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

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 which have characterized properties of the transfer, and quantum monodromy, matrices of the 20- vertex model from the perspective of the quantum inverse scattering method, [54,55] in addition to the structure of nonlocal correlations, [7], the forthcoming approach is an adaptation of a study on the Rational 7-vertex model [3]. For both the 7-vertex and 20-vertex models nonlinearity through the Schrodinger’s equation plays a fundamental role not only in quantitatively describing well-posed initial value problems, but also through the intertwining operations underlying Yang-Baxter equations for the Rational, and Universal, R-Matrices. Given a decomposition of the Universal R-Matrix obtained in [6,7] we formalize a novel application of the quantum inverse scattering method which is related to entanglement, and other behaviors, of nonlinear Schrodinger’s equations [1,8,38,51,53,55,57,65,67,68]. Within the setting of the Rational 7-vertex model, Antonenko and Valinevich demonstrated that the intertwining vectors can be used to transform the R-Matrix of a vertex model into the Boltzmann 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, [55], and by also manipulating q-exponentials, and other related factors from a factorization of the Universal R-Matrix due to Boos et al, [7], 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 Boltzmann 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. (PAC Classification 2010: 03.65.Nk, 02.30Zz, 02.30.Ik, 02.50.Cw, 45.20.Jj, 75.60.Ch, 77.80.Dj).