Integrability and braided tensor categories

Abstract

Many integrable statistical mechanical models possess a fractional-spin conserved current. Such currents have been constructed by utilising quantum-group algebras and ideas from "discrete holomorphicity". I find them naturally and much more generally using a braided tensor category, a topological structure arising in knot invariants, anyons and conformal field theory. I derive a simple constraint on the Boltzmann weights admitting a conserved current, generalising one found using quantum-group algebras. The resulting trigonometric weights are typically those of a critical integrable lattice model, so the method here gives a linear way of "Baxterising", i.e. building a solution of the Yang-Baxter equation out of topological data. It also illuminates why many models do not admit a solution. I discuss many examples in geometric and local models, including (perhaps) a new solution.

arXiv comment23 pages

References in corpus (25)

• Anyons in an exactly solved model and beyond
• Interacting anyons in topological quantum liquids: The golden chain
• Topological Defects on the Lattice I: The Ising model
• Interferometry of non-Abelian Anyons
• Gauge Theory and Integrability, I
• Topological Defects on the Lattice: Dualities and Degeneracies
• LieART 2.0 -- A Mathematica Application for Lie Algebras and Representation Theory
• Parafermionic conformal field theory on the lattice
• On the Construction of Trigonometric Solutions of the Yang-Baxter Equation
• Holomorphic Parafermions in the Potts model and SLE
• The exoticness and realisability of twisted Haagerup-Izumi modular data
• Integrable sigma models and perturbed coset models
• Twisted quantum affine algebras and solutions to the Yang-Baxter equation
• Discretely Holomorphic Parafermions in Lattice Z(N) Models
• Discrete Holomorphicity at Two-Dimensional Critical Points
• An integrable modification of the critical Chalker-Coddington network model
• Discrete holomorphicity and integrability in loop models with open boundaries
• Lie algebra and invariant tensor technology for g2
• Quantum integrability of sigma models on AII and CII symmetric spaces
• Chiral SU(2)_k currents as local operators in vertex models and spin chains
• Critical behaviour of the dilute O(n), Izergin-Korepin and dilute $A_L$ face models: Bulk properties
• Discrete holomorphic parafermions in the Ashkin-Teller model and SLE
• R-matrices and Tensor Product Graph Method
• Integrability as a consequence of discrete holomorphicity: loop models
• Conserved Currents in the Six-Vertex and Trigonometric Solid-On-Solid Models

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• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Topological aspects of the critical three-state Potts model
• Conformal field theory from lattice fermions
• Contrasting pseudo-criticality in the classical two-dimensional Heisenberg and ${\mathrm{RP}}^2$ models: zero-temperature phase transition versus finite-temperature crossover
• Generalizations of Kitaev's honeycomb model from braided fusion categories
• Support theory for Drinfeld doubles of some infinitesimal group schemes
• Les Houches Lecture Notes on Tensor Networks
• Integrability of planar-algebraic models