Intertwining vectors, and Boltzmann weight matrices, of a Solid-on-Solid model from the 20-vertex model

TL;DR

This paper explores the relationship between the 20-vertex model and a Solid-on-Solid (SOS) model by constructing intertwining vectors and Boltzmann weight matrices, extending methods from the 7-vertex model to higher dimensions.

Contribution

The paper develops a new framework for deriving SOS model weights from the 20-vertex model using intertwining vectors, generalizing previous approaches to more complex vertex models.

Findings

Derived a system of 9 equations per R-matrix entry for the 20-vertex model.

Constructed analogs of intertwining vectors for the 20-vertex model.

Showed the asymptotic dependence of the Boltzmann weight matrix on multiple contributions.

Abstract

We initiate a new study on the correspondence between the 20-vertex model and a SOS (Solid-on-Solid) model. In comparison to two previous works of the author in 2024 which characterized properties of the transfer, and quantum monodromy, matrices of the 20-vertex model from the perspective of the quantum inverse scattering method, in addition to the structure of nonlocal correlations, the forthcoming approach is an adaptation of a study on the rational 7-vertex model. For the rational 7-vertex model, Antoenneko and Valinevich demonstrated that the intertwining vectors can be used to transform the R-matrix of a vertex model into the Boltzman weight matrix of an SOS model. To further develop perspectives on classes of higher dimensional SOS models, by leveraging previous computations with L-operators due to the author, and by also manipulating q-exponentials, and other related factors from a factorization of the universal R-matrix due to Boos et al, analogs of intertwining vectors for the 20- vertex model can be obtained. In comparison to the system of equations that have been obtained for the rational 7-vertex model, the system that one obtains for the 20-vertex model consists of 9 equations for each entry of the R-matrix. Furthermore, from the fact that the 20-vertex model contains more possible configurations in the sample space than the rational 7-vertex model does, the Boltzman weight matrix of the corresponding SOS model asymptotically depends upon several contributions, a few of which include tensor products of basis elements, and spectral parameters over the triangular lattice.