A uniform gap for the spin-one Heisenberg chain
On $\mathbb{C}^3$ with basis $|m\rangle$, $m \in \{-1, 0, 1\}$, let $S^z|m\rangle = m|m\rangle$, $S^+|m\rangle = \sqrt{2 - m(m+1)}\,|m+1\rangle$ (zero for $m = 1$), $S^- = (S^+)^*$, $S^x = (S^+ + S^-)/2$ and $S^y = (S^+ - S^-)/(2i)$. For each even integer $N \ge 4$ let $H_N$ be the spin-one Heisenberg Hamiltonian on a periodic chain of $N$ sites, acting on $(\mathbb{C}^3)^{\otimes N}$ by $$H_N = \sum_{j=1}^{N} \sum_{a = x, y, z} S^a_j S^a_{j+1}, \qquad S^a_{N+1} = S^a_1,$$ where $S^a_j$ acts on site $j$, and write $E_0(N) < E_1(N)$ for its two lowest distinct eigenvalues. Haldane predicted, from a continuum description by the O(3) nonlinear sigma model, that antiferromagnetic Heisenberg chains of integer spin have a unique ground state with a gap above it. Tasaki proved that if the finite spin-one chains with free ends and a magnetic field at both end sites have unique ground states with a uniform gap, then the ground state of the infinite chain has a nontrivial topological index.
Problem. Prove or disprove that there is a constant $\Delta > 0$, independent of $N$, such that $E_1(N) - E_0(N) \ge \Delta$ for every even $N \ge 4$.
Source: Haldane (1983), as a physical prediction for antiferromagnetic Heisenberg chains of integer spin; stated by AIM — Open Applied Problems for the spin-one chain on even periodic rings.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MP — Mathematical Physics
- Secondary
- SP
- Keywords
- AIM Open Applied Problems, Haldane gap, Heisenberg chain, quantum spin chains
References
@article{haldane1983,
author = {Haldane, F. D. M.},
title = {Continuum dynamics of the 1-{D} {H}eisenberg antiferromagnet: identification with the {O}(3) nonlinear sigma model},
journal = {Phys. Lett. A},
volume = {93},
number = {9},
year = {1983},
pages = {464--468},
doi = {10.1016/0375-9601(83)90631-X},
}
@article{tasaki2025,
author = {Tasaki, Hal},
title = {Ground state of the {$S=1$} antiferromagnetic {H}eisenberg chain is topologically nontrivial if gapped},
journal = {Phys. Rev. Lett.},
volume = {134},
number = {7},
year = {2025},
pages = {076602},
doi = {10.1103/PhysRevLett.134.076602},
eprint = {2407.17041},
archivePrefix = {arXiv},
}
@misc{aim-haldane-gap,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A uniform gap for the spin-one Heisenberg chain}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/194-haldane-gap.md},
note = {Entry 194 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove a uniform positive spectral gap for the pure antiferromagnetic spin-one Heisenberg chain on even periodic rings, establishing the positive even-periodic formulation of the spin-one Haldane conjecture.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00687, "A uniform gap for the spin-one Heisenberg chain", clopen.solutions.