Geometric Algebra and Algebraic Geometry of Loop and Potts Models

TL;DR

This paper reveals a deep connection between the algebraic structures of integrable models, specifically the affine Temperley-Lieb algebra and Bethe equations, and applies this to compute partition functions of loop and Potts models.

Contribution

It introduces a novel analytical approach linking algebraic geometry with integrable models, enabling new computations of partition functions and zero distributions.

Findings

Partition functions computed for loop and Potts models on finite lattices.

Analytic expressions for condensation curves in the thermodynamic limit.

Demonstration of the algebraic structure's role in integrable model solutions.

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.