Band-geometry criterion for fractional Chern insulator stability
In plain words
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.
Precise statement
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.
What would settle it
Systematic DMRG or exact-diagonalization phase diagrams over band families with tunable geometry, checked against experimental FQAH windows.
Status in the literature
Unverified 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.