Integrability as a consequence of discrete holomorphicity: the Z_N model

I. T. Alam; M. T. Batchelor

doi:10.1088/1751-8113/45/49/494014

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arXiv · 1207.3883

Integrability as a consequence of discrete holomorphicity: the Z_N model

I. T. Alam·M. T. Batchelor

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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.

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BibTeX RIS

I. T. Alam, M. T. Batchelor. 2012-07-17. Integrability as a consequence of discrete holomorphicity: the Z_N model. https://doi.org/10.1088/1751-8113/45/49/494014

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