Variational Mapping of Chern Bands to Landau Levels: Application to Fractional Chern Insulators in Twisted MoTe$_2$

We present a theoretical study of mapping between Chern bands and generalized Landau levels in twisted bilayer MoTe$_2$ ($t$MoTe$_2$), where fractional Chern insulators down to zero magnetic fields have been observed. We construct an exact Landau-level representation of moir\'e bands, where the basis functions, characterized by a uniform quantum geometry, are derived from Landau-level wavefunctions dressed by spinors aligned or antialigned with the layer pseudospin skyrmion field. We further generalize the dressed zeroth Landau level to a variational wavefunction with an ideal yet nonuniform quantum geometry and variationally maximize its weight in the first moir\'e band. The variational wavefunction has a high overlap with the first band and quantitatively captures the exact diagonalization spectra of fractional Chern insulators at hole-filling factors $\nu_h=2/3$ and $3/5$, providing a clear theoretical mechanism for the formation and properties of the fractionalized states. Our work introduces a variational approach to studying fractional states by mapping Chern bands to Landau levels, with application to other systems beyond $t$MoTe$_2$ also demonstrated.