{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"c4eaaa753a6dbb14e02a81284dd9b63876e9185bd5651e28074b7757e13cb4da","created":"2026-10-03T07:17:59Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"e783eca7feb17b83670c78d09f0b95c201d7fa987a13ea3c04f97f7d97effb21","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"classification","assisted_by":[],"external_id":"cm.fractional-chern-insulators.geometry-criterion","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"plain":"Theory says a band that closely mimics a Landau level favours fractional states, but nobody can yet predict from a band's properties alone whether a real material will form a fractional state or an electron crystal.","posed_since":"","precise":"For a partially filled isolated Chern band ($C=1$) of width $W$ with Coulomb scale $U=e^2/(\\epsilon a_M)$, Berry curvature $\\Omega(k)$ and quantum metric $g(k)$, determine the boundary between the $\\nu=1/3$ and $2/3$ fractional Chern insulators and competing charge-density-wave or Fermi-liquid states as a function of $W/U$, the Berry-curvature variance, and the trace-condition violation T = (1/2 pi) integral d^2k [tr g(k) - |Omega(k)|]. An answer is a phase diagram validated by exact diagonalization or DMRG across twisted MoTe2, rhombohedral graphene and model bands.","problem_ref":null,"references":"","settled_by":"Systematic DMRG or exact-diagonalization phase diagrams over band families with tunable geometry, checked against experimental FQAH windows.","status_note":"Ideal bands with $T = 0$ give exact fractional ground states for short-range interactions (results from 2014-2021); a predictive criterion for Coulomb interaction in realistic bands is lacking as of 2026.","title":"Band-geometry criterion for fractional Chern insulator stability","topic_ref":"99dc6b2de04e3a56515c0a5f91773afbf88a5a0f7d59aad4ce980e36539e6e22"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"b410a1c18797ad08f39644bda00ad7fc6f905cb7b9bcf1210f8f4483f83d592d","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"37861e2fc4a1fea94dec298d2f392ca9317b120cc86bbb97224a6ef84b43558a","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"FNdRAKyiuy-zVAzu2SOzSIZkGRe_Y1I0B26M99lxChWEfG_ih60BVOiGS5NSLMu-EbhQHpTrJZR3WF67RYaUAQ"},"schema":"pubphys.envelope/1"},"record_hash":"b410a1c18797ad08f39644bda00ad7fc6f905cb7b9bcf1210f8f4483f83d592d","leaf_index":997}