{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"c4b46bef76e8e73c114d77208943f32bc7d40c469d9da73de097236f27e5f429","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":["ai:gpt-5.5"],"kind":"seed"},"parents":["b410a1c18797ad08f39644bda00ad7fc6f905cb7b9bcf1210f8f4483f83d592d","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"55078d83aa265867dbdb84f9605542a003188324ed7a41bc690fcafe1cb582ff","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"classification","assisted_by":["ai:gpt-5.5"],"external_id":"cm.fractional-chern-insulators.geometry-criterion","kind":"well-posed","literature_status":"partially-resolved","n":"2","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 $T = (1/2 \\pi) \\int \\mathrm{d}^{2}k [\\mathrm{tr}\\,g(k) - \\left| \\Omega(k) \\right|]$. An answer is a phase diagram validated by exact diagonalization or DMRG across twisted $\\mathrm{MoTe_2}$, rhombohedral graphene and model bands.","problem_ref":"b410a1c18797ad08f39644bda00ad7fc6f905cb7b9bcf1210f8f4483f83d592d","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":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"a732162e9603e61876e27632058909fa920591fa4e85206c4e63caee8facb0d1","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"c4784c9b21dc83839e68eebca518c36e7b44f77a7f9d0f43ab6d86a9500f6c60","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"J4g86CS2KjPbTtXRCHknnqFMRzaUxfNROBRVYInkocfV8aGuKC6PEyJtWPMn5ab5f3e8LWXGgC1Q8c8s9qtXCQ"},"schema":"pubphys.envelope/1"},"record_hash":"a732162e9603e61876e27632058909fa920591fa4e85206c4e63caee8facb0d1","leaf_index":2814}