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arXiv:2510.19902 · Journal of High Energy Physics

Temperley-Lieb integrable models and fusion categories

Matthew Blakeney, Luke Corcoran, Marius de Leeuw, Balázs Pozsgay, Eric Vernier

Published Apr 20, 2026• 1 citation
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Abstract

A bstract We show that every fusion category containing a non-invertible, self-dual object a gives rise to an integrable anyonic chain whose Hamiltonian density satisfies the Temperley-Lieb algebra. This spin chain arises by considering the projection onto the identity channel in the fusion process a ⨂ a . We relate these models to Pasquier’s construction of ADE lattice models. We then exploit the underlying Temperley-Lieb structure to discuss the spectrum of these models and argue that these models are gapped when the quantum dimension of a is greater than 2. We show that for fusion categories where the dimension is close to 2, such as the Fib×Fib and Haagerup fusion categories, the finite size effects are large and they can obscure the numerical analysis of the gap.

DOI
10.1007/jhep04(2026)165

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A bstract We show that every fusion category containing a non-invertible, self-dual object a gives rise to an integrable anyonic chain whose Hamiltonian density satisfies the Temperley-Lieb algebra. This spin chain arises by considering the projection onto the identity channel in the fusion process a ⨂ a . We relate these models to Pasquier’s construction of ADE lattice models. We then exploit the underlying Temperley-Lieb structure to discuss the spectrum of these models and argue that these models are gapped when the quantum dimension of a is greater than 2. We show that for fusion categories where the dimension is close to 2, such as the Fib×Fib and Haagerup fusion categories, the finite size effects are large and they can obscure the numerical analysis of the gap.
We review the notion of Yang-Baxter integrability for spin chains that have Hilbert spaces with constraints, such as a Rydberg blockade. We focus on anyonic chains, whose constraints arise from the fusion rules of the fusion categories on which they are based. We discuss the emergence of Temperley-Lieb algebras and present a new result on which types of anyonic chains exhibit them. We then give an overview of known results for integrable anyonic chains and extend them to several fusion categories up to rank 777. Using a modification of the boost operator formalism, we find several new integrable anyonic chains and discuss some of their properties. These include spin-32\frac3223​ models for su(2)k\mathfrak{su}(2)_ksu(2)k​ fusion categories, anyonic chains based on the Tambara-Yamagami fusion categories TY(Zn)(\mathbb{Z}_n)(Zn​), and product fusion categories Fib×\times×Fib and Fib×\times×Ising. We review recent results for spin chains based on the Haagerup-Izumi fusion category HI(Z3)(\mathbb{Z}_3)(Z3​), and present preliminary numerics for a HI(Z5)(\mathbb{Z}_5)(Z5​) model.
In this paper, we review the notion of Yang–Baxter integrability for spin chains that have Hilbert spaces with constraints, such as a Rydberg blockade. We focus on anyonic chains, whose constraints arise from the fusion rules of the fusion categories on which they are based. We discuss the emergence of Temperley–Lieb algebras and present a new result on which types of anyonic chains exhibit them. We then give an overview of known results for integrable anyonic chains and extend them to several fusion categories up to rank [Formula: see text]. By using a modification of the boost operator formalism, we find several new integrable anyonic chains and discuss some of their properties. These include spin-[Formula: see text] models for [Formula: see text] fusion categories, anyonic chains based on the Tambara–Yamagami fusion categories TY[Formula: see text] and product fusion categories Fib[Formula: see text]Fib and Fib[Formula: see text]Ising. We review recent results for spin chains based on the Haagerup–Izumi fusion category HI[Formula: see text] and present preliminary numerics for a HI[Formula: see text] model.
We construct the first integrable models based on the Haagerup fusion category H 3 . We introduce a Haagerup version of the anyonic spin chain and use the boost operator formalism to identify two integrable Hamiltonians of PXP type on this chain. The first of these is an analog of the golden chain; it has a topological symmetry based on H 3 and satisfies the Temperley-Lieb algebra with parameter δ = ( 3 + 13 ) / 2 . We prove its integrability using a Lax formalism, and construct the corresponding solution to the Yang-Baxter equation. We present numerical evidence that this model is gapless with a dynamical critical exponent z ≠ 1 . The second integrable model we find breaks the topological symmetry. We present numerical evidence that this model reduces to a CFT in the large volume limit with central charge c ∼ 3 / 2 .
We introduce a class of 2d2d2d quantum spin systems based on a unitary spherical fusion category with a chosen object. These models have frustration-free, reflection positive, but generally non-commuting local interactions, generalizing the Levin-Wen string-net models. Based on the theory developed in arXiv:2510.20662, we explicitly realize the Temperley-Lieb-Jones algebra, and more generally fusion spin chains as their boundary algebras.
Fusion surface models generalize the concept of anyon chains to 2+1 dimensions, utilizing fusion 2-categories as their input. We investigate bond-algebraic dualities in these systems and show that distinct module tensor categories \mathcal{M} ℳ over the same braided fusion category \mathcal{B} ℬ give rise to dual lattice models. This extends the 1+1d result that dualities in anyon chains are classified by module categories over fusion categories. We analyze two concrete examples: (i) a \text{Rep}(S_3) Rep ( S 3 ) model with a constrained Hilbert space, dual to the spin- \tfrac{1}{2} 1 2 XXZ model on the honeycomb lattice, and (ii) a bilayer Kitaev honeycomb model, dual to a spin- \tfrac{1}{2} 1 2 model with XXZ and Ising interactions. Unlike regular \mathcal{M}=\mathcal{B} ℳ = ℬ fusion surface models, which conserve only 1-form symmetries, models constructed from \mathcal{M} ≠ \mathcal{B} ℳ ≠ ℬ can exhibit both 1-form and 0-form symmetries, including non-invertible ones.
Abstract The equivalent of fusion in boundary conformal field theory (CFT) can be realized quite simply in the context of lattice models by essentially glueing two open spin chains. This has led to many developments, in particular in the context of chiral logarithmic CFT. We consider in this paper a possible generalization of the idea to the case of bulk conformal field theory. This is of course considerably more difficult, since there is no obvious way of merging two closed spin chains into a big one. In an earlier paper, two of us had proposed a “topological” way of performing this operation in the case of models based on the affine Temperley-Lieb (ATL) algebra, by exploiting the associated braid group representation and skein relations. In the present work, we establish — using, in particular, Frobenius reciprocity — the resulting fusion rules for standard modules of ATL in the generic as well as partially degenerate cases. These fusion rules have a simple interpretation in the continuum limit. However, unlike in the chiral case this interpretation does not match the usual fusion in non-chiral CFTs. Rather, it corresponds to the glueing of the right moving component of one conformal field with the left moving component of the other.