Integrability as a consequence of discrete holomorphicity: Loop models

In this paper, we explore the relationship between integrability and the discrete holomorphicity of a class of complex lattice observables in the context of the Potts dense loop model and the O(n) dilute loop model. It is shown that the conditions for integrability, namely, the inversion and Yang-Baxter relations, can be derived from the condition of holomorphicity of the observables. Furthermore, the Z-invariance of the models is shown to result in the invariance of the observables on the boundary of a sublattice under reshuffling of the rhombuses of its planar rhombic embedding.