classifying conformal field theories. This is a preview of subscription content, access via your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any timeSubscribe to this journal Receive 12 print issues and online access $259.00 per yearonly $21.58 per issueBuy this articleUSD 39.95Prices may be subject to local taxes which are calculated during checkoutSource data are provided with this paper.The MATLAB code used to regenerate the phase diagrams in Figs. 4 and 5 from the source data is provided in Supplementary Code 1. Other custom code used in this study is available from the corresponding authors upon reasonable request.Bernard, D. & Felder, G. Quantum group symmetries in 2-D lattice quantum field theory. Nucl. Phys. B 365, 98–120 (1991).Article ADS MathSciNet Google Scholar Rajabpour, M. A. & Cardy, J. Discretely holomorphic parafermions in lattice ZN models. J. Phys. A 40, 14703–14713 (2008).Article ADS Google Scholar Ikhlef, Y. & Weston, R. Discrete holomorphicity in the chiral Potts model. J. Phys. A 48, 294001 (2015).Article MathSciNet Google Scholar Fendley, P. Integrability and braided tensor categories. J. Stat. Phys. 182, 43 (2021).Article ADS MathSciNet Google Scholar Feiguin, A. et al. Interacting anyons in topological quantum liquids: the golden chain. Phys. Rev. Lett. 98, 160409 (2007).Article ADS Google Scholar Kong, L., Lan, T., Wen, X.-G., Zhang, Z.-H. & Zheng, H. Algebraic higher symmetry and categorical symmetry—a holographic and entanglement view of symmetry. Phys. Rev. Res. 2, 043086 (2020).Article Google Scholar Kong, L., Lan, T., Wen, X.-G., Zhang, Z.-H. & Zheng, H. Classification of topological phases with finite internal symmetries in all dimensions. J.

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This research was supported in part by the Perimeter Institute for Theoretical Physics. Research at the Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research, Innovation and Science.These authors contributed equally: Kaixin Ji, Yu Zhao.State Key Laboratory of Surface Physics, Center for Astronomy and Astrophysics, Department of Physics, Center for Field Theory and Particle Physics, and Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai, ChinaKaixin Ji, Yu Zhao & Yidun WanYau Mathematical Sciences Center, Tsinghua University, Beijing, ChinaKaixin Ji & Ling-Yan HungBeijing Institute of Mathematical Sciences and Applications, Beijing, ChinaCe Shen & Ling-Yan HungShanghai Research Center for Quantum Sciences, Shanghai, ChinaYidun WanHefei National Laboratory, Hefei, ChinaYidun WanSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarL.-Y.H. conceived and supervised the project. Y.W. co-supervised the project. L.-Y.H., Y.W. and Y.Z. initiated the early working examples combining the strange correlator and competing anyon condensation approaches. Y.Z. applied the anyon condensation formalism in the SN model. C.S. worked out the classification of algebras and modules, formulated the generalized KW duality and performed the initial numerical verification of key critical points. Y.Z., K.J. and C.S. obtained the algebra and module data used in the numerics. K.J. developed and implemented the symmetry-preserving tensor network algorithms, generated the numerical results and phase diagrams, and derived the equation for predicting the general phase boundaries. All authors discussed the results extensively and contributed to the drafting and revision of the manuscript.Correspondence to Ce Shen, Yidun Wan or Ling-Yan Hung.The authors declare no competing interests.Nature Physics thanks the anonymous reviewers 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.Supplementary Sections 1–6, Supplementary Appendices A–G, Supplementary Figs. 1–22 and Supplementary Tables 1 and 2.MATLAB script that reads Source Data Fig. 4 and Source Data Fig. 5 and regenerates the phase diagrams.Numerical source data and RG-flow values.Numerical source data.Numerical source data.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsJi, K., Zhao, Y., Shen, C. et al. An algorithm to generate two-dimensional critical lattice models using competing anyon condensation. Nat. Phys. (2026). https://doi.org/10.1038/s41567-026-03438-6Download citationReceived: 05 September 2025Accepted: 05 August 2026Published: 14 September 2026Version of record: 14 September 2026DOI: https://doi.org/10.1038/s41567-026-03438-6Anyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative