Geometric Algebra and Algebraic Geometry of Loop and Potts Models

Abstract

We uncover a connection between two seemingly separate subjects in integrable models: the representation theory of the affine Temperley-Lieb algebra, and the algebraic structure of solutions to the Bethe equations of the XXZ spin chain. We study the solution of Bethe equations analytically by computational algebraic geometry, and find that the solution space encodes rich information about the representation theory of Temperley-Lieb algebra. Using these connections, we compute the partition function of the completely-packed loop model and of the closely related random-cluster Potts model, on medium-size lattices with toroidal boundary conditions, by two quite different methods. We consider the partial thermodynamic limit of infinitely long tori and analyze the corresponding condensation curves of the zeros of the partition functions. Two components of these curves are obtained analytically in the full thermodynamic limit.

arXiv comment66 pages

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Cited by in corpus (8)

• Rational $Q$-systems, Higgsing and Mirror Symmetry
• Spaces of states of the two-dimensional O(n) and Potts models
• From combinatorial maps to correlation functions in loop models
• Spin-$s$ Rational $Q$-system
• Rational Q-systems at Root of Unity I. Closed Chains
• Exact Spin Correlators of Integrable Quantum Circuits from Algebraic Geometry
• Two-dimensional massive integrable models on a torus
• Spin-$s$ $Q$-systems: Twist and Open Boundaries