Integrability as a consequence of discrete holomorphicity: the Z_N model

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.

11 pages, 7 figures, some clarifications and a reference added

References in corpus (7)

• Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry
• Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model
• Discretely Holomorphic Parafermions and Integrable Loop Models
• Discrete holomorphicity and integrability in loop models with open boundaries
• Discretely holomorphic parafermions and integrable boundary conditions
• The critical surface fugacity of self-avoiding walks on a rotated honeycomb lattice
• On the theorem of Duminil-Copin and Smirnov about the number of self-avoiding walks in the hexagonal lattice

Cited by in corpus (6)

• Discrete stress-energy tensor in the loop O(n) model
• Discrete holomorphicity and integrability in loop models with open boundaries
• Integrability as a consequence of discrete holomorphicity: loop models
• Discrete Holomorphic Parafermions in the Eight Vertex Model
• On the construction of discrete fermions in the FK-Ising model
• Self-avoiding walk on ${\mathbb{Z}}^2$ with Yang-Baxter weights: universality of critical fugacity and 2-point function