The Haldane gap conjecture for the spin-one Heisenberg antiferromagnetic chain

Claims that the pure bilinear spin-one Heisenberg chain on even periodic rings has a unique ground state for $L\ge60$ and a gap bounded below by explicit constants, so $\Delta_1\ge\log 20/784>0$, with a corresponding lower bound for local excitations in subsequential thermodynamic limits. The companion proves Tasaki's boundary-field gap for odd open chains at $h=3/5$ and the nontrivial index $-1$. The constants are far below the numerical value of about $0.41$. It does NOT treat odd periodic rings, spin above one, or the bilinear-biquadratic family away from the Heisenberg point.

Claims that the pure bilinear spin-one Heisenberg chain on even periodic rings has a unique ground state for $L\ge60$ and a gap bounded below by explicit constants, so $\Delta_1\ge\log 20/784>0$, with a corresponding lower bound for local excitations in subsequential thermodynamic limits. The companion proves Tasaki's boundary-field gap for odd open chains at $h=3/5$ and the nontrivial index $-1$. The constants are far below the numerical value of about $0.41$. It does NOT treat odd periodic rings, spin above one, or the bilinear-biquadratic family away from the Heisenberg point.