bibkey: ahn2022eclectic authors: C. Ahn; L. Corcoran; M. Staudacher year: 2022 title: “Combinatorial Solution of the Eclectic Spin Chain” doi: 10.1007/JHEP03(2022)028 url: https://arxiv.org/abs/2112.04506v1 claim: “We claim that the rank of A^(k) is always maximal: rank(A^(k)) = min(d_{S−k}, d_S).” strata_touched:
- D5/S3/Quantum/SpinChains/HypereclecticNonShortening license: citation-only triage: anchor
Combinatorial Solution of the Eclectic Spin Chain
C. Ahn, L. Corcoran and M. Staudacher, arXiv:2112.04506v1; JHEP 03 (2022) 028.
Appendix A, printed p. 24, introduces the matrix of the power of the Hamiltonian between the elementary-state level spaces and states:
“We claim that the rank of A^(k) is always maximal:”
“rank(A^(k)) = min(d_{S−k}, d_S).” (A.4)
Section 3.2, printed p. 10, states:
“As before we define W_S^{L,M} to be spanned by elementary states with this level S.”
The same page introduces Eq. (3.19) with:
“The Hamiltonian acts on (3.16) as”
In the K = 1 static sector the wall 3 is fixed at the right. The other L−1 letters contain M−1 twos. The level counts the pairs in which a 2 precedes a 1, and the Hamiltonian changes each adjacent 21 into 12 with coefficient one. The rational elementary-state encoding addresses the rank claim for all 1 ≤ M ≤ L and all k ≤ S. The rank of this rational linear map is the rank of its integer matrix over ℚ; comparison with other characteristic-zero fields is a separate scalar-extension step.
C. Ahn and M. Staudacher, “Spectrum of the Hypereclectic Spin Chain and Pólya Counting”, arXiv:2207.02885v1; Physics Letters B, doi: 10.1016/j.physletb.2022.137533. Its introduction, printed p. 1, states:
“They rest on a compelling and extensively checked but unfortunately still unproven non-shortening conjecture.”
The K = 1 rank assertion is distinct from the multi-wall, cyclic and eclectic-universality questions.
Verified locator
- URL: https://arxiv.org/abs/2112.04506v1, §3.2, printed p. 10, Eqs. (3.14)–(3.19), and Appendix A, printed p. 24, Eq. (A.4).
- DOI: https://doi.org/10.1007/JHEP03(2022)028.
- Follow-up URL: https://arxiv.org/abs/2207.02885v1, introduction, printed p. 1.
- Follow-up DOI: https://doi.org/10.1016/j.physletb.2022.137533.
The quotations were checked in the two arXiv v1 PDF texts. The journal versions were not independently compared.