Exact solvability of an Ising-type model, and exact solvability of the 6-vertex, and 8-vertex, models

TL;DR

The paper develops an Ising-type model within the Quantum Inverse Scattering Method framework, using a Bazhanov–Sergeev L-operator to study exact solvability. It constructs transfer-matrix representations from two and three Ising-type L-operators, and demonstrates the existence of Ising-type action-angle coordinates for which the Poisson bracket with their conjugates vanishes, signifying integrability. The work draws explicit connections to vertex-model formalisms via RTT/RLL relations and the Yang-Baxter equation, including domain-wall boundary conditions and weak finite-volume limits to justify asymptotic expansions of transfer and quantum monodromy matrices. Importantly, it argues that an interpolating Ising-type model between the 6-vertex and 8-vertex regimes remains exactly solvable, providing a unified approach across these integrable systems and offering tools for studying discrete probabilistic structures (e.g., DW boundary conditions) in the large-volume limit.

Abstract

We compute the action-angle coordinates for an Ising type model whose L-operator has been previously studied in the literature by Bazhanov and Sergeev. In comparison to computations with such operators that have been examined previously by the author for the 4-vertex, 6-vertex, and 20-vertex, models, computations for asymptotically approximating a collection of sixteen identities with the Poisson bracket, which together constitute the Poisson structure of the Ising type model, exhibit dependencies upon nearest neighbor interactions. Inspite of the fact that L-operators for the 20-vertex model are defined in terms of combinatorial, and algebraic, constituents unlike such operators for the 6-vertex model which are defined in terms of projectors and Pauli basis elements, L-operators for the Ising-type model can be used for concluding that a model which interpolates between the 6-vertex, and 8-vertex, models is exactly solvable.

Figures (7)

• 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 of the 6-vertex model sampled over $\textbf{Z}^2$. The box, whose boundary is outlined in red, is comprised of four equal boxes whose boundaries are also outlined in red within the interior.