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arxiv: 2605.28946 · v1 · pith:2GFT4IZEnew · submitted 2026-05-27 · ✦ hep-th · cond-mat.stat-mech· cond-mat.str-el· math-ph· math.MP· nlin.SI

Constrained integrability and anyonic chains

Matthew Blakeney , Luke Corcoran , Marius de Leeuw This is my paper

Pith reviewed 2026-06-29 10:33 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mechcond-mat.str-elmath-phmath.MPnlin.SI
keywords anyonic chainsYang-Baxter integrabilityfusion categoriesTemperley-Lieb algebrasboost operatorconstrained spin chainssu(2)_kTambara-Yamagami
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The pith

A modified boost operator identifies new integrable anyonic chains for su(2)_k, Tambara-Yamagami, and product fusion categories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reviews Yang-Baxter integrability for spin chains whose Hilbert spaces are restricted by fusion rules, such as those in anyonic systems. It shows how Temperley-Lieb algebras arise in certain constrained chains and classifies which categories produce them. Adapting the boost operator method to these constrained spaces, the authors construct new integrable models for spin-3/2 representations in su(2)_k categories, for Tambara-Yamagami TY(Z_n) categories, and for the products Fib times Fib and Fib times Ising. These add to known integrable anyonic chains and reach categories of rank up to 7. A reader would care because integrability supplies exact conserved quantities and solvable dynamics in systems with built-in constraints.

Core claim

Using a modification of the boost operator formalism that respects the constrained Hilbert spaces and fusion rules, several new integrable anyonic chains are found for su(2)_k fusion categories with spin-3/2, Tambara-Yamagami TY(Z_n), and product categories Fib times Fib and Fib times Ising. The work also reviews results for the Haagerup-Izumi category HI(Z_3) and gives preliminary numerics for HI(Z_5).

What carries the argument

The modified boost operator formalism, which generates an infinite tower of conserved charges on constrained anyonic spaces while preserving the Yang-Baxter equation and fusion rules.

If this is right

  • The new chains possess infinitely many conserved charges and are therefore Yang-Baxter integrable.
  • Exact algebraic solutions become available for the spectrum and dynamics of these constrained models.
  • Temperley-Lieb structure is confirmed for additional fusion categories beyond previously known cases.
  • Numerical studies of the HI(Z_5) model can test the predicted integrability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These models supply concrete test cases for numerical methods that enforce fusion-rule constraints.
  • The same modification technique could be tried on fusion categories of rank greater than 7.
  • Exactly solvable anyonic chains may help isolate universal features of constrained quantum dynamics.
  • Connections between integrability and topological order in these systems remain open for further study.

Load-bearing premise

The boost operator modification remains consistent when projected onto the constrained subspaces defined by the fusion rules.

What would settle it

Explicit computation of the transfer matrix for one newly proposed model, followed by a check that it commutes with the Hamiltonian at every lattice site.

Figures

Figures reproduced from arXiv: 2605.28946 by Luke Corcoran, Marius de Leeuw, Matthew Blakeney.

