{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"cb58b596f6ca1c2d954a55d831a4d0d5bd1618862be847b6e6e1deebcd6ff211","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":["ai:gpt-5.5"],"kind":"seed"},"parents":["427a486851b2a9e4af26d91fa454832221f8ccc16a408e9672d3e1f88649d3f8","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"366d3cf589ae27f1839a271d7a5a2ccc5548cfa660ed248286768bc620c79ef6","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":["ai:gpt-5.5"],"external_id":"mathph.quantum-spin-gaps.haldane-spin1","kind":"well-posed","literature_status":"open","n":"2","parents":[],"plain":"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.","posed_since":"1983","precise":"$H = \\sum_i S_i . S_{i+1}$ 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.","problem_ref":"427a486851b2a9e4af26d91fa454832221f8ccc16a408e9672d3e1f88649d3f8","references":"","settled_by":"A proof of a uniform positive gap, possibly computer-assisted.","status_note":"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.","title":"Haldane gap in the spin-1 Heisenberg chain","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"e1a104a4b04e827e2515a9e71a265b3d28c84b6a3bf4d4b3104cf8e5d220570e","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"8d0e0517d36c579c6168737f9cb95291919ac5ce1c36e602f2e22b73a8607fae","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"Be8_bB4RgfHjJgYB8RDWa1wsfzwI3hKgIpcE8Bxhpc5FvvIrfskkrfBnQraHlNVFXkExT_Eoyt7KpeJqQqPfBg"},"schema":"pubphys.envelope/1"},"record_hash":"e1a104a4b04e827e2515a9e71a265b3d28c84b6a3bf4d4b3104cf8e5d220570e","leaf_index":3450}