Quantum inverse scattering for the 20-vertex model up to Dynkin automorphism: crossing probabilities, 3D Poisson structure, triangular height functions, weak integrability

TL;DR

This paper develops a quantum inverse scattering framework for the 20-vertex triangular lattice, introducing a three-dimensional L-operator and a corresponding three-dimensional transfer matrix up to Dynkin automorphism. By analyzing products of 3D L-operators and their weak infinite-volume limits, it derives a 3D Poisson structure comprising 81 relations among the entries of the 3D transfer matrix, and provides a reparametrization into nine-entry blocks A through I to mirror the 2D case. The main finding is that, unlike the 2D inhomogeneous 6V case with action-angle coordinates, the 20V triangular lattice lacks a complete 3D action-angle structure, signaling a weakened notion of integrability in three dimensions. The work also connects these structural insights to crossing probabilities and limit-shape considerations, suggesting new asymptotic behavior for correlations and height functions in the triangular geometry.

Abstract

We initiate a novel application of the quantum-inverse scattering method for the 20-vertex model, building upon seminal work from Faddeev and Takhtajan on the study of Hamiltonian systems, with applications to crossing probabilities, 3D Poisson structure, triangular height functions, and integrability. In comparison to a previous work of the author in late $2023$ which characterized integrability of a Hamiltonian flow for the 6-vertex model from integrability of inhomogeneous limit shapes, formalized in a work of Keating, Reshetikhin and Sridhar, notions similar to those of integrability can be realized for the 20-vertex model by studying new classes of higher-dimensional L-operators. In comparison to two-dimensional L-operators expressed in terms of Pauli basis elements, three-dimensional L-operators provided by Boos and colleagues have algebraic, combinatorial, and geometric, qualities, all of which impact leading order approximations of correlations, products of L-operators, the transfer matrix, and the quantum monodromy matrix in finite volume. In comparison to the inhomogeneous 6-vertex model, the 20-vertex model does not enjoy as strong of an integrability property through the existence of suitable action-angle variables, which is of interest to further explore, possibly from information on limit shapes given solutions to the three-dimensional Euler-Lagrange equations.

Figures (5)

• Figure 1: Each possible vertex for the six-vertex model, adapted from ${\color{blue}[8]}$.
• Figure 2: Another depiction of each possible vertex for the six-vertex model, adapted from [24].
• Figure 3: A depiction of each possible vertex for the triangular, or three dimensional, six-vertex model, adapted from [15].
• Figure 4: A depiction of each Boltzman weight for the triangular, or three dimensional, six-vertex model, also adapted from [15].
• Figure 5: A depiction of a two-dimensional vertex configuration, reproduced from [36], of the sloped boundary condition six-vertex model over a strip of $\textbf{Z}^2$. $\lambda$ spectral parameters are assigned to ingoing, and outgoing, edges of the configuration along the cluster of frozen faces of the height function.