Physicists have long been fascinated by the strange behavior of matter at a second-order phase transition, the razor-thin tipping point where, for example, a magnet loses its magnetism as temperature rises. At such critical points, fluctuations occur on all length scales at once, and the system is governed by a conformal field theory, a mathematical framework so rigid that its properties can often be catalogued without knowing anything about the underlying material. Yet a stubborn obstacle has stood in the way of turning this catalogue into concrete physics: for many candidate conformal field theories, nobody has known how to write down an actual lattice model, a concrete array of interacting degrees of freedom, whose long-distance behavior realizes the theory. A team of researchers in China now reports a systematic solution, describing an algorithm they call a conformal field theory factory that manufactures two-dimensional critical lattice models on demand.
The work, published in Nature Physics by Kaixin Ji, Yu Zhao, Ce Shen, Yidun Wan and Ling-Yan Hung, draws on some of the deepest ideas in modern condensed matter theory. The authors’ strategy does not start from spins or magnets at all. Instead, they engineer the boundary conditions of three-dimensional topological orders, exotic phases of matter whose excitations, called anyons, can carry quantum statistics that are neither bosonic nor fermionic. These topological orders are described concretely by string-net models, exactly soluble constructions introduced by Michael Levin and Xiao-Gang Wen in 2005, in which the vacuum is pictured as a tangle of fluctuating strings whose allowed patterns are dictated by algebraic data known as a fusion category.
The key innovation lies in how the critical points are created. In a topological phase, certain anyon types can undergo condensation, a process analogous to the condensation of a Bose-Einstein condensate, in which the anyon becomes part of the vacuum and other excitations are reorganized accordingly. When a single set of anyons condenses, the system typically flows from one gapped topological phase to another. The researchers instead arranged for non-commuting anyons to condense in a carefully balanced, commensurate fashion, meaning that two or more condensation channels that cannot coexist in an ordinary gapped phase are forced into competition. The tug-of-war between these incompatible orders prevents the system from settling into any gapped phase, and the resulting critical points flow in the infrared limit to conformal field theories. By tuning the relative weights of the competing condensates, the algorithm generates a lattice Hamiltonian whose low-energy behavior is precisely the desired conformal theory.
The machinery relies on a holographic device known as the strange correlator, a quantity computed as a three-dimensional path integral that maps the boundary lattice model onto the bulk topological order. In this picture, the two-dimensional critical model lives on the boundary of the three-dimensional string-net system, and the algebraic rules governing anyon fusion in the bulk translate directly into the interaction terms of the boundary model. The critical couplings, the parameter values at which the phase transitions occur, are encoded exactly in algebraic data associated with the string-net construction, specifically in the Frobenius algebras that specify which anyons condense. This means that instead of laboriously scanning parameter space numerically to hunt for critical points, physicists can read off where the transitions happen from the underlying category theory, a level of precision control that is rare in the study of strongly correlated systems.
The practical payoff is an infinite family of critical lattice models. The authors demonstrate that their procedure recovers known conformal field theories that preserve the so-called Haagerup symmetries, exotic non-invertible symmetries named after the mathematician Uffe Haagerup, whose fusion categories have intrigued both mathematicians and physicists since the 1990s. Haagerup-symmetric theories have become a testing ground for the emerging theory of categorical symmetry, in which ordinary symmetry groups are replaced by richer algebraic structures. Critical lattice models realizing these symmetries had been proposed only recently, and the new algorithm reproduces them as a special case of a much more general construction, providing independent confirmation of earlier numerical work that had reported evidence for Haagerup conformal field theories in tensor network calculations.
More strikingly, the factory does not merely recycle known results. Among the models it generates, the researchers identified three previously unknown candidate conformal field theories, critical points that had never been observed or catalogued before. These discoveries suggest that the space of two-dimensional conformal field theories is far more densely populated by accessible lattice realizations than the traditional, largely ad hoc methods of statistical mechanics had revealed. Historically, finding a lattice model for a given critical phenomenon was a matter of insight and luck, from Onsager’s solution of the Ising model to the Ashkin-Teller models studied in the early 1980s. The new algorithm replaces that serendipity with a recipe: choose a fusion category, select competing condensable algebras, and compute the resulting boundary model and its phase diagram.
