Exact expressions for correlations in the ground state of the dense O(1) loop model

TL;DR

This paper presents new conjectures and exact expressions for correlation functions in the dense O(1) loop model on semi-infinite lattices with various boundary conditions, including open boundaries, and relates these to Fully Packed Loop models and the XXZ spin chain.

Contribution

It introduces new conjectures for correlations with open boundaries and establishes a mapping between the loop model's ground state and the XXZ spin chain.

Findings

Derived expressions for correlations with open boundaries

Conjectured relations between ground states and Fully Packed Loop models

Mapped the dense O(1) loop model to the XXZ spin chain

Abstract

Conjectures for analytical expressions for correlations in the dense O$(1)$ loop model on semi infinite square lattices are given. We have obtained these results for four types of boundary conditions. Periodic and reflecting boundary conditions have been considered before. We give many new conjectures for these two cases and review some of the existing results. We also consider boundaries on which loops can end. We call such boundaries ''open''. We have obtained expressions for correlations when both boundaries are open, and one is open and the other one is reflecting. Also, we formulate a conjecture relating the ground state of the model with open boundaries to Fully Packed Loop models on a finite square grid. We also review earlier obtained results about this relation for the three other types of boundary conditions. Finally, we construct a mapping between the ground state of the dense O$(1)$ loop model and the XXZ spin chain for the different types of boundary conditions.