Discrete holomorphicity and integrability in loop models with open boundaries

Abstract

We consider boundary conditions compatible with discrete holomorphicity for the dilute O(n) and C_2^(1) loop models. In each model, for a general set of boundary plaquettes, multiple types of loops can appear. A generalisation of Smirnov's parafermionic observable is therefore required in order to maintain the discrete holomorphicity property in the bulk. We show that there exist natural boundary conditions for this observable which are consistent with integrability, that is to say that, by imposing certain boundary conditions, we obtain a set of linear equations whose solutions also satisfy the corresponding reflection equation. In both loop models, several new sets of integrable weights are found using this approach.

23 pages, 27 figures; Minor changes to text, figures improved

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• Conserved Currents in the Six-Vertex and Trigonometric Solid-On-Solid Models
• The dilute Temperley-Lieb O($n=1$) loop model on a semi infinite strip: the ground state
• Currents in the dilute $O(n=1)$ model
• The free-fermionic $C_2^{(1)}$ loop model, double dimers and Kashaev's recurrence
• The ghost algebra and the dilute ghost algebra
• Exact surface energies and boundary excitations of the Izergin-Korepin model with generic boundary fields