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International Journal of Modern Physics A

Constrained integrability and anyonic chains

Matthew Blakeney, Luke Corcoran, Marius de Leeuw

Published Aug 7, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0217751x26300140

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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 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.
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 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 initiate a systematic study of integrable models for spin chains with constrained Hilbert spaces; we focus on spin-1/2 chains with the Rydberg constraint. We extend earlier results for medium-range spin chains to the constrained Hilbert space, and formulate an integrability condition. This enables us to construct new integrable models with fixed interaction ranges. We classify all time- and space-reflection symmetric integrable Rydberg-constrained Hamiltonians of range 3 and 4. At range 3, we find a single family of integrable Hamiltonians: the so-called RSOS quantum chains, which are related to the well-known RSOS models of Andrews, Baxter, and Forrester. At range 4 we find two families of models, the first of which is the constrained XXZ model. We also find a new family of models depending on a single coupling z z . We provide evidence of two critical points related to the golden ratio \phi ϕ , at z=\phi^{-1/2} z = ϕ − 1 / 2 and z=\phi^{3/2} z = ϕ 3 / 2 . We also perform a partial classification of integrable Hamiltonians for range 5.
This article, based on the author's PhD thesis, reviews recent advancements in the field of quantum integrability, in particular the separation of variables (SoV) program for high-rank integrable spin chains and the boost mechanism for solving the Yang-Baxter equation. We begin with a general overview of quantum integrable systems with special emphasis on their description in terms of quantum algebras. We then provide a detailed account of the Yangian of gl(n)\mathfrak{gl}(n)gl(n) in particular the Bethe algebra, fusion, and T- and Q-systems. We then introduce the notion of separation of variables in integrable systems and build on Sklyanin's work in rank 1 models and extend to higher rank. By exploiting a novel link between SoV and quantum algebra representation theory we construct the separated variables for gl(n)\mathfrak{gl}(n)gl(n) spin chains for arbitrary compact representations of the symmetry algebra and develop various new tools along the way. Next, we build on the previous part and develop a new technique for the computation of scalar products in the SoV framework which we call Functional SoV or FSoV. Unlike the work in the previous, operatorial, part this approach is based on the Baxter TQ equations. After developing this technique we supplement it with a new operator construction providing a unified view of functional and operatorial SoV. Then, we generalise the results of the previous part from compact spin chains to non-compact spin chains. The final part of this work is based on the development of tools for solving the Yang- Baxter equation. We develop a bottom-up approach for this based on the so-called Boost automorphism and uses the spin chain Hamiltonian as a starting point. Our approach allows us to classify numerous families of solutions in particular a complete classification of 4 x 4 solutions which preserve fermion number which have applications in the AdS/CFT correspondence.
Anyonic Chains – α\alpha α-Induction – CFT – Defects – Subfactors
Abstract Given a unitary fusion category, one can define the Hilbert space of a so-called “anyonic spin-chain” and nearest neighbor Hamiltonians providing a real-time evolution. There is considerable evidence that suitable scaling limits of such systems can lead to 1+11+11+1 1+1 -dimensional conformal field theories (CFTs), and in fact, can be used potentially to construct novel classes of CFTs. Besides the Hamiltonians and their densities, the spin chain is known to carry an algebra of symmetry operators commuting with the Hamiltonian, and these operators have an interesting representation as matrix-product-operators (MPOs). On the other hand, fusion categories are well-known to arise from a von Neumann algebra-subfactor pair. In this work, we investigate some interesting consequences of such structures for the corresponding anyonic spin-chain model. One of our main results is the construction of a novel algebra of MPOs acting on a bipartite anyonic chain. We show that this algebra is precisely isomorphic to the defect algebra of 1+11+11+1 1+1 CFTs as constructed by Fröhlich et al. and Bischoff et al., even though the model is defined on a finite lattice. We thus conjecture that its central projections are associated with the irreducible vertical (transparent) defects in the scaling limit of the model. Our results partly rely on the observation that MPOs are closely related to the so-called “double triangle algebra” arising in subfactor theory. In our subsequent constructions, we use insights into the structure of the double triangle algebra by Böckenhauer et al. based on the braided structure of the categories and on α\alpha α α -induction. The introductory section of this paper to subfactors and fusion categories has the character of a review.