{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"8eede3be1653882bf12b1aeb7a2d39017e5ed86cd2d4d5345ec9a713b3361953","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"1810a26c36012b592e43125441415c54871637b1d0064fd1857f16a37833a8bd","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.bose-condensation.hardcore-off-half-filling","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"For bosons hopping on a 3D lattice with at most one particle per site, condensation is proven only when exactly half the sites are filled, because the proof uses a mirror symmetry that holds only there. At any other filling it is unproven.","posed_since":"1978","precise":"Hard-core bosons on $Z^3$, H = -sum_<xy> (a_x^+ a_y + a_y^+ a_x), equivalently the spin-$1/2$ XY model in a uniform field along z, at density $\\rho \\text{ in } (0,1), \\rho != 1/2$. Prove off-diagonal long-range order, $\\operatorname{lim}_{\\mid x-y\\mid \\to \\infty} \\langle a_x^{+} a_y\\rangle > 0$, for $T$ below some $T_c(\\rho) > 0$. Answer: a proof for an open interval of $\\rho$.","problem_ref":null,"references":"","settled_by":"A proof of off-diagonal long-range order at some density $\\rho \\ne 1/2$ that does not rely on reflection positivity.","status_note":"The half-filled case follows from reflection positivity (Dyson, Lieb and Simon, Journal of Statistical Physics 1978); a nonzero field breaks reflection positivity.","title":"Condensation of lattice hard-core bosons away from half filling","topic_ref":"8e2da899d10157a0800115bf0ebd7035da124ea4f6901ed1432508c7509786fe"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"eb5d83994809f68c6dd6d9ecc3ef5b4ce329d4ecf68c2796904942e7706eefeb","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"997b6acd6b5c0d678f49b81a5efb1f28bd0f065631bfff217e068245ebd6f3cd","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"x2g6cfs06MQgHvnwX15UjqFWp6yhUFgS5qpKc-2IgLnW-OzL5uk7LWm96FTEX9z8_qnNHtGMIkfTB8Y21jgkAQ"},"schema":"pubphys.envelope/1"},"record_hash":"eb5d83994809f68c6dd6d9ecc3ef5b4ce329d4ecf68c2796904942e7706eefeb","leaf_index":1599}