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arXiv:2507.14280 · SciPost Physics

Irrational CFTs from coupled anyon chains with non-invertible symmetries?

António Antunes, Junchen Rong

Published Jul 6, 2026• 2 citations
arXivHTMLPDF

Abstract

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 .

DOI
10.21468/scipostphys.21.1.005

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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 .
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.
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.
An anyon-chain-like lattice model with symmetry described by the Ising fusion category is studied. Combining numerical and analytical studies, we uncover a rich phase diagram that contains three phases: a symmetric critical phase and two categorical symmetry breaking phases. The symmetric phase lies in the same universality class as the usual critical Ising model. The first symmetry-breaking phase, dubbed the \emph{categorical ferromagnetic} phase, has the Ising fusion category fully broken and exhibits a threefold ground-state degeneracy, as expected from the generalized Landau paradigm. The other symmetry-breaking phase is analogous to a conventional antiferromagnet: it breaks lattice translation and part of the Ising fusion category, and therefore is termed the \emph{categorical antiferromagnetic} phase. Unlike ordinary antiferromagnetic states associated with finite invertible symmetry breaking, this phase itself is critical, being described by a fourfold degenerate Ising conformal field theory. We argue more generally that antiferromagnetic states associated with broken non-invertible symmetries have a large low-energy manifold that grows exponentially in system size, due to the greater-than-one quantum dimension of domain walls. We also numerically study the transitions between the three phases. The transition between the symmetric and categorical ferromagnetic phase is described by the c=7/10c=7/10c=7/10 tricritical Ising CFT, while the transition between the symmetric and categorical antiferromagnetic phases is less understood. Our numerical data suggest that the latter transition is continuous and described by a conformal field theory with central charge c=3/2c=3/2c=3/2.
We consider a generic Fibonacci topological wave function on a square lattice, and the norm of this wave function can be mapped into the partition function of two-coupled ϕ2ϕ^{2}ϕ2-state Potts models with ϕ=(5+1)/2ϕ=(\sqrt{5}+1)/2ϕ=(5​+1)/2 as the golden ratio. A global phase diagram is thus established to display non-abelian topological phase transitions. The Fibonacci topological phase corresponds to an emergent new phase of the two-coupled Potts models, and continuously change into two non-topological phases separately, which are dual each other and divided by a first-order phase transition line. Under the self-duality, the Fibonacci topological state enters into the first-order transition state at a quantum tricritical point, where two continuous quantum phase transitions bifurcate. All the topological phase transitions are driven by condensation of anyonic bosons consisting of Fibonacci anyon and its conjugate. However, a fractional supersymmetry is displayed at the quantum tricritical point, characterized by a coset conformal field theory.
Dualities of quantum field theories are challenging to realize in lattice models of qubits. In this work, we explore one of the simplest dualities, T-duality of the compact boson CFT, and its realization in quantum spin chains. In the special case of the XX model, we uncover an exact lattice T-duality, which is associated with a non-invertible symmetry that exchanges two lattice U(1) symmetries. The latter symmetries flow to the momentum and winding U(1) symmetries with a mixed anomaly in the CFT. However, the charge operators of the two U(1) symmetries do not commute on the lattice and instead generate the Onsager algebra. We discuss how some of the anomalies in the CFT are nonetheless still exactly realized on the lattice and how the lattice U(1) symmetries enforce gaplessness. We further explore lattice deformations preserving both U(1) symmetries and find a rich gapless phase diagram with special \mathrm{Spin}(2k)_1 S p i n ( 2 k ) 1 WZW model points and whose phase transitions all have dynamical exponent z>1 z > 1 .
Coupling N N large m m minimal models and flowing to IR fixed points is a systematic way to build new classes of compact unitary 2d CFTs which are likely to be irrational, and potentially have a positive Virasoro twist gap above the vacuum. In this paper, we build on the construction of [A. Antunes and C. Behan, Phys. Rev. Lett. 130, 071602 (2023)], establishing that, for spins less than 10, additional currents transforming in non-trivial irreducible representations of the permutation symmetry S_N S N are not conserved at the IR fixed points. Along the way, we develop a finer understanding of the spectrum of these theories, of the special properties of the N=4 N = 4 case and of non-invertible symmetries that constrain them. We also discuss variations of the original setup of [A. Antunes and C. Behan, Phys. Rev. Lett. 130, 071602 (2023)], some of which can exist for smaller values of the UV central charge.