{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"89a36f14c894f163bd09caf0ba3fc398ea8730476acc217e31e227bc188a8c7e","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"2f37c124cd023bf326e6faecc0739dc8d55a782c072d6cdab0ed63c45adb4780","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.quantum-spin-gaps.haldane-spin1","kind":"well-posed","literature_status":"open","n":"1","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 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":null,"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":"2277c2056f9b4652a96eb984886b4114703ad1bdf3de5eaf3cbb73c0aab421ae"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"427a486851b2a9e4af26d91fa454832221f8ccc16a408e9672d3e1f88649d3f8","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"1a6fbec0041958631593e03ea91b166e09cf789fdd44cb023964dec5680b6eed","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"HJgip8rHwLIvOJ2yrljopgnoO-vYLJBJYhZd_AogxxioR_vELSHlFYRSW0UG7YIzIplIQ6A8rdnKYqSzHQZ7Dg"},"schema":"pubphys.envelope/1"},"record_hash":"427a486851b2a9e4af26d91fa454832221f8ccc16a408e9672d3e1f88649d3f8","leaf_index":1653}