Figure 1
Figure 1. Figure 1: An anyonic chain fusion diagram with the degrees of freedom, [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A pair of measurements on three particles, and the corresponding eigenstate in the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A different measurement basis for the fusion of the same three particles. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Adjacency rules for the Fibonacci Hilbert space. Nodes represent simple objects. If [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Half-chain entanglement entropy and gap for the anti-ferromagnetic case [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Half-chain entanglement entropy and gap for the ferromagnetic case [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Adjacency graph for the Ising fusion category with external object [PITH_FULL_IMAGE:figures/full_fig_p032_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Half-chain entanglement entropy and gap for Ising anyonic chain ( [PITH_FULL_IMAGE:figures/full_fig_p033_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Adjacency graph for the su(2)3 fusion category with external object 1 2 . 0 1 1 2 3 2 [PITH_FULL_IMAGE:figures/full_fig_p034_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Adjacency graph for the su(2)3 fusion category with external object 1. 6.3 su(2)4 ⊗ 0 1 2 1 3 2 2 0 0 1 2 1 3 2 2 1 2 1 2 0 ⊕ 1 1 2 ⊕ 3 2 1 ⊕ 2 3 2 1 1 1 2 ⊕ 3 2 0 ⊕ 1 ⊕ 2 1 2 ⊕ 3 2 1 3 2 3 2 1 ⊕ 2 1 2 ⊕ 3 2 0 ⊕ 1 1 2 2 2 3 2 1 1 2 0 [PITH_FULL_IMAGE:figures/full_fig_p034_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Adjacency graph for the su(2)4 fusion category with external object 1 2 . a = 1. This choice of external object leads to a constrained Hilbert space described by the adjacency graph in figure 12. We note that there are two connected components, and so the Hilbert space decomposes V 1 su(2)4 = Vint ⊕ Vhalf-int into integer and half-integer sectors. The half￾integer sector is fully connected with two nodes,… view at source ↗
Figure 12
Figure 12. Figure 12: Adjacency graph for the su(2)4 fusion category with external object 1. 6.4 su(2)5 = Z2 × psu(2)5 Due to the decomposition su(2)5 = Z2 × psu(2)5, it suffices to consider the integer sector psu(2)5 = {0, 1, 2}, whose fusion rules are shown in table 2. In this case there is a single unitary solution to the pentagon equations, and the quantum dimensions of the objects are given by (d0, d1, d2) =  1, 1 + 2 co… view at source ↗
Figure 13
Figure 13. Figure 13: Adjacency graph for the psu(2)5 fusion category with external object 1. 6.5 su(2)6 The case k = 6 contains seven objects 0, 1 2 , 1, 3 2 , 2, 5 2 , 6, whose quantum dimensions are given by (d0, d1/2, d1, d3/2, d2, d5/2, d6) =  1, 1 + √ 2, q 2 + √ 2, q 4 + 2√ 2, q 2 + √ 2, 1 + √ 2, 1  . (99) There are three inequivalent non-trivial choices of external object to define an anyonic chain in this case. The c… view at source ↗
Figure 14
Figure 14. Figure 14: Adjacency graph for the su(2)6 fusion category with external object 3 2 . 6.6 su(2)7 = Z2 × psu(2)7 In this case we can consider psu(2)7 = {0, 1, 2, 3}, where the quantum dimensions are (d0, d1, d2, d3) =  1, 1 + 2 cos 2π 9 , 1 + 2 cos π 9 , 2 cos π 9  ∼ (1, 1.88, 2.88, 2.53). (105) Due to (86) the choice a = 2 leads to a spin- 3 2 anyonic chain. The corresponding constrained Hilbert space of dimension … view at source ↗
Figure 15
Figure 15. Figure 15: Adjacency graph for the psu(2)7 fusion category with external object 2. 7 Other fusion categories In this section we continue our review of anyonic chains beyond the su(2)k case. 10We find three isolated points numerically in the psu(2)8 and psu(2)9 spin- 3 2 case. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Adjacency graph for the so(5)2 fusion category with external object ψ5. 7.2 HI(Zn) A large class of fusion rings labelled by a finite group G was introduced by Izumi [105, 106, 107], dubbed Haagerup-Izumi fusion rings HI(G). In recent years there has been a surge of interest in models defined from the Haagerup fusion ring, which is the case G = Z3 and the simplest example of a fusion category not related … view at source ↗
Figure 17
Figure 17. Figure 17: Adjacency rules for the Haagerup Hilbert space. [PITH_FULL_IMAGE:figures/full_fig_p042_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Adjacency rules for the HI(Z5) Hilbert space with external object ρ. 7.3 TY(Zn) The Tambara-Yamagami fusion category over Zn consists of a non-invertible object ρ, to￾gether with n invertible objects α0, . . . αn−1 [114]. The objects αi form a Zn subalgebra: αi ⊗ αj = αi+j mod n (120) 43 [PITH_FULL_IMAGE:figures/full_fig_p043_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Half-chain entanglement entropy and gap for the HI [PITH_FULL_IMAGE:figures/full_fig_p044_19.png] view at source ↗
Figure 21
Figure 21. Figure 21: Half-chain entanglement entropy and gap for the TY [PITH_FULL_IMAGE:figures/full_fig_p045_21.png] view at source ↗
Figure 20
Figure 20. Figure 20: Adjacency graph for TY(Zn) with external object ρ. 7.4 Fib × Fib Recently there has been interest in models constructed by considering products of Fusion categories. One of the simplest examples is the choice Fib×Fib [34, 26, 117]. In this fusion category the objects are constructed as products of Fib objects {1 × 1, 1 × τ, τ × 1, τ × τ} := {1, 2, 3, 4}. The fusion rules can be derived from the Fib rules,… view at source ↗
Figure 22
Figure 22. Figure 22: Adjacency graph for Fib × Fib with external object 4. 7.5 Fib × Ising As a final example, we briefly consider the Fib × Ising fusion category, dubbed in Anyonwiki as TriCritIsing [32]. In this case there are six objects {1 × 1, 1 × ψ, 1 × σ, τ × 1, τ × ψ, τ × σ} := {1, 2, 3, 4, 5, 6}. The fusion rules for this model are given in table 7, and the quantum dimensions of the objects are (d1, d2, d3, d4, d5, d… view at source ↗
Figure 23
Figure 23. Figure 23: Half-chain entanglement entropy and gap for the Fib [PITH_FULL_IMAGE:figures/full_fig_p048_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Half-chain entanglement entropy and gap for the Fib [PITH_FULL_IMAGE:figures/full_fig_p048_24.png] view at source ↗
read the original abstract

