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Nature Physics

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

Kaixin Ji, Yu Zhao, Ce Shen, Yidun Wan, Ling-Yan Hung

Published Sep 14, 2026• 1 citation
Publisher

Abstract

Abstract In this paper, we introduce a "\textbf{CFT factory}" -- a novel algorithm of methodically generating 2D lattice models that would flow to conformal field theories (CFTs) in the infrared. We realise these models by engineering the boundary conditions of 3D topological orders (SymTOs) 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 the $A$-series minimal models, and uncovers previously unknown critical points. Notably, we discover at least three novel CFTs (with central charge $c\approx1.3$, $1.8$, and $2.5$) that preserves the Haagerup symmetries, in addition to recovering previously reported examples. The non-invertible symmetries preserved at these points are dictated by a novel "refined condensation tree". The condensation trees, together with a generalised Kramers-Wannier (KW) duality, predict large swathes of phase boundaries and sieves out second order phase transitions. This predictive power is illustrated not only in well-studied examples, such as the 8-vertex model associated with the $A_5$ category, but also in new cases involving Haagerup symmetries, validated by an improved symmetry-preserving tensor-network renormalization group method. The critical couplings are precisely encoded in algebraic data (the Frobenius algebras and quantum dimensions of unitary fusion categories), thereby establishing a powerful and systematic route to the discovery and potential classification of new CFTs.

