Integrability as a consequence of discrete holomorphicity: the Z_N model

TL;DR

This paper demonstrates that discrete holomorphicity conditions in the Z_N model lead to integrability, deriving key relations like inversion and star-triangle from linear equations related to Boltzmann weights.

Contribution

It extends the connection between discrete holomorphicity and integrability to the Z_N model, explicitly deriving key equations from geometric considerations.

Findings

Discrete holomorphicity yields quadratic and cubic equations for Boltzmann weights.

Inversion relations follow from the two-rhombus equation.

Star-triangle relation is derived from the three-rhombus equation.

Abstract

It has recently been established that imposing the condition of discrete holomorphicity on a lattice parafermionic observable leads to the critical Boltzmann weights in a number of lattice models. Remarkably, the solutions of these linear equations also solve the Yang-Baxter equations. We extend this analysis for the Z_N model by explicitly considering the condition of discrete holomorphicity on two and three adjacent rhombi. For two rhombi this leads to a quadratic equation in the Boltzmann weights and for three rhombi a cubic equation. The two-rhombus equation implies the inversion relations. The star-triangle relation follows from the three-rhombus equation. We also show that these weights are self-dual as a consequence of discrete holomorphicity.