Discrete holomorphicity and integrability in loop models with open boundaries

TL;DR

This paper explores boundary conditions compatible with discrete holomorphicity in loop models, leading to new integrable weights and solutions that satisfy reflection equations, advancing understanding of boundary integrability in statistical models.

Contribution

It introduces a generalized parafermionic observable for loop models with open boundaries, establishing boundary conditions that preserve integrability and discovering new integrable weights.

Findings

Existence of boundary conditions compatible with discrete holomorphicity.

Derivation of new integrable weights for loop models.

Solutions satisfy reflection equations, confirming integrability.

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.