Haldane gap in the spin-1 Heisenberg chain
In plain words
Haldane predicted that a chain of spin-1 magnets with the simplest antiparallel coupling has a finite energy gap above its ground state. Computers confirm it, but there is no proof.
Precise statement
$H = \sum_i S_i . S_{i+1}$ with $S = 1$ on a ring of L sites. Prove $\mathrm{inf}_{L} (E_{1}(L) - E_{0}(L)) > 0$ (numerically the gap is approximately $0.4105 J$), together with a unique infinite-volume ground state with exponentially decaying correlations. Answer: a proof.
What would settle it
A proof of a uniform positive gap, possibly computer-assisted.
Status in the literature
Unverified note: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
Tasaki (Physical Review Letters, 2025; arXiv 2407.17041) proved that the ground state is in a nontrivial symmetry-protected phase if the gap exists; the gap itself is unproven.
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