Poisson structure and Integrability of a Hamiltonian flow for the inhomogeneous six-vertex model
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866908625483595776 |
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| author | Rigas, Pete |
| author_facet | Rigas, Pete |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_15181 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Poisson structure and Integrability of a Hamiltonian flow for the inhomogeneous six-vertex model Rigas, Pete Mathematical Physics Statistical Mechanics Probability Exactly Solvable and Integrable Systems 82B23, 82C10, 81U02, 34L25, 60K35, 82B20, 82B27 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. |
| title | Poisson structure and Integrability of a Hamiltonian flow for the inhomogeneous six-vertex model |
| topic | Mathematical Physics Statistical Mechanics Probability Exactly Solvable and Integrable Systems 82B23, 82C10, 81U02, 34L25, 60K35, 82B20, 82B27 |
| url | https://arxiv.org/abs/2310.15181 |