A Simple Necessary and Sufficient Condition for Yang--Baxter Integrability

@misc{sanatani2026simple,
  title = {{A Simple Necessary and Sufficient Condition for Yang--Baxter Integrability}},
  author = {Mizuki Sanatani and Naoto Shiraishi and Fuga Ishii},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.29660},
  eprint = {2607.29660},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.29660},
}

TY  - UNPB
TI  - A Simple Necessary and Sufficient Condition for Yang--Baxter Integrability
AU  - Mizuki Sanatani
AU  - Naoto Shiraishi
AU  - Fuga Ishii
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2607.29660
UR  - https://arxiv.org/abs/2607.29660
AB  - Quantum integrability is a cornerstone of the exact theory of interacting quantum spin chains. In its standard formulation, however, one starts from R-matrices satisfying the Yang--Baxter equation, rather than from the Hamiltonian itself. It has therefore remained unclear how Yang--Baxter solvability can be characterized directly at the Hamiltonian level, and how it is related to the existence of local conservation laws. Here we prove that, in a broad standard setting, the Reshetikhin condition is not only necessary but also sufficient for Yang--Baxter integrability, thereby reducing the hidden algebraic structure of integrability to a Hamiltonian-level conservation law. Since the Reshetikhin condition is equivalent to conservation of the total energy current, this Hamiltonian-level criterion is also experimentally accessible. This result establishes a quantum counterpart of the Liouville--Arnold theorem for isotropic spin chains, stating that Yang--Baxter solvability is equivalent to an infinite hierarchy of local conserved quantities. Our result also simplifies substantially the search for integrable spin chains by replacing the search for R-matrices with a direct criterion on local Hamiltonians.
ER  -