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arXiv:2310.15181

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

Pete Rigas

Published Oct 20, 2023• 0 citations
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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.

DOI
10.48550/arxiv.2310.15181

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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.
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The inhomogeneous six-vertex model is a 2D multiparametric integrable statistical system.In the scaling limit it is expected to cover different classes of critical behaviour which, for the most part, have remained unexplored.For general values of the parameters and twisted boundary conditions the model possesses U(1) invariance.In this paper we discuss the restrictions imposed on the parameters for which additional global symmetries arise that are consistent with the integrable structure.These include the lattice counterparts of C, P and T as well as translational invariance.The special properties of the lattice system that possesses an additional Z r invariance are considered.We also describe the Hermitian structures, which are consistent with the integrable one.The analysis lays the groundwork for studying the scaling limit of the inhomogeneous six-vertex model.