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

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.

Template (169 pages, added in exposition of the QISM approach, clarified statement of main results). Video presentation overview available at: https://www.youtube.com/watch?v=tIO0AwbQp6Q, while related topics are discussed at: https://youtu.be/wDGeDqJvM9E , https://youtu.be/AZZy9NuF7kw