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Peer-reviewedCC BY 4.0Journal of High Energy Physics

Bethe ansatz without nesting

  • Gleb Arutyunov1
  • Hrachya Babujian2,3
  • Minghao Gao1
  1. 1UniversitΓ€t Hamburg
  2. 2Yerevan Physics Institute
  3. 3Beijing Institute of Mathematical Sciences and Applications
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Abstract

A bstract : We develop a non-nested Bethe ansatz description of rational 𝔀𝔩 β„“ spin chains in the vector representation. Starting from the quantum spectral curve and the separation-of-variables framework, we derive closed systems of Bethe equations involving only the momentum-carrying Bethe roots. The construction is worked out explicitly for the 𝔀𝔩 ₃ and 𝔀𝔩 β‚„ spin chains and then generalized to arbitrary rank. A central result of this work is the identification of a recursive hierarchy associated with the fundamental transfer matrices. The hierarchy is generated by regularity conditions of the lower transfer matrices and closes through a universal rank- β„“ equation β„› β„“ = 0. This equation replaces the final level of the conventional nested Bethe ansatz and eliminates all auxiliary Bethe roots. Consequently, the complete spectral data of an eigenstate are encoded solely in the first Baxter polynomial Q ₁ ( u ). We further obtain explicit expressions for the eigenvalues of all fundamental transfer matrices in terms of the momentum-carrying roots alone. The resulting formulation provides a compact characterization of the spectrum of rational 𝔀𝔩 β„“ spin chains and reveals a direct connection between the quantum spectral curve, transfer-matrix fusion relations, and a truncated Q -system underlying the non-nested description. Finally, we investigate the quasi-classical (Gaudin) limit of the non-nested Bethe equations. For the 𝔀𝔩 ₃ spin chain, we show that the leading non-trivial contribution gives rise to Gaudin equations whose pole-free form naturally defines a scalar third-order 𝔀𝔩 ₃ oper.

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Cite

BibTeXRIS
BibTeX
@article{arutyunov2026bethe,
  title = {{Bethe ansatz without nesting}},
  author = {Gleb Arutyunov and Hrachya Babujian and Minghao Gao},
  journal = {Journal of High Energy Physics},
  year = {2026},
  month = oct,
  doi = {10.1007/jhep10(2026)092},
  url = {https://link.springer.com/10.1007/JHEP10(2026)092},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Bethe ansatz without nesting
AU  - Gleb Arutyunov
AU  - Hrachya Babujian
AU  - Minghao Gao
PY  - 2026
DA  - 2026/10/08
JO  - Journal of High Energy Physics
DO  - 10.1007/jhep10(2026)092
UR  - https://link.springer.com/10.1007/JHEP10(2026)092
AB  - A bstract : We develop a non-nested Bethe ansatz description of rational 𝔀𝔩 β„“ spin chains in the vector representation. Starting from the quantum spectral curve and the separation-of-variables framework, we derive closed systems of Bethe equations involving only the momentum-carrying Bethe roots. The construction is worked out explicitly for the 𝔀𝔩 ₃ and 𝔀𝔩 β‚„ spin chains and then generalized to arbitrary rank. A central result of this work is the identification of a recursive hierarchy associated with the fundamental transfer matrices. The hierarchy is generated by regularity conditions of the lower transfer matrices and closes through a universal rank- β„“ equation β„› β„“ = 0. This equation replaces the final level of the conventional nested Bethe ansatz and eliminates all auxiliary Bethe roots. Consequently, the complete spectral data of an eigenstate are encoded solely in the first Baxter polynomial Q ₁ ( u ). We further obtain explicit expressions for the eigenvalues of all fundamental transfer matrices in terms of the momentum-carrying roots alone. The resulting formulation provides a compact characterization of the spectrum of rational 𝔀𝔩 β„“ spin chains and reveals a direct connection between the quantum spectral curve, transfer-matrix fusion relations, and a truncated Q -system underlying the non-nested description. Finally, we investigate the quasi-classical (Gaudin) limit of the non-nested Bethe equations. For the 𝔀𝔩 ₃ spin chain, we show that the leading non-trivial contribution gives rise to Gaudin equations whose pole-free form naturally defines a scalar third-order 𝔀𝔩 ₃ oper.
SN  - 1029-8479
ER  -