{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"20e563c15b9cf7115b4673974862473ac57341fd421ee27f4a8d3a0a6e26b5a1","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"e5783e439b339e280b0d18ed71d2e5cbfa822ffbbc510d26c72e0e14099c9b8a","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.quantum-spin-gaps.quantum-ferromagnet-3d","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"A cubic lattice of quantum magnets with parallel-favouring coupling should be magnetized at low temperature, as real iron is. The proof works for classical spins and for the antiparallel quantum case, but not for the quantum ferromagnet.","posed_since":"","precise":"H = -sum_<xy> S_x . S_y on $Z^3$ with spin $S = 1/2$ (or any S). Prove that for T below some $T_c > 0$ the infinite-volume Gibbs states have spontaneous magnetization, e.g. $\\lim_{L} L^{-6} \\langle(\\sum_x S_x)^2\\rangle > 0$ in the box of side L. Answer: a proof.","problem_ref":null,"references":"","settled_by":"A proof of long-range order at positive temperature for the spin-$1/2$ quantum Heisenberg ferromagnet on $Z^{3}$.","status_note":"Reflection positivity gives order for the classical ferromagnet (Frohlich, Simon and Spencer, 1976) and the quantum antiferromagnet (Dyson, Lieb and Simon, 1978; Kennedy, Lieb and Shastry, 1988) but fails here; the spin-wave free energy is proven at low T (Correggi, Giuliani and Seiringer, Communications in Mathematical Physics 2015, arXiv 1312.7873); a 2017 preprint claim (Suto, arXiv 1710.04441) lists no journal publication.","title":"Spontaneous magnetization of the 3D quantum Heisenberg ferromagnet","topic_ref":"2277c2056f9b4652a96eb984886b4114703ad1bdf3de5eaf3cbb73c0aab421ae"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"50f68f24a16d2f3b86acfc08cb6fe0d3ac6f27f48f5c3d34cf82b8aeb3afc39a","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"e53015006c44e9e641b7f9e49e975175300a919efb0a7279646f5e1456289269","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"xbuO7ADwjwSyJ7zXBJcMdIgqTmdFnPDn8C3cE8_wfFl5xjHytRfqlQv2roy0aCeIIEFp8-tToM6jmlxLHEXGAQ"},"schema":"pubphys.envelope/1"},"record_hash":"50f68f24a16d2f3b86acfc08cb6fe0d3ac6f27f48f5c3d34cf82b8aeb3afc39a","leaf_index":1656}