{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"43dd88c044cbef6e2b31463b9dad0d882cefc47c7337059ed99f573fa7bacbae","created":"2026-10-03T07:18:05Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"463ab406645c3a3a0ecbdc2ca3608175ed8c8af692d242d008dd5934d1c878ae","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.crystal-order-temperature.positional-power-law-2d","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Theory predicts that in a 2D crystal at low temperature positional order fades slowly, as a power of distance, instead of quickly. Prove this power law for a particle system.","posed_since":"1968","precise":"For a 2D continuum particle system with triangular ground state at low T, prove that the translational correlation $\\langle \\rho_G(x) \\rho_G(0)*\\rangle$, $G$ a reciprocal lattice vector of the triangular lattice, decays as $\\mid x\\mid^{-\\eta_G(T)}$ with $\\eta_G(T)\\to 0$ as $T\\to 0$ (quasi-long-range order), with matching upper and lower power-law bounds.","problem_ref":null,"references":"","settled_by":"Upper and lower power-law bounds on positional correlations with exponent vanishing as $T \\to 0$.","status_note":"Absence of true positional order for 2D continuum particles is proved (Richthammer 2007).","title":"Power-law decay of positional correlations in 2D crystals","topic_ref":"ea8bde04090399eb2ce07a2138c4a5deafce80c26031655b989bc3381e0879d5"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"ba72841724fdf20600cea18da230e91c11afad0e2845bdc2cd9863cfde2454fa","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"6c544f0c7a87de3b2bbd5c828754afee0da002e435bf858952244eeeabe352e0","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"26pUz0rvHW2PPJ-uBDuJ6CrNLBN2OzFzItPn-2StDpca-kLQKaBehKtDv93R_shp-oOFgFU31cawgI25yrI-CQ"},"schema":"pubphys.envelope/1"},"record_hash":"ba72841724fdf20600cea18da230e91c11afad0e2845bdc2cd9863cfde2454fa","leaf_index":1606}