DOI
10.1038/s41567-026-03438-6

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Abstract In this paper, we introduce a "\textbf{CFT factory}" -- a novel algorithm of methodically generating 2D lattice models that would flow to conformal field theories (CFTs) in the infrared. We realise these models by engineering the boundary conditions of 3D topological orders (SymTOs) 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 the AAA-series minimal models, and uncovers previously unknown critical points. Notably, we discover at least three novel CFTs (with central charge c≈1.3c\approx1.3c≈1.3, 1.81.81.8, and 2.52.52.5) that preserves the Haagerup symmetries, in addition to recovering previously reported examples. The non-invertible symmetries preserved at these points are dictated by a novel "refined condensation tree". The condensation trees, together with a generalised Kramers-Wannier (KW) duality, predict large swathes of phase boundaries and sieves out second order phase transitions. This predictive power is illustrated not only in well-studied examples, such as the 8-vertex model associated with the A5A_5A5​ category, but also in new cases involving Haagerup symmetries, validated by an improved symmetry-preserving tensor-network renormalization group method. The critical couplings are precisely encoded in algebraic data (the Frobenius algebras and quantum dimensions of unitary fusion categories), thereby establishing a powerful and systematic route to the discovery and potential classification of new CFTs.
In this paper, we introduce a ``CFT factory'' : a novel algorithm of methodically generating 2D lattice models that would flow to 2D conformal fixed points in the infrared. These 2D models are realised by giving critical boundary conditions to 3D topological orders (symTOs/symTFTs) described by string-net models, often called the strange correlators. We engineer these critical boundary conditions by introducing a commensurate amount of non-commuting anyon condensates. The non-invertible symmetries preserved at the critical point can be controlled by studying a novel ``refined condensation tree''. Our structured method generates an infinite family of critical lattice models, including the A-series minimal models, and uncovers previously unknown critical points. Notably, we find at least three novel critical points (c≈1.3\approx 1.3≈1.3, 1.81.81.8, and 2.52.52.5 respectively) preserving the Haagerup symmetries, in addition to recovering previously reported ones. The condensation tree, together with a generalised Kramers-Wannier duality, predicts precisely large swathes of phase boundaries, fixes almost completely the global phase diagram, and sieves out second order phase transitions. This is not only illustrated in well-known examples (such as the 8-vertex model related to the A5A_5A5​ category) but also further verified with precision numerics, using our improved (non-invertible) symmetry-preserving tensor-network RG, in novel examples involving the Haagerup symmetries. We show that critical couplings can be precisely encoded in the categorical data (Frobenius algebras and quantum dimensions in unitary fusion categories), thus establishing a powerful, systematic route to discovering and potentially classifying new conformal field theories.
A bstract The coset construction of two-dimensional conformal field theory (2D CFT) defines a 2D CFT by taking the quotient of two previously known chiral algebras. In this work, we use the methods of non-abelian (non-invertible) anyon condensation to describe 2D topological cosets, defined by the special case where the quotient of chiral algebras is a conformal embedding. In this case, the coset has zero central charge, and the coset theory is thus purely topological. Using non-abelian anyon condensation we describe in general the spectrum of line and local operators as well as their fusion, operator product expansion, and the action of the lines on local operators. An important application of our results is to QCD 2 with massless fermions in any representation that leads to a gapped phase, where topological cosets (conjecturally) describe the infrared fixed point. We discuss several such examples in detail. For instance, we find that the Spin (8) 1 / SU (3) 3 and Spin (16) 1 / Spin (9) 2 topological cosets appearing at the infrared fixed point of appropriate QCD 2 theories are described by ℤ 2 × ℤ 2 triality and ℤ 2 × Rep(S 3 ) fusion categories respectively. Additionally, using this setup, we argue that chiral Spin (8) QCD 2 with massless chiral fermions in the vectorial and spinorial representations is not only gapped, but moreover trivially gapped, with a unique ground state.
Understanding the universality class of continuous phase transition is of central importance in condensed matter physics. In one spatial dimension, the universal properties are encoded in the conformal field theory (CFT), which is in turn specified by the conformal data. In this thesis, we propose a systematic method to extract complete and accurate conformal data from the critical quantum spin chain based on the operator-state correspondence, thus completing the project initiated by Cardy and others in the 80's. \n \nOur method is based on the low-energy eigenstates of the critical quantum spin chain with periodic boundary conditions. First, scaling dimensions and conformal spins are extracted by the energies and momenta. Second, the primary states and conformal towers are identified by using the Koo-Saleur lattice Virasoro generators. Third, we propose a systematic way of identifying lattice operators with CFT operators, which enables us to compute operator product expansion coefficients from the low-energy eigenstates. Finally, the whole approach is generalized to critical quantum spin chains with antiperiodic boundary condition. In order to reduce finite-size corrections, we put forward the periodic uniform matrix product state (puMPS) algorithm which enables us to compute low-energy eigenstates of a critical quantum spin chain up to several hundreds of spins. Our method also enables us to study nonperturbatively the renormalization group flow between two CFTs as well as explore the emergence of extended symmetries beyond conformal symmetry. \n \nWe test our method with the Ising model and its generalization due to O'Brien and Fendley. The latter model is featured by a tricritical point described by the tricritical Ising CFT and a line of critical points interpolating between the Ising CFT and the tricritical Ising CFT. We extract complete conformal data from the two models and find excellent agreement with analytical results. Furthermore, we study the spectral renormalization group flow between the two CFTs nonperturbatively. At the tricritical point, the underlying CFT has an extended symmetry, the superconformal symmetry. We propose lattice operators that correspond to supervirasoro generators and verify their action on low-energy eigenstates. In this way we study the emergence of superconformal symmetry in critical quantum spin chains.
Self-dual S3S_3S3​ gauge theory in 2+1d: lattice model and topological phase transitions
Electric-magnetic self-duality of the Z2\mathbb{Z}_2Z2​ gauge theory, realized microscopically as a half-lattice-translation exchanging electric charge and magnetic flux, has been an influential example of a duality symmetry with an exact lattice realization. We construct the first non-Abelian generalization of this construction: a lattice model of the S3S_3S3​ quantum double D(S3)\mathcal{D}(S_3)D(S3​) on a tensor product Hilbert space in which the Z2em\mathbb{Z}^{\mathrm{em}}_2Z2em​ anyon-permutation symmetry, exchanging the non-Abelian chargeon CCC and fluxon FFF, is realized via lattice translation. Consequently we find that the zigzag boundary termination of the model realizes, without fine-tuning, a gapless critical edge state described by the tetracritical Ising CFT. The bulk admits three independent Z2em\mathbb{Z}_2^{\mathrm{em}}Z2em​-preserving bosonic perturbations, driving D(S3)\mathcal{D}(S_3)D(S3​) into distinct gapped phases. We analyze these transitions by three independent methods: category-theoretic anyon condensation, microscopic lattice Hamiltonians, and Chern-Simons-Higgs theory, which all agree, yielding a unified picture. These examples motivate a minimal-condensation principle: proliferating a bosonic anyon generically drives condensation of a minimal condensable algebra containing it, with symmetry-related condensates appearing as degenerate vacua that spontaneously break the anyon-permutation symmetry. Our model construction extends to an infinite family of self-dual dihedral quantum doubles D(D2n)\mathcal{D}(D_{2n})D(D2n​). Notably, each model is sign-problem-free, opening the door to large-scale numerical exploration of the phases of non-Abelian Chern-Simons-Higgs theories.
We use the formalism of strange correlators to construct a critical classical lattice model in two dimensions with the Haagerup fusion category H_{3} as input data. We present compelling numerical evidence in the form of finite entanglement scaling to support a Haagerup conformal field theory (CFT) with central charge c=2. Generalized twisted CFT spectra are numerically obtained through exact diagonalization of the transfer matrix, and the conformal towers are separated in the spectra through their identification with the topological sectors. It is further argued that our model can be obtained through an orbifold procedure from a larger lattice model with input Z(H_{3}), which is the simplest modular tensor category that does not admit an algebraic construction. This provides a counterexample for the conjecture that all rational CFT can be constructed from standard methods.
Irrational CFTs in 1+1d with a discrete spectrum and no conserved currents other than the stress-tensor are expected to be generic, unsolvable by standard methods, and hard to construct explicitly. We introduce a lattice model that realizes a candidate for such a CFT as a conformal phase of matter without fine-tuning. The model is constructed by coupling N≥3 N ≥ 3 golden anyon chains together, preserving N N copies of the Fibonacci non-invertible symmetry. We use the MPS/DMRG approach to study this model numerically, which allows us to calculate the corresponding conformal data, obtaining hints of its irrationality. Along the way, we characterize the phase diagram for N=2 N = 2 coupled chains where we identify a weakly first-order phase transition as well as critical points that we are able to identify with known rational CFTs, except for one case. We also provide an extensive list of rational CFTs with 1<c<2.1 1 < c < 2.1 .