The numerical verification of the construction is itself technically notable. The team developed symmetry-preserving tensor network algorithms to map out the phase diagrams of their models, coloring the parameter space by the numerically determined central charge, a fundamental invariant of a conformal field theory that measures the number of its degrees of freedom. In the phase diagrams, regions corresponding to different condensed anyon orders meet along critical lines and surfaces, and the interpolation between multiple competing condensates can be visualized in ternary diagrams representing three-condensate mixtures. The agreement between the predicted critical couplings extracted from the algebraic data and the numerical scans provides a stringent consistency check of the entire framework, and the MATLAB code and source data used to regenerate the phase diagrams have been made available with the paper.
The broader implications extend beyond two-dimensional statistical mechanics. Conformal field theories occupy a central role in high-energy theoretical physics as well, appearing as limits of quantum field theories, as building blocks of string theory, and through the AdS/CFT correspondence as dual descriptions of quantum gravity. A systematic method for discretizing conformal field theories onto lattices offers a potential route to studying them with the numerical tools of condensed matter, including tensor networks and quantum simulation. The authors and other researchers in the field have also drawn connections to topological holography and the idea that symmetries themselves can be understood as shadows of topological order, suggesting that the factory could illuminate how generalized, non-invertible symmetries emerge at quantum critical points.
The work also raises tantalizing prospects for classification. One of the great unsolved problems in theoretical physics is to classify all possible conformal field theories, a task that has proved formidable even in two dimensions where the machinery is most powerful. By establishing a structured scheme in which critical theories arise from combinatorial algebraic data, the conformal field theory factory provides a framework for discovering and potentially organizing these theories in families. If every entry in a suitable catalogue of fusion categories and condensable algebras yields a critical model, physicists may be able to enumerate, or at least systematically sample, far more of the landscape of critical behavior than ever before. For a field that has spent half a century stitching together critical phenomena one painstaking example at a time, the prospect of a factory that produces them by the dozen marks a genuine shift in method, and the three brand-new candidate theories that emerged from its first run hint at how much of that landscape still lies unexplored.
Subject of Research: An algorithm generating two-dimensional critical lattice models from competing anyon condensation in three-dimensional topological orders
Article Title: An algorithm to generate two-dimensional critical lattice models using competing anyon condensation
Article References: Ji, K., Zhao, Y., Shen, C., Wan, Y., & Hung, L.-Y. (2026). An algorithm to generate two-dimensional critical lattice models using competing anyon condensation. Nature Physics. https://doi.org/10.1038/s41567-026-03438-6
Image Credits: AI Generated
DOI: 10.1038/s41567-026-03438-6
Keywords: conformal field theory, anyon condensation, topological order, string-net models, critical phenomena, lattice models, Haagerup symmetry, phase transitions, fusion categories, tensor networks, categorical symmetry, theoretical physics
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Denise Maddox. (September 21, 2026). New Algorithm Generates Critical Lattice Models Through Competing Anyon Condensation. Scienmag. https://scienmag.com/new-algorithm-generates-critical-lattice-models-through-competing-anyon-condensation/
Denise Maddox. “New Algorithm Generates Critical Lattice Models Through Competing Anyon Condensation.” Scienmag, 21 September 2026, https://scienmag.com/new-algorithm-generates-critical-lattice-models-through-competing-anyon-condensation/. Accessed 21 September 2026.
Denise Maddox. “New Algorithm Generates Critical Lattice Models Through Competing Anyon Condensation.” Scienmag. September 21, 2026. https://scienmag.com/new-algorithm-generates-critical-lattice-models-through-competing-anyon-condensation/
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Tags: anyon condensationcategorical symmetryconformal field theorycritical phenomenafusion categoriesHaagerup symmetrylattice modelsphase transitionsstring-net modelstensor networksTheoretical Physicstopological order
