Finite-gap potentials as a semiclassical limit of the thermodynamic Bethe Ansatz
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| Format: | Preprint |
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2025
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| _version_ | 1866911612411052032 |
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| author | Melin, Valdemar Wiegmann, Paul Zarembo, Konstantin |
| author_facet | Melin, Valdemar Wiegmann, Paul Zarembo, Konstantin |
| contents | We show that the semiclassical limit of thermodynamic Bethe Ansatz equations naturally reconstructs the algebro-geometric spectra of finite-gap periodic potentials. This correspondence is illustrated using the traveling-wave (snoidal) solution of the defocusing modified Korteweg--de Vries equation. In this framework, the Bethe-root distribution of the associated quantum field theory yields an Abelian differential of the second kind on the elliptic Riemann surface specified by the spectral endpoints, a structure central to the algebro-geometric theory of solitons. The semiclassical parameter is identified with the large-rank limit of the internal symmetry group ($O(2N)$) of the underlying quantum field theory (the Gross-Neveu model with a chemical potential). Our analysis indicates that the analytic structure of the spectrum is dictated solely by the Dynkin diagram ($D_N$) and its large-rank limit ($D_\infty$), independently of the particular integrable model used to realize it. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_19655 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite-gap potentials as a semiclassical limit of the thermodynamic Bethe Ansatz Melin, Valdemar Wiegmann, Paul Zarembo, Konstantin High Energy Physics - Theory Strongly Correlated Electrons Mathematical Physics We show that the semiclassical limit of thermodynamic Bethe Ansatz equations naturally reconstructs the algebro-geometric spectra of finite-gap periodic potentials. This correspondence is illustrated using the traveling-wave (snoidal) solution of the defocusing modified Korteweg--de Vries equation. In this framework, the Bethe-root distribution of the associated quantum field theory yields an Abelian differential of the second kind on the elliptic Riemann surface specified by the spectral endpoints, a structure central to the algebro-geometric theory of solitons. The semiclassical parameter is identified with the large-rank limit of the internal symmetry group ($O(2N)$) of the underlying quantum field theory (the Gross-Neveu model with a chemical potential). Our analysis indicates that the analytic structure of the spectrum is dictated solely by the Dynkin diagram ($D_N$) and its large-rank limit ($D_\infty$), independently of the particular integrable model used to realize it. |
| title | Finite-gap potentials as a semiclassical limit of the thermodynamic Bethe Ansatz |
| topic | High Energy Physics - Theory Strongly Correlated Electrons Mathematical Physics |
| url | https://arxiv.org/abs/2512.19655 |