Let's Baxterise

TL;DR

This paper explores Baxterisation as a method to analyze R-matrices and related structures, enabling efficient identification of integrable models by studying algebraic varieties through birational transformations.

Contribution

It introduces a generalized Baxterisation framework applicable beyond Yang-Baxter structures, focusing on algebraic varieties and iteration properties to distinguish integrable cases.

Findings

Baxterisation helps identify integrable models via algebraic variety analysis.

Finite order and Abelian varieties correspond to integrable cases.

Complete classification of finite order iterations for the Baxter model.

Abstract

We recall the concept of Baxterisation of an R-matrix, or of a monodromy matrix, which corresponds to build, from one point in the $ R$-matrix parameter space, the algebraic variety where the spectral parameter(s) live. We show that the Baxterisation, which amounts to studying the iteration of a birational transformation, is a ``win-win'' strategy: it enables to discard efficiently the non-integrable situations, focusing directly on the two interesting cases where the algebraic varieties are of the so-called ``general type'' (finite order iteration) or are Abelian varieties (infinite order iteration). We emphasize the heuristic example of the sixteen vertex model and provide a complete description of the finite order iterations situations for the Baxter model. We show that the Baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying Yang-Baxter structure is not used: we Baxterise L-operators, local quantum Lax matrices, and quantum Hamiltonians.