{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"7b2b515242e3d5131dad8e22f572fed0138c0c004102ed6df7b006478407fea1","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"6878a53d7ea90d917eb21fbec6bf82693881eaf2fb448abce380b240df05d034","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.fermi-liquids-bcs.fermi-liquid-3d","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Show that electrons with weak repulsion in a 3D continuum act like slightly modified free particles down to very low temperatures. At the lowest temperatures they are expected to pair, so the statement must stop above that scale.","posed_since":"","precise":"Spin-$1/2$ fermions in $R^{3}$ with dispersion $k^{2}/(2m)$ and weak short-range repulsion $\\lambda v$. Prove that for T above the pairing scale ($T \\ge \\exp(-c/\\lambda^{2}) E_F$, say) the momentum distribution has a jump of size $Z = 1 - \\operatorname{O}(\\lambda^{2})$ smeared only on scale T, and $\\operatorname{Im} \\Sigma(\\omega, k_F) \\sim \\omega^{2} + (\\pi T)^{2}$. Answer: a proof with explicit constants.","problem_ref":null,"references":"","settled_by":"A convergent multiscale (renormalization-group) proof of Fermi-liquid bounds in 3D down to an exponentially small temperature.","status_note":"In $2\\mathrm{D}$, Disertori and Rivasseau (2000) controlled jellium above an exponentially small temperature, and Feldman, Knorrer and Trubowitz (2003-2004) proved a $T = 0$ Fermi liquid for Fermi surfaces without inversion symmetry.","title":"Fermi-liquid behavior of weakly interacting fermions above the pairing scale","topic_ref":"092ec16641325783d3e30e41c35811e9c4984811735d26f9a5dda60705ccb883"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"ee5ca2f0b9b5a97ad65a6e9a166e58c47dde572c59c8a2e34309ebb4c6d1a5d3","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"0e948a8e7d5306357f9e4d46f2e4fd54060314a183f18412ed1d15687824fe3f","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"JAgA-VcsLiohTF_rbSYf2onnZJNclitIb0j9HZam5N4FJ7RyeE1djZ2Lrmy170MXiLDlmWRNAm9tUhmZCuMJCg"},"schema":"pubphys.envelope/1"},"record_hash":"ee5ca2f0b9b5a97ad65a6e9a166e58c47dde572c59c8a2e34309ebb4c6d1a5d3","leaf_index":1613}