{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"b847106dd8e0dc808f0ff200c054eaa7c968a641c86240fbe5be7aa6fdc62d53","created":"2026-10-03T07:18:11Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"440bd5eb0d5d0ec211c03a0a5f83a94c88bdbd8f89495dde63cea38cfa7da462","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"stat.spin-glass-finite-d.lower-critical-dimension","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Below a certain space dimension, magnets with random-sign interactions never freeze into a glass at any nonzero temperature. That dimension lies between two and three, and a conjecture says it is exactly two and a half.","posed_since":"1994","precise":"For the Edwards-Anderson Ising model with symmetric couplings in zero field, continued to non-integer dimension d (hierarchical lattices, long prisms, or long-range one-dimensional proxies), determine the lower critical dimension $d_l$ below which $T_c = 0$, defined by the vanishing of the domain-wall stiffness exponent $\\theta(d)$. Test the conjecture $d_l = 5/2$ exactly, which follows from replica-symmetry-breaking interface arguments (Franz, Parisi, Virasoro 1994). An answer is $d_l$ with error below 0.02, or a derivation of its exact value.","problem_ref":null,"references":"","settled_by":"A computation of $\\theta(d)$ on a continuous family of geometries interpolating between $d = 2$ and $d = 3$ with controlled finite-size corrections, or an RG derivation of $d_l$.","status_note":"Domain-wall stiffness fits give $d_{\\ell}$ close to $5/2$ (Boettcher, Phys. Rev. Lett. 95, 197205, 2005) and 2026 long-prism simulations agree with the replica prediction (arXiv:2601.07926), while a 2026 rigorous result finds no spin-glass order on Migdal-Kadanoff hierarchical lattices of fractal dimension just below 3 (arXiv:2607.15676).","title":"Exact lower critical dimension of the Ising spin glass","topic_ref":"2fabbf2d62cd6cfeaad5bb54ef05b26854ae4b38a41f86b17737f97acef6a87f"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"05686d3f10aff57eeae5877a0736b832f6284a3814f25c038cda84962f9133d9","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"49019dbaea8e31f025772eaa5219e2dcf060f157da78e23b413ce6e1d652f618","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"1OYI5eDSB1p-eTU0cQkSPQY2VX7NW4HQTHCOz9GlSI9m8JoIf2GdSYxOCA85QXB7-FalqxlA-BzR-9xW-E11Dw"},"schema":"pubphys.envelope/1"},"record_hash":"05686d3f10aff57eeae5877a0736b832f6284a3814f25c038cda84962f9133d9","leaf_index":2114}