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 $7$. Using a modification of the boost operator formalism, we find several new integrable anyonic chains and discuss some of their properties. These include spin-$\frac32$ models for $\mathfrak{su}(2)_k$ fusion categories, anyonic chains based on the Tambara-Yamagami fusion categories TY$(\mathbb{Z}_n)$, 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$(\mathbb{Z}_3)$, and present preliminary numerics for a HI$(\mathbb{Z}_5)$ model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper reviews Yang-Baxter integrability for constrained spin chains, focusing on anyonic chains whose Hilbert spaces are restricted by fusion rules of fusion categories. It discusses the emergence of Temperley-Lieb algebras in this setting and presents a new result on which anyonic chains exhibit them. The manuscript reviews known integrable anyonic chains, extends results to fusion categories of rank up to 7, and uses a modification of the boost operator formalism to construct several new integrable models, including spin-3/2 chains for su(2)_k categories, chains based on Tambara-Yamagami TY(Z_n) categories, and product categories Fib×Fib and Fib×Ising. It also reviews recent results on Haagerup-Izumi HI(Z_3) and provides preliminary numerics for HI(Z_5).

Significance. If the modified boost operator is shown to preserve Yang-Baxter integrability on the fusion-constrained subspaces, the new models would constitute concrete additions to the catalog of integrable anyonic chains, potentially useful for studying constrained quantum dynamics and anyonic statistics. The discussion of Temperley-Lieb algebras and the extension to product categories could help classify integrable structures in fusion-category-based systems.

major comments (2)
  1. [Section introducing the modified boost operator and the new models] The central claim relies on a modification of the boost operator producing Hamiltonians that satisfy the Yang-Baxter equation when restricted to the fusion-rule subspace. The manuscript must explicitly demonstrate (e.g., via direct computation of the R-matrix or commutator with fusion projectors) that the modification commutes with the constraints for the listed categories; without this, the integrability of the new su(2)_k, TY(Z_n), Fib×Fib, and Fib×Ising chains remains unverified.
  2. [Results on new integrable anyonic chains] For the spin-3/2 su(2)_k models and the product-category examples, the paper should provide the explicit local Hamiltonian terms (or at least the two-site operators) and verify that they obey the fusion rules while satisfying the integrability condition; the abstract alone does not supply this check.
minor comments (2)
  1. Clarify the precise form of the boost-operator modification (e.g., which terms are altered and why) early in the text so that the subsequent claims can be followed without ambiguity.
  2. The preliminary numerics for HI(Z_5) would benefit from a brief statement of the system sizes used and the observable whose convergence is being checked.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying areas where additional explicit verification would strengthen the presentation of the new integrable models. We address each major comment below and will incorporate the requested clarifications and explicit computations in a revised version.

read point-by-point responses

  1. Referee: [Section introducing the modified boost operator and the new models] The central claim relies on a modification of the boost operator producing Hamiltonians that satisfy the Yang-Baxter equation when restricted to the fusion-rule subspace. The manuscript must explicitly demonstrate (e.g., via direct computation of the R-matrix or commutator with fusion projectors) that the modification commutes with the constraints for the listed categories; without this, the integrability of the new su(2)_k, TY(Z_n), Fib×Fib, and Fib×Ising chains remains unverified.

    Authors: We agree that an explicit demonstration of commutation with the fusion projectors is required to rigorously establish that the modified boost operator preserves the constrained subspace. The current manuscript introduces the modified formalism and states the resulting integrability but does not include the direct commutator calculations for each category. In the revision we will add a new subsection containing these computations (via commutators with the fusion projectors) for the su(2)_k, TY(Z_n), Fib×Fib, and Fib×Ising cases, confirming that the Hamiltonians remain within the fusion-rule subspace and satisfy the Yang-Baxter equation there. revision: yes

  2. Referee: [Results on new integrable anyonic chains] For the spin-3/2 su(2)_k models and the product-category examples, the paper should provide the explicit local Hamiltonian terms (or at least the two-site operators) and verify that they obey the fusion rules while satisfying the integrability condition; the abstract alone does not supply this check.

    Authors: We acknowledge that the explicit two-site operators and the accompanying verification steps are necessary for the reader to reproduce and confirm the claims. While the manuscript presents the construction via the modified boost operator and discusses the resulting models, the explicit local terms for the spin-3/2 su(2)_k and product-category (Fib×Fib, Fib×Ising) examples are only summarized rather than written out. In the revised manuscript we will include these explicit operators in the main text (or a dedicated appendix), together with direct checks that they respect the fusion rules and satisfy the integrability condition on the constrained space. revision: yes

Circularity Check

0 steps flagged

Modification of boost operator applied to external fusion categories; no load-bearing self-definition or fitted predictions

full rationale

The paper reviews constrained YB integrability, presents a new result on Temperley-Lieb algebras for anyonic chains, extends known results to rank-7 categories, and uses a modification of the boost operator to obtain new models for su(2)_k, TY(Z_n), Fib×Fib and Fib×Ising. No quoted step reduces a claimed prediction to a fitted input by construction, nor does any central claim rest on a self-citation chain that itself lacks independent verification. The formalism is applied to external fusion-rule data; the abstract supplies no explicit equations showing the modification is defined in terms of the output Hamiltonians. This is the normal non-circular case.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone; the work relies on standard fusion category data and prior integrability techniques.

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