The spin-one Haldane gap. Proves the spin-one Haldane gap conjecture for the pure antiferromagnetic Heisenberg chain on even periodic rings: the spectral gap stays uniformly positive as the chain grows. A companion establishes a gap for odd open chains with endpoint field $h=3/5$ and gives boundary-selected infinite-volume states with topological index −1.
released 2026-09-24 | 1 theorem · 10 lemmas · 16 proofs · 10,481 words |
PLAY LEVEL 1 »(pdf)
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.
released 2026-09-24 | 1 theorem · 7 lemmas · 16 proofs · 9,635 words |
PLAY LEVEL 2 »(pdf)
We prove a uniform spectral gap for odd open spin-one antiferromagnetic Heisenberg chains with the same fixed magnetic field at both endpoints. The field $h=3/5$ gives a gap greater than $\log(10)/392$ on every chain of $2L+1$ sites with L ≥ 960. Tasaki's index theorem then gives index −1 for every subsequential local limit of these boundary-selected ground states.