Poisson structure and Integrability of a Hamiltonian flow for the inhomogeneous six-vertex model

TL;DR

This work establishes the integrability of the Hamiltonian flow for the inhomogeneous six-vertex model by constructing action-angle variables from a finite-volume Poisson-bracket framework tied to the quantum inverse scattering method. Building on the Δ < -1 regime and inhomogeneities, the authors derive explicit L-operator recursions and reduce the monodromy to a Hamiltonian system, revealing a 2D Poisson structure with a nine-term decomposition of {A(u), A(u')}. The action-angle coordinates are defined via ϕ(u) = -arg(B(u)) and ρ(u) = (1/π) log[1 + ε|B(u)|^2], with a lemma establishing the canonical conjugacy, and a theorem showing that {ϕ(u1), ϕ(u2)} = {ρ(u1), ρ(u2)} = 0, confirming integrability. By translating Bethe-ansatz-like structures into a Hamiltonian setting and demonstrating the vanishing brackets in canonical coordinates, the paper links domain-wall L-operators, monodromy matrices, and weak-volume asymptotics to a linearized, integrable flow, with potential extensions to higher vertex models and inhomogeneous limit shapes.

Abstract

We compute the action-angle variables for a Hamiltonian flow of the inhomogeneous six-vertex model, from a formulation introduced in a 2022 work due to Keating, Reshetikhin, and Sridhar, hence confirming a conjecture of the authors as to whether the Hamiltonian flow is integrable. To demonstrate that such an integrability property of the Hamiltonian holds from the action-angle variables, we make use of previous methods for studying Hamiltonian systems, implemented by Faddeev and Takhtajan, in which it was shown that integrability of a Hamiltonian system holds for the nonlinear Schrodinger's equation by computing action-angle variables from the Poisson bracket, which is connected to the analysis of entries of the monodromy and transfer matrices. For the inhomogeneous six-vertex model, an approach for computing the action-angle variables is possible through formulating several relations between entries of the quantum monodromy, and transfer, matrices, which can be not only be further examined from the structure of $L$ operators, but also from computing several Poisson brackets parameterized from entries of the monodromy matrix.

Figures (4)

• 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 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.
• Figure 4: A depiction of taking the weak finite volume limit towards $- \infty$ along $\textbf{Z}^2$ for the height function of the 6-vertex model, from an adaptation presented in previous work of the author, [10]. The collection of faces highlighted in yellow above can be used to construct longer paths with faces highlighted in grey.